Nonlinear elliptic eigenvalue problems in cylindrical domains becoming unbounded in one direction

Abstract

The aim of this work is to characterize the asymptotic behaviour of the first eigenfunction of the generalised p-Laplace operator with mixed (Dirichlet and Neumann) boundary conditions in cylindrical domains when the length of the cylindrical domains tends to infinity. This generalises an earlier work of Chipot et al. (Asymptot. Anal. 85(3–4) (2013) 199–227) where the linear case $p = 2$ is studied. Asymptotic behavior of all the higher eigenvalues of the linear case and the second eigenvalues of general case (using topological degree) for such problems is also studied.

Keywords

p-Laplacian uniform elliptcity Poincaré inequality Krasnoselskii’s genus

1. Introduction

The asymptotic analysis of problems set in cylindrical domains which become unbounded in one or several directions for various partial differential equations has been carried out by several authors since the pioneering work of Chipot [15]. In this paper, we revisit one of these problems which is set in cylindrical domains becoming unbounded in one direction for the generalised p-Laplacian with mixed boundary conditions. This continues the work in [15], which concerns the asymptotic analysis of the linear case $p = 2$ with mixed (Dirichlet-Neumann) boundary conditions. We now briefly describe the setting.

Consider $Ω_{ℓ} = (- ℓ, ℓ) \times ω$ , where ℓ is a positive real number and ω is an open bounded set in $R^{n - 1}$ , $n ⩾ 2$ . Any generic point $x \in R^{n}$ will be denoted by $x = (x_{1}, X_{2})$ , where $x_{1} \in R$ and $X_{2} \in R^{n - 1}$ . Then for each $x_{1} \in (- ℓ, ℓ)$ , ω is the cross-section of $Ω_{ℓ}$ orthogonal to $x_{1}$ -direction. Let A be a $n \times n$ symmetric matrix of the form $\begin{matrix} A (X_{2}) : = (\begin{array}{cc} a_{11} (X_{2}) & A_{12} (X_{2}) \\ A_{12}^{t} (X_{2}) & A_{22} (X_{2}) \end{array}), X_{2} \in ω, \end{matrix}$ where $A_{22}$ is an $(n - 1) \times (n - 1)$ matrix and assume that A is uniformly bounded and uniformly elliptic on $R \times ω$ (precise definitions are given in next section). For $p ⩾ 2$ consider the following eigenvalue problem on $Ω_{ℓ}$ for the generalised p-Laplacian with Dirichlet boundary condition $\begin{matrix} (1.1) & \{\begin{array}{l} - div (| A (X_{2}) \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} A (X_{2}) \nabla u_{ℓ}) = λ_{D} (Ω_{ℓ}) | u_{ℓ} |^{p - 2} u_{ℓ} in Ω_{ℓ}, \\ u_{ℓ} = 0 on \partial Ω_{ℓ}, \end{array} \end{matrix}$ and the corresponding eigenvalue problem $\begin{matrix} (1.2) & \{\begin{array}{l} - div (| A_{22} (X_{2}) \nabla_{X_{2}} u \cdot \nabla_{X_{2}} u |^{\frac{p - 2}{2}} A_{22} (X_{2}) \nabla_{X_{2}} u) = μ (ω) | u |^{p - 2} u in ω, \\ u = 0 on \partial ω, \end{array} \end{matrix}$ defined on the cross section ω, where $\nabla_{X_{2}} = (\frac{\partial}{\partial x_{2}}, \frac{\partial}{\partial x_{3}}, \dots, \frac{\partial}{\partial x_{n}})$ .

For the linear case $p = 2$ , it is well known that each of the problems (1.1) and (1.2) admit an infinite set of positive discrete eigenvalues tending to infinity (see [25], Theorem 3.6.1). Let $λ_{D}^{k} (Ω_{ℓ})$ and $μ^{k} (ω)$ denote the k-th eigenvalues of (1.1) and (1.2) respectively in this case. In [14], Theorem 2.4, it was shown that as ℓ goes to infinity, the k-th eigenvalue $λ_{D}^{k} (Ω_{ℓ})$ of (1.1) converges to the first eigenvalue $μ^{1} (ω)$ of (1.2) with the optimal convergence rate of $\frac{1}{ℓ^{2}}$ . It was further proved that under appropriate scaling, the eigenfunctions corresponding to the first eigenvalues of (1.1) converge to the eigenfunction corresponding to the first eigenvalue of (1.2) in the appropriate function space (see [14], Theorem 3.4). In general for $p ⩾ 2$ , the first eigenvalue of (1.1) and (1.2) can be obtained by appropriate minimization problems (described later in (2.6) and (2.7)), we continue to denote them by $λ_{D}^{1} (Ω_{ℓ})$ and $μ^{1} (ω)$ respectively, the value of p will be clear from the context. In [23], the following theorem was proved.

Theorem 1.
[23] Let $p ⩾ 2$ . Then ∃ a constant C depends only on A, ω and p, such that $\begin{matrix} (1.3) & μ^{1} (ω) ⩽ λ_{D}^{1} (Ω_{ℓ}) ⩽ μ^{1} (ω) + \frac{C}{ℓ} . \end{matrix}$

We now describe the problem which will be our main focus in this paper. Consider the following eigenvalue problem for the generalised p-Laplacian with mixed (Dirichlet and Neumann) boundary conditions: $\begin{matrix} (1.4) & \{\begin{array}{l} - div (| A (X_{2}) \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} A (X_{2}) \nabla u_{ℓ}) = λ_{M} (Ω_{ℓ}) | u_{ℓ} |^{p - 2} u_{ℓ} in Ω_{ℓ}, \\ u_{ℓ} = 0 on γ_{ℓ} : = (- ℓ, ℓ) \times \partial ω, \\ (A (X_{2}) \nabla u_{ℓ}) \cdot ν = 0 on Γ_{ℓ} : = {- ℓ, ℓ} \times ω, \end{array} \end{matrix}$ where ν denotes the outward unit normal to $Γ_{ℓ}$ .

In the linear case $p = 2$ , where the set of all eigenvalues of (1.4) is well understood (see [29]) a similar asymptotic analysis of the first and second eigenvalue was carried out in [15]. It was shown there that under an additional condition, the first eigenvalue $λ_{M}^{1} (Ω_{ℓ})$ of (1.4) converges to a limit as ℓ goes to infinity but in this case the limit is strictly smaller than the first eigenvalue $μ^{1} (ω)$ of (1.2). In particular there is a gap in the limiting behaviour of $λ_{M}^{1} (Ω_{ℓ})$ and $μ^{1} (ω)$ . This is in sharp contrast to the limiting behavior of eigenvalue problem with Dirichlet boundary condition (1.1) as discussed before. Further it was proved that under a symmetry condition on the matrix A, the second eigenvalue $λ_{M}^{2} (Ω_{ℓ})$ of (1.4) has the same limit as that of the first eigenvalue $λ_{M}^{1} (Ω_{ℓ})$ of (1.4) as ℓ goes to infinity.

This was followed up in [23] where it was shown that the gap phenomenon continues to hold for the limit superior of the first eigenvalues of (1.4) for the nonlinear case $p > 2$ under a similar additional condition. To summarize these results:
Theorem 2.
[15,23] Let $p ⩾ 2$ and let W be the eigenfunction corresponding to the eigenvalue $μ^{1} (ω)$ such that $\int_{ω} | W |^{p} = 1$ . Then

(i) For $p = 2$ , $\begin{matrix} (1.5) & lim_{ℓ \to \infty} λ_{M}^{1} (Ω_{ℓ}) < μ^{1} (ω), \end{matrix}$ provided $A_{12} \cdot \nabla_{X_{2}} W ≢ 0$ a.e. on ω, otherwise $λ_{M}^{1} (Ω_{ℓ}) = μ^{1} (ω)$ for all $ℓ > 0$ . Furthermore, if A satisfies the symmetry (S) (see Definition 1 later), then $\begin{matrix} lim_{ℓ \to \infty} λ_{M}^{2} (Ω_{ℓ}) = lim_{ℓ \to \infty} λ_{M}^{1} (Ω_{ℓ}) . \end{matrix}$ (ii) For $p > 2$ , one has $\begin{matrix} (1.6) & \underset{ℓ \to \infty}{lim sup} λ_{M}^{1} (Ω_{ℓ}) < μ^{1} (ω) \end{matrix}$ provided $A_{12} \cdot \nabla_{X_{2}} W ≢ 0$ a.e. on ω, otherwise $λ_{M}^{1} (Ω_{ℓ}) = μ^{1} (ω)$ for all $ℓ > 0$ .

A detailed discussion on the condition $A_{12} \cdot \nabla_{X_{2}} W ≢ 0$ is done in [15] [see, Remark 4.2].

On the way to the proof of the gap phenomenon in Theorem 2 (i), several other interesting results were obtained in [15]. For example, behavior of the first normalised eigenfunction of (1.4) near the end (on sets like $Ω_{ℓ} ∖ Ω_{ℓ - 1}$ ), in the middle of the cylinder (sets like $Ω_{ℓ / 2}$ ) were established. Also the value $Λ : = {lim sup}_{ℓ \to \infty} λ_{M}^{1} (Ω_{ℓ})$ is identified with an appropriate problem on the semi-infinite cylinders.

Similar questions remained unanswered for the case $p > 2$ , which we wish to take up in this paper. Let $u_{ℓ}$ be the normalised eigenfunction corresponding to the first eigenvalue $λ_{M}^{1} (Ω_{ℓ})$ of (1.4). In our first result, we analyse the behavior of $u_{ℓ}$ as ℓ goes to infinity and show that as in the case $p = 2$ these eigenfunctions decay to zero in the middle of the cylinder and their mass concentrates near the base of the cylinder. Throughout this paper, $[x]$ will denote the integer part of $x \in R$ .
Theorem 1.1.
Assume that $p ⩾ 2$ and $A_{12} \cdot \nabla_{X_{2}} W ≢ 0$ a.e in ω. Then ∃ constants $α \in (0, 1)$ and $C_{1}, C_{2} > 0$ depend only on A, ω and p such that for $ℓ > ℓ_{0}$ we have for every $0 < r ⩽ ℓ - 1$ , $\begin{matrix} (1.7) & \int_{Ω_{r}} | \nabla u_{ℓ} |^{p} ⩽ C_{1} α^{[ℓ - r]}, \end{matrix}$ and $\begin{matrix} (1.8) & \int_{Ω_{r}} | u_{ℓ} |^{p} ⩽ C_{2} α^{[ℓ - r]} . \end{matrix}$

The following identity $\begin{matrix} A \nabla (u v) \cdot \nabla (u v) - u^{2} (A \nabla v \cdot \nabla v) = A \nabla u \cdot \nabla (u v^{2}) \end{matrix}$ plays a crucial role in proofs for the case $p = 2$ . For general p, we do not have this kind of identity. To overcome this difficulty we use the uniformly ellipticity condition on the matrix A and the Poincaré inequality (see (2.3) and (2.1) in Section 2). Other complication arises because of the fact, that for $p = 2$ the spaces involved are Hilbert spaces which is not the case for general p. Our next result is crucial for understanding the value $Λ : = {lim sup}_{ℓ \to \infty} λ_{M}^{1} (Ω_{ℓ})$ .

Consider the semi-infinite cylinders $\begin{matrix} Ω_{\infty}^{+} = (0, \infty) \times ω and Ω_{\infty}^{-} = (- \infty, 0) \times ω \end{matrix}$ with the boundaries $\begin{matrix} γ_{\infty}^{+} = (0, \infty) \times \partial ω, γ_{\infty}^{-} = (- \infty, 0) \times \partial ω, \end{matrix}$ and the quantities $\begin{matrix} (1.9) & ν_{\infty}^{\pm} = inf_{0 \neq u \in V (Ω_{\infty}^{\pm})} \frac{\int_{Ω_{\infty}^{\pm}} | A \nabla u \cdot \nabla u |^{\frac{p}{2}}}{\int_{Ω_{\infty}^{\pm}} | u |^{p}}, \end{matrix}$ where the spaces $V (Ω_{\infty}^{\pm})$ are defined by $\begin{matrix} V (Ω_{\infty}^{\pm}) = {u \in W^{1, p} (Ω_{\infty}^{\pm}) : u = 0 on γ_{\infty}^{\pm}} . \end{matrix}$ Detailed description of these spaces is given in Section 2.
Theorem 1.2.
Assume that $p ⩾ 2$ . Then $\begin{matrix} (1.10) & lim_{ℓ \to \infty} λ_{M}^{1} (Ω_{ℓ}) = min {ν_{\infty}^{+}, ν_{\infty}^{-}} . \end{matrix}$

We further provide a condition for which the gaps hold between $ν_{\infty}^{\pm}$ and $μ^{1} (ω)$ and another condition for which the gaps fail. This was known for $p = 2$ from [15].
Theorem 1.3.
(i) Assume that $A_{12} \cdot \nabla_{X_{2}} W ≢ 0$ . If $\begin{matrix} (1.11) & \int_{ω} | A_{22} \nabla_{X_{2}} W \cdot \nabla_{X_{2}} W |^{\frac{p - 2}{2}} (A_{12} \cdot \nabla_{X_{2}} W) W ⩾ 0, \end{matrix}$ then $ν_{\infty}^{+} < μ_{1} (ω)$ .

(ii) If $\begin{matrix} (1.12) & A_{12} \cdot \nabla_{X_{2}} W ⩽ 0 a.e in ω, \end{matrix}$ then $ν_{\infty}^{+} = μ_{1} (ω)$ . Moreover in this case there is no minimizer realizing $ν_{\infty}^{+}$ .

We conclude the paper with some results about the convergence of the other eigenvalues of problem (1.4), these results may be compared with the ones in [14,15]. Unlike the linear case $p = 2$ , there is no specific description of the second and higher eigenvalues available for $p > 2$ for the eigenvalue problem (1.4). Although there is a description available of what we call Krasnoselskii eigenvalues, through the concept of Krasnoselskii’s genus. Let ${β_{k} (Ω_{ℓ})}$ denote these Krasnoselskii eigenvalues of (1.4) which we instead use to analyze the asymptotic behaviour. We discuss these eigenvalues briefly in the next section. For $p = 2$ therefore, we have two descriptions of eigenvalues of the problem (1.4), one through classical methods using Hilbert space structure and the other one using Krasnoselskii’s genus, and these two descriptions coincide in this case (see Lemma 6.1). For $p > 2$ it is not clear if Krasnoselskii eigenvalues exhaust the set of all eigenvalues of (1.4). However the first Krasnoselskii eigenvalue $β_{1} (Ω_{ℓ})$ coincides with the usual first eigenvalue $λ_{M}^{1} (Ω_{ℓ})$ (see Corollary 2).

In the literature there are other descriptions of eigenvalues of (1.4) available, interested readers may look at [22] for example.
Definition 1.
We shall say that A satisfies the symmetry (S) if the set ω is symmetric w.r.t the origin, i.e. $ω = - ω$ and $A (- X_{2}) = A (X_{2})$ .

The following theorem is an extension of Theorem 7.1 in [15] for higher eigenvalues. Only the asymptotic behavior of second eigenvalue was considered there.
Theorem 1.4.
Let A satisfies the symmetry (S). Then for $p = 2$ and for any $k ⩾ 2$ , $\begin{matrix} (1.13) & lim_{ℓ \to \infty} λ_{M}^{k} (Ω_{ℓ}) = lim_{ℓ \to \infty} λ_{M}^{1} (Ω_{ℓ}) . \end{matrix}$
Theorem 1.5.
Let A satisfies the symmetry (S). Then for $p > 2$ , $\begin{matrix} (1.14) & lim_{ℓ \to \infty} β_{2} (Ω_{ℓ}) = lim_{ℓ \to \infty} λ_{M}^{1} (Ω_{ℓ}) . \end{matrix}$

For other related work on asymptotic behavior of problems defined on cylindrical domains interested readers may look at [4,9,13,16,18,24,30,31] for study of elliptic problems, [11,12,17] for parabolic problems. The behavior of Stokes equations has been studied in [7,8], whereas purely variational problems have been studied in [2,10,27]. Asymptotic problems of the types mentioned above have been summarized and various physical applications are discussed in the books [3,5] by Chipot. Recently, problems involving nonlocal operators have also been considered, e.g., see [1,19–21,33]. For the first time in such literature semilinear problems are studied in [6].

This paper is organized as follow: In Section 2 we briefly review the notion of Krasnoselskii’s genus and related facts, underlying function spaces, uniform Poincaré inequality, properties of the matrix A and other relevant known results. Section 3 contains the proof of Theorem 1.1. In Section 4 we prove Theorem 1.2, which identifies the limit of $λ_{M}^{1} (Ω_{ℓ})$ as ℓ goes to infinity. In Section 5 gap phenomenon on semi infinite cylinder (Theorem 1.3) is studied. In Section 6 we conclude the paper with the convergence results (Theorem 1.4 and 1.5) concerning the k-th eigenvalue for $p = 2$ and the second Krasnoselskii’s eigenvalue for $p > 2$ of the problem (1.4).
2. Preliminaries

Let V be a Banach space. Define the collection $\begin{matrix} A = {A \subset V : A is closed and A = - A} \end{matrix}$ of all closed and symmetric subsets of V. Let $γ : A \to N \cup {0, \infty}$ be defined by $\begin{matrix} γ (A) = \{\begin{array}{l} inf {m : \exists h : A \to R^{m} ∖ {0}, h is continuous and h (- u) = - h (u)} \\ \infty, if {\dots} = ϕ, in particular if A ∋ 0, \end{array} \end{matrix}$ for $A \neq ϕ$ and $γ (ϕ) = 0$ . $γ (A)$ is called the Krasnoselskii’s genus of the set A. We summarise some important properties of the Krasnoselskii’s genus in the following:

Proposition 2.1.
(i) γ is a monotone sub-additive continuous map and $γ (A)$ is finite for any compact symmetric subset of $V ∖ {0}$ .

(ii) Let Ω be any bounded symmetric neighborhood of the origin in $R^{m}$ . Then $γ (\partial Ω) = m$ .

(iii) Let A be a compact symmetric subset of a Hilbert space V with the inner product ${(\cdot, \cdot)}_{V}$ and let $γ (A) = m < \infty$ . Then A contains at least m mutually orthogonal vectors, i.e. $\exists u_{1}, u_{2}, \dots, u_{m} \in A$ such that ${(u_{j}, u_{k})}_{V} = 0$ for $j \neq k$ .

Let us define the following terminologies:
Definition 2.
(i) Suppose $E \in C^{1} (V)$ is a functional on V. A sequence $(u_{m})$ is called a Palais-Smale sequence for E, if $| E (u_{m}) | ⩽ c$ uniformly on m, while $‖ DE (u_{m}) ‖ \to 0$ as $m \to \infty$ , where $DE (u)$ is the Fréchet derivative of the functional E at u.

(ii) E is said to satisfy the Palais-Smale (P.S) condition if any Palais-Smale sequence has a strongly convergent subsequence.
Definition 3.
Let M be a Hausdorff topological space. Assume that there exists an open covering ${U_{i} : i \in I}$ of M and homeomorphisms $ϕ_{i}$ from $U_{i}$ onto an open subset $ϕ_{i} (U_{i})$ of some Banach space $B_{i}$ . Further assume that $ϕ_{i} (U_{i} \cap U_{j})$ is open in $B_{i}$ and whenever $U_{i} \cap U_{j} \neq ϕ$ for $i, j \in I$ , then the map $\begin{matrix} ϕ_{i} \circ ϕ_{j}^{- 1} : ϕ_{j} (U_{i} \cap U_{j}) \to ϕ_{i} (U_{i} \cap U_{j}) \end{matrix}$ is $C^{k, 1}$ for some $k \in N$ , where the space $C^{k, 1}$ consists of those functions whose k-th order Fréchet derivative is Lipschitz. Then the object $(M, {(U_{i}, ϕ_{i}) : i \in I})$ is called a $C^{k, 1}$ -Banach manifold.

By an even functional E on V, we mean $E (- u) = E (u)$ , $u \in V$ . The following theorem assures the existence of the critical points and characterizes the critical values of E.
Theorem 2.2.
(i) Suppose E is an even functional of class $C^{1}$ on a complete symmetric $C^{1, 1}$ -manifold $M \subset V ∖ {0}$ in some Banach space V. Also suppose E satisfies (P.S) condition and is bounded from below on M. Let $\hat{γ} (M) = sup {γ (K) : K \subset M compact and symmetric}$ . Then the functional E possesses at least $\hat{γ} (M) ⩽ \infty$ pairs of critical points.

(ii) Let $A$ be defined as above. For any $k ⩽ \hat{γ} (M)$ consider the family $\begin{matrix} F_{k} = {A \in A : A \subset M, γ (A) ⩾ k} . \end{matrix}$ If $\begin{matrix} β_{k} = inf_{A \in F_{k}} sup_{u \in A} E (u) \end{matrix}$ is finite, then $β_{k}$ is a critical value of E.

We refer to [32] (Chapter II, Section 5) for more details about the Krasnoselskii’s genus and the proofs of Proposition 2.1 and Theorem 2.2.

For an open set Ω in $R^{n}$ and $1 < p < \infty$ , $W^{1, p} (Ω)$ is the usual Sobolev space defined by $\begin{matrix} W^{1, p} (Ω) = {v \in L^{p} (Ω) : \partial_{x_{i}} v \in L^{p} (Ω), i = 1, 2 \dots, n}, \end{matrix}$ with the norm $\begin{matrix} ‖ v ‖_{W^{1, p} (Ω)} = {\int_{Ω} (| v |^{p} + | \nabla v |^{p})}^{1 / p} . \end{matrix}$ As before, let $Ω_{ℓ} = (- ℓ, ℓ) \times ω$ on $R^{n}$ , where $ℓ > 0$ and ω is an open bounded set in $R^{n - 1}$ and we define the space $\begin{matrix} W_{0}^{1, p} (Ω_{ℓ}) = {v \in W^{1, p} (Ω_{ℓ}) : v = 0 on \partial Ω_{ℓ}} . \end{matrix}$ We denote the boundary of $Ω_{ℓ}$ by $γ_{ℓ} \cup Γ_{ℓ}$ , where $γ_{ℓ} = (- ℓ, ℓ) \times \partial ω$ and $Γ_{ℓ} = {- ℓ, ℓ} \times ω$ . To analyze the eigenvalues and eigenfunctions of the problem (1.4), we define a suitable space $\begin{matrix} V (Ω_{ℓ}) = {v \in W^{1, p} (Ω_{ℓ}) : v = 0 on γ_{ℓ}} . \end{matrix}$ The boundary conditions are considered in the sense of traces. The following version of Poincaré inequality is used in this work.
Lemma 2.1.
(Poincaré inequality) Let $Ω = U \times ω$ , where U is any open subset of $R$ , ω is an open bounded set in $R^{n - 1}$ . Let $u \in V (Ω)$ , where $V (Ω) = {u \in W^{1, p} (Ω) : u = 0 on U \times \partial ω}$ . Then we have $\begin{matrix} (2.1) & ‖ u ‖_{L^{p} (Ω)} ⩽ C_{p} ‖ \nabla u ‖_{L^{p} (Ω)} \end{matrix}$ where $C_{p} > 0$ is a constant depends only on p and ω.

Using Lemma 2.1 it can be easily verified that $‖ v ‖_{p, Ω_{ℓ}}^{p} = \int_{Ω_{ℓ}} | \nabla v |^{p}$ is a norm on $V (Ω_{ℓ})$ and $V (Ω_{ℓ})$ is a Banach space with respect to this norm.

We consider the $n \times n$ symmetric matrix A of the form $\begin{matrix} A (X_{2}) = (\begin{array}{cc} a_{11} (X_{2}) & A_{12} (X_{2}) \\ A_{12}^{t} (X_{2}) & A_{22} (X_{2}) \end{array}) \end{matrix}$ where $a_{11} (X_{2}) \in R$ , $A_{12} (X_{2})$ is a $1 \times (n - 1)$ matrix and $A_{22} (X_{2})$ is a $(n - 1) \times (n - 1)$ matrix. The components of $A (X_{2})$ are assumed to be bounded measurable functions on ω and we assume that A is an uniformly bounded and uniformly positive definite matrix i.e., there exists positive constants C and λ such that $\begin{matrix} (2.2) & ‖ A (X_{2}) ‖ ⩽ C a.e X_{2} \in ω, \end{matrix}$ and $\begin{matrix} (2.3) & A (X_{2}) ξ \cdot ξ ⩾ λ | ξ |^{2} a.e X_{2} \in ω, \forall ξ \in R^{n} . \end{matrix}$ Here $‖ \cdot ‖$ denotes the norm of matrices, $| . |$ denotes the euclidean norm and $X \cdot Y$ denotes the usual inner product of two vectors X and Y in $R^{n}$ .

To obtain the Krasnoselskii eigenvalues of (1.4), we define a closed symmetric $C^{1, 1}$ -manifold $M_{ℓ}$ of the Banach space $V (Ω_{ℓ})$ as $\begin{matrix} M_{ℓ} = {u \in V (Ω_{ℓ}) : \int_{Ω_{ℓ}} | u |^{p} = 1}, \end{matrix}$ and an even functional $E_{ℓ}$ on $M_{ℓ}$ by $\begin{matrix} E_{ℓ} (u) = \int_{Ω_{ℓ}} | A \nabla u \cdot \nabla u |^{\frac{p}{2}} . \end{matrix}$ Then $E_{ℓ}$ satisfies (P.S) condition, as shown in [22]. For $k \in N$ , let ${v_{1}, v_{2}, \dots, v_{k}}$ be a linearly independent set in $M_{ℓ}$ and let $K = {v = \sum_{i = 1}^{k} α_{i} v_{i} : α_{i} \in R, 1 ⩽ i ⩽ k and \int_{Ω_{ℓ}} | v |^{p} = 1}$ . Then K is a compact and symmetric subset of a k-dimensional subspace of $L^{p}$ and is homeomorphic to $S^{k - 1}$ , the unit sphere in $R^{k}$ . Therefore by Proposition 2.1 (ii), $γ (K) = k$ and hence $\hat{γ} (M_{ℓ}) = \infty$ . Theorem 2.2 (i) assures that there are infinitely many critical values in both the cases $p = 2$ and $p > 2$ . For $p = 2$ , these critical values of (1.4) coincide with the usual eigenvalues (as discussed earlier) and therefore we continue to denote them by $λ_{M}^{k} (Ω_{ℓ})$ and for $p > 2$ , we call these critical values as Krasnoselskii eigenvalues of the problem (1.4) and denote them by $β_{k} (Ω_{ℓ})$ .

Multiplying the equations (1.1) and (1.2) by functions from the corresponding underlying spaces and then integrating by parts, we formulate the corresponding weak version of the Dirichlet problems (1.1) and (1.2) respectively as follows: $\begin{array}{c} (2.4) & \{\begin{array}{l} u \in W_{0}^{1, p} (Ω_{ℓ}), \\ \int_{Ω_{ℓ}} | A \nabla u \cdot \nabla u |^{\frac{p - 2}{2}} A \nabla u \cdot \nabla v = λ_{D} (Ω_{ℓ}) \int_{Ω_{ℓ}} | u |^{p - 2} u v, \forall v \in W_{0}^{1, p} (Ω_{ℓ}) . \end{array} \\ (2.5) & \{\begin{array}{l} u \in W_{0}^{1, p} (ω), \\ \int_{ω} | A_{22} \nabla_{X_{2}} u \cdot \nabla_{X_{2}} u |^{\frac{p - 2}{2}} A \nabla_{X_{2}} u \cdot \nabla_{X_{2}} v = μ (ω) \int_{ω} | u |^{p - 2} u v, \forall v \in W_{0}^{1, p} (ω) . \end{array} \end{array}$ The following characterization of the first eigenvalues $λ_{D}^{1} (Ω_{ℓ})$ of (2.4) and $μ^{1} (ω)$ of (2.5) is well known: $\begin{array}{c} (2.6) & λ_{D}^{1} (Ω_{ℓ}) = inf_{u \in W_{0}^{1, p} (Ω_{ℓ}), u \neq 0} \frac{\int_{Ω_{ℓ}} | A \nabla u \cdot \nabla u |^{\frac{p}{2}}}{\int_{Ω_{ℓ}} | u |^{p}} . \\ (2.7) & μ^{1} (ω) = inf_{u \in W_{0}^{1, p} (ω), u \neq 0} \frac{\int_{ω} | A_{22} \nabla_{X_{2}} u \cdot \nabla_{X_{2}} u |^{\frac{p}{2}}}{\int_{ω} | u |^{p}} = \frac{\int_{ω} | A_{22} \nabla_{X_{2}} W \cdot \nabla_{X_{2}} W |^{\frac{p}{2}}}{\int_{ω} | W |^{p}} . \end{array}$ It is also a known fact (see [26,28]) that both $λ_{D}^{1} (Ω_{ℓ})$ and $μ^{1} (ω)$ are simple and the corresponding eigenfunctions do not change sign. Here W is the first normalised non-negative eigenfunction of (2.5), considered earlier in Theorem 2.

The corresponding weak form of the mixed boundary value problem (1.4) is the following: $\begin{matrix} (2.8) & \{\begin{array}{l} u \in V (Ω_{ℓ}), \\ \int_{Ω_{ℓ}} | A \nabla u \cdot \nabla u |^{\frac{p - 2}{2}} A \nabla u \cdot \nabla v = λ_{M} (Ω_{ℓ}) \int_{Ω_{ℓ}} | u |^{p - 2} u v, \forall v \in V (Ω_{ℓ}) \end{array} \end{matrix}$ and the first eigenvalue is given in terms of the Rayleigh quotient as $\begin{matrix} (2.9) & λ_{M}^{1} (Ω_{ℓ}) = inf_{u \in V (Ω_{ℓ}), u \neq 0} \frac{\int_{Ω_{ℓ}} | A \nabla u \cdot \nabla u |^{\frac{p}{2}}}{\int_{Ω_{ℓ}} | u |^{p}} \end{matrix}$ In this case as well, the eigenvalue $λ_{M}^{1} (Ω_{ℓ})$ is simple and the corresponding eigenfunction $u_{ℓ}$ have constant sign.
3. Main result on the asymptotic behavior of the first eigenfunction

Proof of Theorem 1.1.
Let $0 < ℓ^{'} ⩽ ℓ - 1$ . Define the function $ρ_{ℓ^{'}} = ρ_{ℓ^{'}} (x_{1})$ by $\begin{matrix} ρ_{ℓ^{'}} (x_{1}) = \{\begin{array}{ll} 1 & if | x_{1} | ⩽ ℓ^{'}, \\ ℓ^{'} + 1 - | x_{1} | & if | x_{1} | \in (ℓ^{'}, ℓ^{'} + 1), \\ 0 & if | x_{1} | ⩾ ℓ^{'} + 1 . \end{array} \end{matrix}$ Clearly, $ρ_{ℓ^{'}} u_{ℓ} \in W_{0}^{1, p} (Ω_{ℓ})$ . From (2.6) and using the fact that A is symmetric, we get $\begin{aligned} λ_{D}^{1} (Ω_{ℓ}) \int_{Ω_{ℓ}} ρ_{ℓ^{'}}^{p} | u_{ℓ} |^{p} & ⩽ \int_{Ω_{ℓ}} {| A \nabla (ρ_{ℓ^{'}} u_{ℓ}) \cdot \nabla (ρ_{ℓ^{'}} u_{ℓ}) |}^{\frac{p}{2}} \\ = \int_{Ω_{ℓ}} {| ρ_{ℓ^{'}}^{2} A \nabla u_{ℓ} \cdot \nabla u_{ℓ} + 2 ρ_{ℓ^{'}} u_{ℓ} A \nabla u_{ℓ} \cdot \nabla ρ_{ℓ^{'}} + u_{ℓ}^{2} A \nabla ρ_{ℓ^{'}} \cdot \nabla ρ_{ℓ^{'}} |}^{\frac{p}{2}} \\ = \int_{Ω_{ℓ}} {| ρ_{ℓ^{'}}^{2} A \nabla u_{ℓ} \cdot \nabla u_{ℓ} + A \nabla (ρ_{ℓ^{'}} u_{ℓ}^{2}) \cdot \nabla ρ_{ℓ^{'}} |}^{\frac{p}{2}} . \end{aligned}$ On R.H.S we use the inequality ${(a + b)}^{q} ⩽ a^{q} + q 2^{q - 1} (b^{q} + a^{q - 1} b)$ for $a, b ⩾ 0$ , $q ⩾ 1$ , which gives $\begin{array}{c} λ_{D}^{1} (Ω_{ℓ}) \int_{Ω_{ℓ}} ρ_{ℓ^{'}}^{p} | u_{ℓ} |^{p} \\ ⩽ \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p}{2}} ρ_{ℓ^{'}}^{p} \\ (3.1) & + \frac{p}{2} C_{0} \int_{Ω_{ℓ}} ({| A \nabla (ρ_{ℓ^{'}} u_{ℓ}^{2}) \cdot \nabla ρ_{ℓ^{'}} |}^{\frac{p}{2}} + | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} | A \nabla (ρ_{ℓ^{'}} u_{ℓ}^{2}) \cdot \nabla ρ_{ℓ^{'}} |), \end{array}$ where $C_{0} = 2^{\frac{p - 2}{2}}$ . We estimate the integrals on R.H.S separately. $\begin{array}{c} \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p}{2}} ρ_{ℓ^{'}}^{p} \\ = \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} (ρ_{ℓ^{'}}^{p} A \nabla u_{ℓ} \cdot \nabla u_{ℓ} + p ρ_{ℓ^{'}}^{p - 1} u_{ℓ} A \nabla u_{ℓ} \cdot \nabla ρ_{ℓ^{'}} - p ρ_{ℓ^{'}}^{p - 1} u_{ℓ} A \nabla u_{ℓ} \cdot \nabla ρ_{ℓ^{'}}) \\ = \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} A \nabla u_{ℓ} \cdot \nabla (ρ_{ℓ^{'}}^{p} u_{ℓ}) - p \int_{Ω_{ℓ}} ρ_{ℓ^{'}}^{p - 1} u_{ℓ} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} A \nabla u_{ℓ} \cdot \nabla ρ_{ℓ^{'}} \end{array}$ Using the fact that $u_{ℓ}$ is the first eigenfunction of (2.8) and taking modulus in the last integral, we get $\begin{matrix} (3.2) & \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p}{2}} ρ_{ℓ^{'}}^{p} ⩽ λ_{M}^{1} (Ω_{ℓ}) \int_{Ω_{ℓ}} ρ_{ℓ^{'}}^{p} | u_{ℓ} |^{p} + p \int_{Ω_{ℓ}} ρ_{ℓ^{'}}^{p - 1} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} | A \nabla u_{ℓ} \cdot \nabla ρ_{ℓ^{'}} | | u_{ℓ} | . \end{matrix}$ Now we estimate the second integral on R.H.S of (3.1). $\begin{array}{c} \int_{Ω_{ℓ}} {| A \nabla (ρ_{ℓ^{'}} u_{ℓ}^{2}) \cdot \nabla ρ_{ℓ^{'}} |}^{\frac{p}{2}} \\ = \int_{Ω_{ℓ}} {| 2 ρ_{ℓ^{'}} u_{ℓ} A \nabla u_{ℓ} \cdot \nabla ρ_{ℓ^{'}} + u_{ℓ}^{2} A \nabla ρ_{ℓ^{'}} \cdot \nabla ρ_{ℓ^{'}} |}^{\frac{p}{2}} \\ (3.3) & ⩽ 2^{\frac{p}{2}} C_{0} \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla ρ_{ℓ^{'}} |^{\frac{p}{2}} ρ_{ℓ^{'}}^{\frac{p}{2}} | u_{ℓ} |^{\frac{p}{2}} + C_{0} \int_{Ω_{ℓ}} | A \nabla ρ_{ℓ^{'}} \cdot \nabla ρ_{ℓ^{'}} |^{\frac{p}{2}} | u_{ℓ} |^{p} . \end{array}$ Finally the last integral on R.H.S of (3.1) gives $\begin{array}{c} \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} | A \nabla (ρ_{ℓ^{'}} u_{ℓ}^{2}) \cdot \nabla ρ_{ℓ^{'}} | \\ = \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} | 2 ρ_{ℓ^{'}} u_{ℓ} A \nabla u_{ℓ} \cdot \nabla ρ_{ℓ^{'}} + u_{ℓ}^{2} A \nabla ρ_{ℓ^{'}} \cdot \nabla ρ_{ℓ^{'}} | \\ (3.4) & ⩽ 2 \int_{Ω_{ℓ}} ρ_{ℓ^{'}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} | A \nabla u_{ℓ} \cdot \nabla ρ_{ℓ^{'}} | | u_{ℓ} | + \int_{Ω_{ℓ}} u_{ℓ}^{2} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} | A \nabla ρ_{ℓ^{'}} \cdot \nabla ρ_{ℓ^{'}} | . \end{array}$ Combining (3.1), (3.2), (3.3) and (3.4) and using $0 ⩽ ρ_{ℓ^{'}} ⩽ 1$ we obtain $\begin{array}{c} (λ_{D}^{1} (Ω_{ℓ}) - λ_{M}^{1} (Ω_{ℓ})) \int_{Ω_{ℓ}} ρ_{ℓ^{'}}^{p} | u_{ℓ} |^{p} \\ ⩽ p \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} | A \nabla u_{ℓ} \cdot \nabla ρ_{ℓ^{'}} | | u_{ℓ} | \\ + p C_{0}^{3} \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla ρ_{ℓ^{'}} |^{\frac{p}{2}} | u_{ℓ} |^{\frac{p}{2}} + \frac{p}{2} C_{0}^{2} \int_{Ω_{ℓ}} | A \nabla ρ_{ℓ^{'}} \cdot \nabla ρ_{ℓ^{'}} |^{\frac{p}{2}} | u_{ℓ} |^{p} \\ + p C_{0} \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} | A \nabla u_{ℓ} \cdot \nabla ρ_{ℓ^{'}} | | u_{ℓ} | + \frac{p}{2} C_{0} \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} | A \nabla ρ_{ℓ^{'}} \cdot \nabla ρ_{ℓ^{'}} | u_{ℓ}^{2} \\ (3.5) & = p (C_{1} I_{1} + C_{2} I_{2} + C_{3} I_{3} + C_{4} I_{4}), \end{array}$ where $\begin{array}{c} I_{1} = \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} | A \nabla u_{ℓ} \cdot \nabla ρ_{ℓ^{'}} | | u_{ℓ} |, \\ I_{2} = \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla ρ_{ℓ^{'}} |^{\frac{p}{2}} | u_{ℓ} |^{\frac{p}{2}}, \\ I_{3} = \int_{Ω_{ℓ}} | A \nabla ρ_{ℓ^{'}} \cdot \nabla ρ_{ℓ^{'}} |^{\frac{p}{2}} | u_{ℓ} |^{p}, \\ I_{4} = \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} | A \nabla ρ_{ℓ^{'}} \cdot \nabla ρ_{ℓ^{'}} | u_{ℓ}^{2} \end{array}$ and $C_{1} = (1 + C_{0})$ , $C_{2} = C_{0}^{3}$ , $C_{3} = \frac{1}{2} C_{0}^{2}$ and $C_{4} = \frac{1}{2} C_{0}$ .

Let $D_{ℓ^{'}} = Ω_{ℓ^{'} + 1} ∖ Ω_{ℓ^{'}}$ . Note that $| \nabla ρ_{ℓ^{'}} | = χ_{D_{ℓ^{'}}}$ . We now estimate the integrals $I_{1}$ , $I_{2}$ , $I_{3}$ and $I_{4}$ respectively.

Estimate for $I_{1}$ : Using Cauchy-Schwarz inequality, we obtain $\begin{matrix} I_{1} ⩽ \int_{Ω_{ℓ}} | A \nabla u_{ℓ} |^{\frac{p}{2}} | \nabla u_{ℓ} |^{\frac{p - 2}{2}} | \nabla ρ_{ℓ^{'}} | | u_{ℓ} | ⩽ ‖ A ‖_{\infty}^{\frac{p}{2}} \int_{D_{ℓ^{'}}} | \nabla u_{ℓ} |^{p - 1} | u_{ℓ} | . \end{matrix}$ By the Hölder’s inequality, and then by the Poincaré inequality (2.1), we obtain $\begin{matrix} (3.6) & I_{1} ⩽ ‖ A ‖_{\infty}^{\frac{p}{2}} {(\int_{D_{ℓ^{'}}} | \nabla u_{ℓ} |^{p})}^{\frac{p - 1}{p}} {(\int_{D_{ℓ^{'}}} | u_{ℓ} |^{p})}^{\frac{1}{p}} ⩽ ‖ A ‖_{\infty}^{\frac{p}{2}} C_{p} \int_{D_{ℓ^{'}}} | \nabla u_{ℓ} |^{p} . \end{matrix}$

Estimate for $I_{2}$ : Cauchy-Schwarz inequality gives $\begin{matrix} I_{2} ⩽ \int_{Ω_{ℓ}} | A \nabla u_{ℓ} |^{\frac{p}{2}} | \nabla ρ_{ℓ^{'}} |^{\frac{p}{2}} | u_{ℓ} |^{\frac{p}{2}} ⩽ ‖ A ‖_{\infty}^{\frac{p}{2}} \int_{D_{ℓ^{'}}} | \nabla u_{ℓ} |^{\frac{p}{2}} | u_{ℓ} |^{\frac{p}{2}} . \end{matrix}$ Using (2.2) and then the Hölder’s inequality and the Poincaré inequality (2.1), we obtain $\begin{matrix} (3.7) & I_{2} ⩽ ‖ A ‖_{\infty}^{\frac{p}{2}} {(\int_{D_{ℓ^{'}}} | \nabla u_{ℓ} |^{p})}^{\frac{1}{2}} {(\int_{D_{ℓ^{'}}} | u_{ℓ} |^{p})}^{\frac{1}{2}} ⩽ ‖ A ‖_{\infty}^{\frac{p}{2}} C_{p}^{\frac{p}{2}} \int_{D_{ℓ^{'}}} | \nabla u_{ℓ} |^{p} . \end{matrix}$

Estimate for $I_{3}$ : Using the Poincaré inequality (2.1), we estimate the integral $\begin{matrix} (3.8) & I_{3} ⩽ ‖ A ‖_{\infty}^{\frac{p}{2}} \int_{D_{ℓ^{'}}} | u_{ℓ} |^{p} ⩽ ‖ A ‖_{\infty}^{\frac{p}{2}} C_{p}^{p} \int_{D_{ℓ^{'}}} | \nabla u_{ℓ} |^{p} . \end{matrix}$

Estimate for $I_{4}$ : Similarly we estimate the integral $I_{4}$ as follows. $\begin{matrix} I_{4} ⩽ ‖ A ‖_{\infty}^{\frac{p}{2}} \int_{D_{ℓ^{'}}} | \nabla u_{ℓ} |^{p - 2} | u_{ℓ} |^{2} . \end{matrix}$ Again we use here Hölder’s inequality, and then the Poincaré inequality (2.1) to obtain $\begin{matrix} (3.9) & I_{4} ⩽ ‖ A ‖_{\infty}^{\frac{p}{2}} {(\int_{D_{ℓ^{'}}} | \nabla u_{ℓ} |^{p})}^{\frac{p - 2}{p}} {(\int_{D_{ℓ^{'}}} | u_{ℓ} |^{p})}^{\frac{2}{p}} ⩽ ‖ A ‖_{\infty}^{\frac{p}{2}} C_{p}^{2} \int_{D_{ℓ^{'}}} | \nabla u_{ℓ} |^{p} . \end{matrix}$ Substituting (3.6), (3.7), (3.8) and (3.9) in (3.5), we get $\begin{matrix} (3.10) & (λ_{D}^{1} (Ω_{ℓ}) - λ_{M}^{1} (Ω_{ℓ})) \int_{Ω_{ℓ}} ρ_{ℓ^{'}}^{p} | u_{ℓ} |^{p} ⩽ p ‖ A ‖_{\infty}^{\frac{p}{2}} C^{'} \int_{D_{ℓ^{'}}} | \nabla u_{ℓ} |^{p} \end{matrix}$ where $C^{'} = (C_{1} C_{p} + C_{2} C_{p}^{\frac{p}{2}} + C_{3} C_{p}^{p} + C_{4} C_{p}^{2})$ . Notice that, (1.6) together with (1.3) gives a $β^{'} > 0$ such that for $ℓ > ℓ_{0}$ we have $λ_{D}^{1} (Ω_{ℓ}) - λ_{M}^{1} (Ω_{ℓ}) > β^{'}$ and (3.10) gives $\begin{matrix} (3.11) & β^{'} \int_{Ω_{ℓ}} ρ_{ℓ^{'}}^{p} | u_{ℓ} |^{p} ⩽ p ‖ A ‖_{\infty}^{\frac{p}{2}} C^{'} \int_{D_{ℓ^{'}}} | \nabla u_{ℓ} |^{p} . \end{matrix}$

Since $ρ_{ℓ^{'}}^{p} u_{ℓ} \in V (Ω_{ℓ})$ , from (2.8) we get $\begin{array}{rcl} λ_{M}^{1} (Ω_{ℓ}) \int_{Ω_{ℓ}} ρ_{ℓ^{'}}^{p} | u_{ℓ} |^{p} & = & \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} A \nabla u_{ℓ} \cdot \nabla (ρ_{ℓ^{'}}^{p} u_{ℓ}) \\ = & \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p}{2}} ρ_{ℓ^{'}}^{p} + p \int_{Ω_{ℓ}} ρ_{ℓ^{'}}^{p - 1} u_{ℓ} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} A \nabla u_{ℓ} \cdot \nabla ρ_{ℓ^{'}} . \end{array}$ Using the inequality $a - | b | ⩽ a + b$ for $a, b \in R$ , we obtain $\begin{matrix} λ_{M}^{1} (Ω_{ℓ}) \int_{Ω_{ℓ}} ρ_{ℓ^{'}}^{p} | u_{ℓ} |^{p} ⩾ \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p}{2}} ρ_{ℓ^{'}}^{p} - p \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p - 2}{2}} | A \nabla u_{ℓ} \cdot \nabla ρ_{ℓ^{'}} | ρ_{ℓ^{'}}^{p - 1} | u_{ℓ} | . \end{matrix}$ Using (2.3) and $ρ_{ℓ^{'}} = 1$ in $Ω_{ℓ^{'}}$ in the first integral, $0 ⩽ ρ_{ℓ^{'}} ⩽ 1$ and $| \nabla ρ_{ℓ^{'}} | = χ_{D_{ℓ^{'}}}$ and then the Hölder’s inequality in the second integral of the R.H.S, we get $\begin{array}{rcl} λ_{M}^{1} (Ω_{ℓ}) \int_{Ω_{ℓ}} ρ_{ℓ^{'}}^{p} | u_{ℓ} |^{p} & ⩾ & λ^{\frac{p}{2}} \int_{Ω_{ℓ^{'}}} | \nabla u_{ℓ} |^{p} - p ‖ A ‖_{\infty}^{\frac{p}{2}} \int_{D_{ℓ^{'}}} | \nabla u_{ℓ} |^{p - 1} | u_{ℓ} | \\ ⩾ & λ^{\frac{p}{2}} \int_{Ω_{ℓ^{'}}} | \nabla u_{ℓ} |^{p} - p ‖ A ‖_{\infty}^{\frac{p}{2}} {(\int_{D_{ℓ^{'}}} | \nabla u_{ℓ} |^{p})}^{\frac{p - 1}{p}} {(\int_{D_{ℓ^{'}}} | u_{ℓ} |^{p})}^{\frac{1}{p}} . \end{array}$ Finally the Poincaré inequality (2.1) on the last integral gives $\begin{matrix} (3.12) & λ_{M}^{1} (Ω_{ℓ}) \int_{Ω_{ℓ}} ρ_{ℓ^{'}}^{p} | u_{ℓ} |^{p} ⩾ λ^{\frac{p}{2}} \int_{Ω_{ℓ^{'}}} | \nabla u_{ℓ} |^{p} - p ‖ A ‖_{\infty}^{\frac{p}{2}} C_{p} \int_{D_{ℓ^{'}}} | \nabla u_{ℓ} |^{p} . \end{matrix}$

Combining (3.12) with (3.11), we get $\begin{matrix} β^{'} λ^{\frac{p}{2}} \int_{Ω_{ℓ^{'}}} | \nabla u_{ℓ} |^{p} ⩽ p ‖ A ‖_{\infty}^{\frac{p}{2}} (λ_{M}^{1} (Ω_{ℓ}) C^{'} + β^{'} C_{p}) \int_{D_{ℓ^{'}}} | \nabla u_{ℓ} |^{p}, \end{matrix}$ which gives $\begin{matrix} (3.13) & β \int_{Ω_{ℓ^{'}}} | \nabla u_{ℓ} |^{p} ⩽ C \int_{D_{ℓ^{'}}} | \nabla u_{ℓ} |^{p}, \end{matrix}$ where $β = β^{'} λ^{\frac{p}{2}}$ , and $C = p ‖ A ‖_{\infty}^{\frac{p}{2}} (μ^{1} (ω) C^{'} + β^{'} C_{p})$ . Here we use the fact that $λ_{M}^{1} (Ω_{ℓ}) ⩽ μ^{1} (ω)$ , which follows from (2.9) by using $u (x) = W (X_{2}) \in V (Ω_{ℓ})$ as a test function. Therefore, from (3.13) we deduce that $\begin{matrix} (β + C) \int_{Ω_{ℓ^{'}}} | \nabla u_{ℓ} |^{p} ⩽ C \int_{Ω_{ℓ^{'} + 1}} | \nabla u_{ℓ} |^{p}, \end{matrix}$ and we have $\begin{matrix} \int_{Ω_{ℓ^{'}}} | \nabla u_{ℓ} |^{p} ⩽ α \int_{Ω_{ℓ^{'} + 1}} | \nabla u_{ℓ} |^{p}, \end{matrix}$ where $α = \frac{C}{β + C} < 1$ . Applying this procedure successively for $ℓ^{'} = r, r + 1, \dots, r + [ℓ - r] - 1$ , we get $\begin{aligned} \int_{Ω_{r}} | \nabla u_{ℓ} |^{p} & ⩽ α^{[ℓ - r]} \int_{Ω_{ℓ}} | \nabla u_{ℓ} |^{p} ⩽ \frac{α^{[ℓ - r]}}{λ^{\frac{p}{2}}} \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p}{2}} \\ (3.14) & = \frac{λ_{M}^{1} (Ω_{ℓ})}{λ^{\frac{p}{2}}} α^{[ℓ - r]} ⩽ \frac{μ^{1} (ω)}{λ^{\frac{p}{2}}} α^{[ℓ - r]}, \end{aligned}$ which proves (1.7). To prove (1.8), we use the Poincaré inequality (2.1) and (3.14) and we get $\begin{matrix} (3.15) & \int_{Ω_{r}} | u_{ℓ} |^{p} ⩽ C_{p}^{p} \int_{Ω_{r}} | \nabla u_{ℓ} |^{p} ⩽ \frac{μ^{1} (ω) C_{p}^{p}}{λ^{\frac{p}{2}}} α^{[ℓ - r]} . \end{matrix}$ (3.14) and (3.15) together completes the proof. □

An immediate corollary of this theorem is the following, which gives the concentration of the masses near the ends of the cylinder. Corollary 1.
If $A_{12} \cdot \nabla_{X_{2}} W ≢ 0$ a.e. in ω, then for large ℓ and for any $0 < r ⩽ ℓ - 1$ , we have $\begin{matrix} (3.16) & \int_{Ω_{ℓ} ∖ Ω_{r}} | \nabla u_{ℓ} |^{p} ⩾ 1 - C_{1} α^{[ℓ - r]} and \int_{Ω_{ℓ} ∖ Ω_{r}} | u_{ℓ} |^{p} ⩾ 1 - C_{2} α^{[ℓ - r]} . \end{matrix}$

4. Connection with the problem on semi-infinite cylinder

Although, unlike the first Dirichlet eigenvalues it is not known whether the function $ℓ \to λ_{M}^{1} (Ω_{ℓ})$ is monotone or not, we show that the limit of $λ_{M}^{1} (Ω_{ℓ})$ exists as ℓ goes to infinity (Theorem 1.2). To identify the limit we introduce the variational problems (1.9) on semi-infinite cylinders $Ω_{\infty}^{\pm}$ . In next lemma, we find the possible range of the quantities $ν_{\infty}^{\pm}$ defined in (1.9) and then (in Lemma 4.2) we identify $ν_{\infty}^{\pm}$ with the limits of specific minimization problems (defined later in (4.4)) defined on the half-cylinders $Ω_{ℓ}^{\pm}$ , which we will use to prove Theorem 1.2.

Remark 1.
If A has the symmetry (S), then it is easy to check that $ν_{\infty}^{+} = ν_{\infty}^{-}$ , since $v (- x_{1}, - X_{2}) \in V (Ω_{\infty}^{-})$ for any $v (x_{1}, X_{2}) \in V (Ω_{\infty}^{+})$ and vice versa, and clearly both give the same value in (1.9). Otherwise, we use $ν_{\infty}^{-} = {\tilde{ν}}_{\infty}^{+}$ , where ${\tilde{ν}}_{\infty}^{+}$ is defined similarly as $ν_{\infty}^{+}$ , but in place of $A (X_{2})$ we put $\begin{matrix} \tilde{A} (X_{2}) = (\begin{array}{cc} a_{11} (X_{2}) & - A_{12} (X_{2}) \\ - A_{12}^{t} (X_{2}) & A_{22} (X_{2}) \end{array}) . \end{matrix}$
Lemma 4.1.
We have $\begin{matrix} (4.1) & λ C_{p}^{p} ⩽ ν_{\infty}^{\pm} ⩽ μ^{1} (ω) \end{matrix}$ where λ is the constant given in ( 2.3 ) and $C_{p}$ is the constant of Poincaré inequality ( 2.1 ).
Proof.
Since ${\tilde{A}}_{22} (X_{2}) = A_{22} (X_{2})$ as shown in Remark 1, and $μ^{1} (ω)$ depends only on $A_{22} (X_{2})$ , it is enough to prove (4.1) for $ν_{\infty}^{+}$ . We get the lower bound $ν_{\infty}^{+} ⩾ λ C_{p}^{p}$ by using the uniform ellipticity (2.3) of A and the Poincaré inequality (2.1). To show the upper bound $ν_{\infty}^{+} ⩽ μ^{1} (ω)$ , we first fix $ϵ > 0$ . For $x = (x_{1}, X_{2}) \in Ω_{\infty}^{+}$ define the function $v_{ϵ}$ in $V (Ω_{\infty}^{+})$ as $\begin{matrix} v_{ϵ} (x) = e^{- ϵ x_{1}} W (X_{2}) . \end{matrix}$ We have $\begin{array}{rcl} \int_{Ω_{\infty}^{+}} | A \nabla v_{ϵ} \cdot \nabla v_{ϵ} |^{\frac{p}{2}} & = & \int_{Ω_{\infty}^{+}} {| e^{- 2 ϵ x_{1}} (ϵ^{2} a_{11} W^{2} - 2 ϵ (A_{12} \cdot \nabla_{X_{2}} W) W + A_{22} \nabla_{X_{2}} W \cdot \nabla_{X_{2}} W) |}^{\frac{p}{2}} \\ ⩽ & (\int_{0}^{\infty} e^{- p ϵ x_{1}}) \int_{ω} {(| A_{22} \nabla_{X_{2}} W \cdot \nabla_{X_{2}} W | + ϵ | ϵ a_{11} W^{2} - 2 (A_{12} \cdot \nabla_{X_{2}} W) W |)}^{\frac{p}{2}} . \end{array}$ Splitting the integral on ω by using the inequality ${(a + b)}^{q} ⩽ a^{q} + q 2^{q - 1} (a^{q - 1} b + b^{q})$ for $a, b ⩾ 0$ , $q ⩾ 1$ , and then using (2.7) and the fact that W is normalised we get $\begin{array}{rcl} \int_{Ω_{\infty}^{+}} | A \nabla v_{ϵ} \cdot \nabla v_{ϵ} |^{\frac{p}{2}} & ⩽ & (\int_{0}^{\infty} e^{- p ϵ x_{1}}) [μ^{1} (ω) + ϵ C \int_{ω} | A_{22} \nabla_{X_{2}} W \cdot \nabla_{X_{2}} W |^{\frac{p - 2}{2}} \\ \times | ϵ a_{11} W^{2} - 2 (A_{12} \cdot \nabla_{X_{2}} W) W | \\ (4.2) & + ϵ^{\frac{p}{2}} C \int_{ω} {| ϵ a_{11} W^{2} - 2 (A_{12} \cdot \nabla_{X_{2}} W) W |}^{\frac{p}{2}}] . \end{array}$ As $\begin{matrix} (4.3) & \int_{Ω_{\infty}^{+}} | v_{ϵ} |^{p} = \int_{0}^{\infty} e^{- p ϵ x_{1}}, \end{matrix}$ substituting (4.2), (4.3) in (1.9), we get $\begin{array}{rcl} ν_{\infty}^{+} & ⩽ & μ^{1} (ω) + ϵ C \int_{ω} | A_{22} \nabla_{X_{2}} W \cdot \nabla_{X_{2}} W |^{\frac{p - 2}{2}} | a_{11} ϵ W^{2} - 2 (A_{12} \cdot \nabla_{X_{2}} W) W | \\ + ϵ^{\frac{p}{2}} C \int_{ω} {| a_{11} ϵ W^{2} - 2 (A_{12} \cdot \nabla_{X_{2}} W) W |}^{\frac{p}{2}} . \end{array}$ Passing to the limit $ϵ \to 0$ we get the result. □

We now consider the minimization problems $\begin{matrix} (4.4) & {\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{\pm}) = inf {\int_{Ω_{ℓ}^{\pm}} | A \nabla u \cdot \nabla u |^{\frac{p}{2}} : u \in W^{1, p} (Ω_{ℓ}^{\pm}), \int_{Ω_{ℓ}^{\pm}} | u |^{p} = 1, u = 0 on γ_{ℓ}^{\pm} \cup Γ_{ℓ}^{\pm}} \end{matrix}$ in the half cylinders $\begin{matrix} (4.5) & Ω_{ℓ}^{+} = (0, ℓ) \times ω and Ω_{ℓ}^{-} = (- ℓ, 0) \times ω, \end{matrix}$ where the boundaries are defined by $\begin{matrix} (4.6) & Γ_{ℓ}^{\pm} = {\pm ℓ} \times ω, γ_{ℓ}^{+} = (0, ℓ) \times \partial ω and γ_{ℓ}^{-} = (- ℓ, 0) \times \partial ω . \end{matrix}$ In the following lemma, we identify the values $ν_{\infty}^{\pm}$ with the limits (as ℓ goes to infinity) of the quantities defined in (4.4).
Remark 2.
Since for any fixed $ℓ > 0$ the domains are bounded, it is a well known fact that the infimum in (4.4) is attained. We denote by ${\tilde{u}}_{ℓ}^{\pm}$ , the positive normalized minimizers corresponding to ${\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{\pm})$ .
Lemma 4.2.
We have $\begin{matrix} (4.7) & ν_{\infty}^{\pm} = lim_{ℓ \to \infty} {\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{\pm}) . \end{matrix}$
Proof.
It is enough to prove (4.7) only for ${\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{+})$ , proof for ${\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{-})$ is similar. If $ℓ_{1} < ℓ_{2}$ , then any admissible function in (4.4) for ${\tilde{λ}}_{M}^{1} (Ω_{ℓ_{1}}^{+})$ can be extended to an admissible function for ${\tilde{λ}}_{M}^{1} (Ω_{ℓ_{2}}^{+})$ by setting it zero on $Ω_{ℓ_{2}}^{+} ∖ Ω_{ℓ_{1}}^{+}$ , which shows that ${\tilde{λ}}_{M}^{1} (Ω_{ℓ_{1}}^{+}) ⩾ {\tilde{λ}}_{M}^{1} (Ω_{ℓ_{2}}^{+})$ . Therefore, monotonicity of the function $ℓ \mapsto {\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{+})$ implies that ${lim}_{ℓ \to \infty} {\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{+})$ exists.

A similar argument shows that ${\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{+}) ⩾ ν_{\infty}^{+}$ , ∀ $ℓ > 0$ .

The following space $\begin{matrix} (4.8) & V_{s} (Ω_{\infty}^{+}) = {u \in C^{\infty} (Ω_{\infty}^{+}) \cap V (Ω_{\infty}^{+}) : \exists M = M (u) > 0 such that u = 0 on (M, \infty) \times ω} \end{matrix}$ is clearly dense in $V (Ω_{\infty}^{+})$ . Let $\tilde{u} = \frac{u}{‖ u ‖_{L^{p}}}$ , where $u \in V (Ω_{\infty}^{+}) ∖ {0}$ . Then $\exists {v_{n}} \in V_{s} (Ω_{\infty}^{+}) ∖ {0}$ with $supp (v_{n}) \subset V (Ω_{ℓ_{n}}^{+})$ such that $v_{n} \to \tilde{u}$ in $V (Ω_{\infty}^{+})$ as $n \to \infty$ . In particular ${v_{n}}$ is bounded in $V (Ω_{\infty}^{+})$ and $‖ v_{n} ‖_{L^{p} (Ω_{\infty}^{+})} \to 1$ .

Define $\tilde{v_{n}} = \frac{v_{n}}{‖ v_{n} ‖_{L^{p}}}$ and let $ϵ > 0$ . Then for all $n ⩾ n_{0}$ , $\begin{array}{rcl} ‖ \tilde{u} - \tilde{v_{n}} ‖_{W^{1, p} (Ω_{\infty}^{+})} & ⩽ & ‖ \tilde{u} - v_{n} ‖_{W^{1, p} (Ω_{\infty}^{+})} + {‖ v_{n} - \frac{v_{n}}{‖ v_{n} ‖_{L^{p}}} ‖}_{W^{1, p} (Ω_{\infty}^{+})} \\ (4.9) & = & ‖ \tilde{u} - v_{n} ‖_{W^{1, p} (Ω_{\infty}^{+})} + | 1 - \frac{1}{‖ v_{n} ‖_{L^{p}}} | ‖ v_{n} ‖_{W^{1, p} (Ω_{\infty}^{+})} < ϵ . \end{array}$ Using the inequality $| a^{p} - b^{p} | ⩽ 2 p | a - b | max {a^{p - 1}, b^{p - 1}}$ for $a, b ⩾ 0$ and $p ⩾ 1$ and then Holder’s inequality we obtain $\begin{array}{rcl} I_{n} & = & | \int_{Ω_{\infty}^{+}} (| A \nabla \tilde{u} \cdot \nabla \tilde{u} |^{\frac{p}{2}} - | A \nabla \tilde{v_{n}} \cdot \nabla \tilde{v_{n}} |^{\frac{p}{2}}) | \\ ⩽ & p \int_{Ω_{\infty}^{+}} | A \nabla \tilde{u} \cdot \nabla \tilde{u} - A \nabla \tilde{v_{n}} \cdot \nabla \tilde{v_{n}} | max {| A \nabla \tilde{u} \cdot \nabla \tilde{u} |^{\frac{p - 2}{2}}, | A \nabla \tilde{v_{n}} \cdot \nabla \tilde{v_{n}} |^{\frac{p - 2}{2}}} \\ ⩽ & p {(\int_{Ω_{\infty}^{+}} | A \nabla \tilde{u} \cdot \nabla \tilde{u} - A \nabla \tilde{v_{n}} \cdot \nabla \tilde{v_{n}} |^{\frac{p}{2}})}^{\frac{2}{p}} \\ (4.10) & \times {(\int_{Ω_{\infty}^{+}} max {| A \nabla \tilde{u} \cdot \nabla \tilde{u} |^{\frac{p}{2}}, | A \nabla \tilde{v_{n}} \cdot \nabla \tilde{v_{n}} |^{\frac{p}{2}}})}^{\frac{p - 2}{p}} \end{array}$ The second integral in R.H.S of (4.10) is clearly bounded. Using the identity $A \nabla u \cdot \nabla u - A \nabla v \cdot \nabla v = A \nabla (u - v) \cdot \nabla (u - v) + 2 A \nabla (u - v) \cdot \nabla v$ in (4.10), we get $\begin{array}{rcl} I_{n} & ⩽ & p C_{1} {(\int_{Ω_{\infty}^{+}} {| A \nabla (\tilde{u} - \tilde{v_{n}}) \cdot \nabla (\tilde{u} - \tilde{v_{n}}) + 2 A \nabla (\tilde{u} - \tilde{v_{n}}) \cdot \nabla \tilde{v_{n}} |}^{\frac{p}{2}})}^{\frac{2}{p}} \\ (4.11) & ⩽ & p C_{1} {(2^{\frac{p - 2}{2}} \int_{Ω_{\infty}^{+}} {| A \nabla (\tilde{u} - \tilde{v_{n}}) \cdot \nabla (\tilde{u} - \tilde{v_{n}}) |}^{\frac{p}{2}} + 2^{p - 1} \int_{Ω_{\infty}^{+}} {| A \nabla (\tilde{u} - \tilde{v_{n}}) \cdot \nabla \tilde{v_{n}} |}^{\frac{p}{2}})}^{\frac{2}{p}} \end{array}$ Using (4.9), we estimate the first integral of (4.11) as $\begin{matrix} (4.12) & \int_{Ω_{\infty}^{+}} {| A \nabla (\tilde{u} - \tilde{v_{n}}) \cdot \nabla (\tilde{u} - \tilde{v_{n}}) |}^{\frac{p}{2}} ⩽ ‖ A ‖_{\infty}^{\frac{p}{2}} ϵ^{p} . \end{matrix}$ Hölder’s inequality together with (4.9) gives $\begin{matrix} (4.13) & \int_{Ω_{\infty}^{+}} {| A \nabla (\tilde{u} - \tilde{v_{n}}) \cdot \nabla \tilde{v_{n}} |}^{\frac{p}{2}} ⩽ C_{2} ‖ A ‖_{\infty}^{\frac{p}{2}} ϵ^{\frac{p}{2}} . \end{matrix}$ Combining (4.12), (4.13) with (4.11) we get for $n ⩾ n_{0}$ , $\begin{matrix} (4.14) & I_{n} = | \frac{\int_{Ω_{\infty}^{+}} | A \nabla u \cdot \nabla u |^{\frac{p}{2}}}{\int_{Ω_{\infty}^{+}} | u |^{p}} - \frac{\int_{Ω_{\infty}^{+}} | A \nabla v_{n} \cdot \nabla v_{n} |^{\frac{p}{2}}}{\int_{Ω_{\infty}^{+}} | v_{n} |^{p}} | ⩽ C ϵ . \end{matrix}$ This proves (4.7). □
Proof of Theorem 1.2.
(i) First we show that $\begin{matrix} (4.15) & \underset{ℓ \to \infty}{lim sup} λ_{M}^{1} (Ω_{ℓ}) ⩽ min {ν_{\infty}^{+}, ν_{\infty}^{-}} . \end{matrix}$ Without loss of generality we may assume that $ν_{\infty}^{+} = min {ν_{\infty}^{+}, ν_{\infty}^{-}}$ . For any $ℓ > 0$ the function ${\tilde{u}}_{ℓ}^{+} (x_{1} - \frac{ℓ}{2}, X_{2}) \in V (Ω_{ℓ / 2})$ , where ${\tilde{u}}_{ℓ}^{+} (x_{1}, X_{2})$ is the minimizer for ${\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{+})$ (see Remark 2), is a suitable test function in (2.9) and we have $λ_{M}^{1} (Ω_{ℓ / 2}) ⩽ {\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{+})$ . On the other hand, for any $ϵ > 0$ , Lemma 4.2 gives an $ℓ_{ϵ} > 0$ such that ${\tilde{λ}}_{M}^{1} (Ω_{2 ℓ_{ϵ}}^{+}) < ν_{\infty}^{+} + ϵ$ . Monotonicity of $ℓ \mapsto {\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{+})$ gives ${sup}_{ℓ ⩾ ℓ_{ϵ}} λ_{M}^{1} (Ω_{ℓ}) ⩽ ν_{\infty}^{+} + ϵ$ and (4.15) holds.

(ii) For the reverse inequality first we consider the case $A_{12} \cdot \nabla W ≢ 0$ a.e. in ω. Let $ρ (x_{1})$ be the function defined by $\begin{matrix} ρ (x_{1}) = \{\begin{array}{ll} 0 & if x_{1} ⩽ - 1, \\ 1 + x_{1} & if x_{1} \in (- 1, 0), \\ 1 & if x_{1} ⩾ 0 . \end{array} \end{matrix}$ Define $w_{ℓ + 1} (x_{1}, X_{2}) = ρ (x_{1} + ℓ) u_{ℓ} (x_{1} + ℓ, X_{2})$ on $Ω_{ℓ + 1}^{-}$ , where $u_{ℓ}$ is the minimizer in (2.9). It is easy to see that $w_{ℓ + 1}$ is a suitable test function for defining ${\tilde{λ}}_{M}^{1} (Ω_{ℓ + 1}^{-})$ in (4.4). Note that $| \nabla ρ | = 1$ in $(- 1, 0) \times ω$ . Hence $\begin{array}{rcl} \int_{Ω_{ℓ + 1}^{-}} | A \nabla w_{ℓ + 1} \cdot \nabla w_{ℓ + 1} |^{\frac{p}{2}} & ⩽ & \int_{Ω_{ℓ}^{+}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p}{2}} + \int_{(- 1, 0) \times ω} {| A \nabla (ρ u_{ℓ}) \cdot \nabla (ρ u_{ℓ}) |}^{\frac{p}{2}} \\ (4.16) & ⩽ & \int_{Ω_{ℓ}^{+}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p}{2}} + 2^{p - 1} ‖ A ‖_{\infty}^{\frac{p}{2}} \int_{(- 1, 0) \times ω} (| u_{ℓ} |^{p} + | \nabla u_{ℓ} |^{p}) . \end{array}$ Substituting (1.7), (1.8) in (4.16), we get a constant $C > 0$ such that $\begin{matrix} (4.17) & \int_{Ω_{ℓ + 1}^{-}} | A \nabla w_{ℓ + 1} \cdot \nabla w_{ℓ + 1} |^{\frac{p}{2}} ⩽ \int_{Ω_{ℓ}^{+}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p}{2}} + C α^{[ℓ - 1]} . \end{matrix}$ We denote by $N_{ℓ}^{\pm}$ and $D_{ℓ}^{\pm}$ the quantities $\begin{matrix} N_{ℓ}^{\pm} = \int_{Ω_{ℓ}^{\pm}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p}{2}}, D_{ℓ}^{\pm} = \int_{Ω_{ℓ}^{\pm}} | u_{ℓ} |^{p} . \end{matrix}$ Clearly $\begin{matrix} (4.18) & N_{ℓ}^{+} + N_{ℓ}^{-} = λ_{M}^{1} (Ω_{ℓ}) and D_{ℓ}^{+} + D_{ℓ}^{-} = 1 . \end{matrix}$ By the same computation for $Ω_{ℓ + 1}^{+}$ and then using (4.18) we get $\begin{matrix} (4.19) & {\tilde{λ}}_{M}^{1} (Ω_{ℓ + 1}^{-}) ⩽ \frac{N_{ℓ}^{+} + C α^{[ℓ - 1]}}{D_{ℓ}^{+}} and {\tilde{λ}}_{M}^{1} (Ω_{ℓ + 1}^{+}) ⩽ \frac{N_{ℓ}^{-} + C α^{[ℓ - 1]}}{D_{ℓ}^{-}} . \end{matrix}$ By (4.19) and (4.18), we have $\begin{matrix} (4.20) & min {{\tilde{λ}}_{M}^{1} (Ω_{ℓ + 1}^{-}), {\tilde{λ}}_{M}^{1} (Ω_{ℓ + 1}^{+})} ⩽ D_{ℓ}^{+} {\tilde{λ}}_{M}^{1} (Ω_{ℓ + 1}^{-}) + D_{ℓ}^{-} {\tilde{λ}}_{M}^{1} (Ω_{ℓ + 1}^{+}) ⩽ λ_{M}^{1} (Ω_{ℓ}) + 2 C α^{[ℓ - 1]} . \end{matrix}$ Passing to the limit as $ℓ \to \infty$ on both sides, and using Lemma 4.2 it follows that $\begin{matrix} (4.21) & min {ν_{\infty}^{+}, ν_{\infty}^{-}} ⩽ \underset{ℓ \to \infty}{lim inf} λ_{M}^{1} (Ω_{ℓ}), \end{matrix}$ which together with (4.15) proves the theorem in this case.

(iii) We now consider the case $A_{12} \cdot \nabla W \equiv 0$ . In this case it is already known from Theorem 2 that $λ_{M}^{1} (Ω_{ℓ}) = μ^{1} (ω)$ for all $ℓ > 0$ . On the other hand we see in Theorem 1.3 (ii) that $ν_{\infty}^{\pm} = μ^{1} (ω)$ in this case. This completes the proof. □

In the next proposition we show that the first eigenfunction for the problem (1.4) on $Ω_{ℓ}$ decays to 0 on that semi-infinite cylinder which corresponds to the strictly smaller of the two values $ν_{\infty}^{+}$ and $ν_{\infty}^{-}$ . Proposition 4.1.
If $ν_{\infty}^{+} < ν_{\infty}^{-}$ , then ${lim}_{ℓ \to \infty} \int_{Ω_{ℓ}^{+}} | \nabla u_{ℓ} |^{p} + | u_{ℓ} |^{p} = 0$ .
Proof.
Taking the limit $ℓ \to \infty$ in (4.20) and using (4.7) and (1.10), we obtain $\begin{matrix} (\underset{ℓ \to \infty}{lim sup} D_{ℓ}^{+}) ν_{\infty}^{-} + (1 - \underset{ℓ \to \infty}{lim sup} D_{ℓ}^{+}) ν_{\infty}^{+} ⩽ ν_{\infty}^{+} \end{matrix}$ which gives $\begin{matrix} \underset{ℓ \to \infty}{lim sup} D_{ℓ}^{+} (ν_{\infty}^{-} - ν_{\infty}^{+}) ⩽ 0, \end{matrix}$ and we must have ${lim sup}_{ℓ \to \infty} D_{ℓ}^{+} = 0$ . Again by (4.19) we have, $\begin{array}{rcl} \frac{N_{ℓ}^{+}}{D_{ℓ}^{-}} + {\tilde{λ}}_{M}^{1} (Ω_{ℓ + 1}^{+}) - \frac{C α^{[ℓ - 1]}}{D_{ℓ}^{-}} & ⩽ & \frac{N_{ℓ}^{+} + N_{ℓ}^{-}}{D_{ℓ}^{-}} \\ = & \frac{N_{ℓ}^{+} + N_{ℓ}^{-}}{D_{ℓ}^{+} + D_{ℓ}^{-}} + \frac{(N_{ℓ}^{+} + N_{ℓ}^{-}) D_{ℓ}^{+}}{(D_{ℓ}^{+} + D_{ℓ}^{-}) D_{ℓ}^{-}} \\ (4.22) & = & λ_{M}^{1} (Ω_{ℓ}) + \frac{D_{ℓ}^{+}}{D_{ℓ}^{-}} (N_{ℓ}^{+} + N_{ℓ}^{-}) . \end{array}$ Since ${lim}_{ℓ \to \infty} D_{ℓ}^{-} = 1$ , passing to the limit $ℓ \to \infty$ in (4.22) and using (4.7) and (1.10), we get ${lim}_{ℓ \to \infty} N_{ℓ}^{+} = 0$ . Finally ${lim}_{ℓ \to \infty} \int_{Ω_{ℓ}^{+}} | \nabla u_{ℓ} |^{p} = 0$ follows using (2.3). □

5. Gap phenomenon on semi-infinite cylinder

In this section we revisit the problem (1.9) on the semi-infinite cylinders. We first investigate when the infimum in (1.9) is attained and follow it up with Theorem 1.3 where we investigate when the gap between $ν_{\infty}^{\pm}$ and $μ^{1} (ω)$ holds and when does not hold.

Proposition 5.1.
If $\begin{matrix} (5.1) & ν_{\infty}^{+} < μ^{1} (ω), \end{matrix}$ then $ν_{\infty}^{+}$ is attained. The minimizer ${\tilde{u}}^{+}$ in ( 1.9 ) is unique upto multiplication by a constant, has constant sign and satisfies $\begin{matrix} (5.2) & \{\begin{array}{l} - div (| A \nabla {\tilde{u}}^{+} \cdot \nabla {\tilde{u}}^{+} |^{\frac{p - 2}{2}} A \nabla {\tilde{u}}^{+}) = ν_{\infty}^{+} | {\tilde{u}}^{+} |^{p - 2} {\tilde{u}}^{+} in Ω_{\infty}^{+}, \\ {\tilde{u}}^{+} = 0 on γ_{\infty}^{+}, \\ (A \nabla {\tilde{u}}^{+}) \cdot ν = 0 on {0} \times ω . \end{array} \end{matrix}$
Proof.
For any $ℓ > 0$ the function ${\tilde{u}}_{ℓ}^{+}$ (defined in Remark 2) can be extended by zero to a function in $W^{1, p} (Ω_{\infty}^{+})$ . Since the sequence ${{\tilde{u}}_{ℓ}^{+}}$ is bounded in $W^{1, p} (Ω_{\infty}^{+})$ , there is a subsequence ${{\tilde{u}}_{ℓ_{k}}^{+}}$ which converges weakly to a limit ${\tilde{u}}^{+} \in W^{1, p} (Ω_{\infty}^{+})$ . Definition of ${\tilde{λ}}_{M}^{1} (Ω_{ℓ_{k}}^{+})$ gives the following identity: $\begin{matrix} (5.3) & \int_{Ω_{\infty}^{+}} {| A \nabla {\tilde{u}}_{ℓ_{k}}^{+} \cdot \nabla {\tilde{u}}_{ℓ_{k}}^{+} |}^{\frac{p}{2}} = {\tilde{λ}}_{M}^{1} (Ω_{ℓ_{k}}^{+}) \int_{Ω_{\infty}^{+}} {| {\tilde{u}}_{ℓ_{k}}^{+} |}^{p} . \end{matrix}$ We claim that $\begin{matrix} (5.4) & \int_{Ω_{\infty}^{+}} {| A \nabla {\tilde{u}}^{+} \cdot \nabla {\tilde{u}}^{+} |}^{\frac{p}{2}} = ν_{\infty}^{+} \int_{Ω_{\infty}^{+}} {| {\tilde{u}}^{+} |}^{p} . \end{matrix}$ Clearly ${\tilde{u}}^{+} \in V (Ω_{\infty}^{+})$ . From (1.9) we have $\begin{matrix} (5.5) & \int_{Ω_{\infty}^{+}} {| A \nabla {\tilde{u}}^{+} \cdot \nabla {\tilde{u}}^{+} |}^{\frac{p}{2}} ⩾ ν_{\infty}^{+} \int_{Ω_{\infty}^{+}} {| {\tilde{u}}^{+} |}^{p} . \end{matrix}$ By Fatou’s lemma and by (5.3), we get $\begin{matrix} (5.6) & \int_{Ω_{\infty}^{+}} {| A \nabla {\tilde{u}}^{+} \cdot \nabla {\tilde{u}}^{+} |}^{\frac{p}{2}} ⩽ \underset{k \to \infty}{lim inf} \int_{Ω_{\infty}^{+}} {| A \nabla {\tilde{u}}_{ℓ_{k}}^{+} \cdot \nabla {\tilde{u}}_{ℓ_{k}}^{+} |}^{\frac{p}{2}} = ν_{\infty}^{+} (\underset{k \to \infty}{lim inf} \int_{Ω_{\infty}^{+}} {| {\tilde{u}}_{ℓ_{k}}^{+} |}^{p}) = ν_{\infty}^{+} . \end{matrix}$ To prove our claim, it is enough to show that $\int_{Ω_{\infty}^{+}} | {\tilde{u}}^{+} |^{p} = 1$ . In addition we will show that ${\tilde{u}}_{ℓ}^{+}$ decays to zero for large $x_{1}$ , and that implies concentration near $x_{1} = 0$ , using the same technique as in the proof of Theorem 1.1.

Let ℓ and $ℓ^{'}$ be such that $0 < ℓ^{'} ⩽ ℓ - 1$ . Define ${\tilde{ρ}}_{ℓ^{'}} = {\tilde{ρ}}_{ℓ^{'}} (x_{1})$ by $\begin{matrix} {\tilde{ρ}}_{ℓ^{'}} (x_{1}) = \{\begin{array}{ll} 0 & x_{1} ⩽ ℓ^{'} \\ x_{1} - ℓ^{'} & x_{1} \in (ℓ^{'}, ℓ^{'} + 1) \\ 1 & x_{1} ⩾ ℓ^{'} + 1 . \end{array} \end{matrix}$ Clearly ${\tilde{u}}_{ℓ}^{+} {\tilde{ρ}}_{ℓ^{'}}$ satisfies Dirichlet boundary condition on $\partial Ω_{ℓ}^{+}$ . Translating this function in $Ω_{ℓ / 2}$ and using definition of $λ_{D}^{1} (Ω_{ℓ / 2})$ and ${\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{+})$ respectively, we get $\begin{matrix} λ_{D}^{1} (Ω_{ℓ / 2}) \int_{Ω_{ℓ}^{+}} {| {\tilde{u}}_{ℓ}^{+} |}^{p} {\tilde{ρ}}_{ℓ^{'}}^{p} ⩽ \int_{Ω_{ℓ}^{+}} {| A \nabla ({\tilde{u}}_{ℓ}^{+} {\tilde{ρ}}_{ℓ^{'}}) \cdot \nabla ({\tilde{u}}_{ℓ}^{+} {\tilde{ρ}}_{ℓ^{'}}) |}^{\frac{p}{2}} \end{matrix}$ and $\begin{matrix} \int_{Ω_{ℓ}^{+}} {| A \nabla {\tilde{u}}_{ℓ}^{+} \cdot \nabla {\tilde{u}}_{ℓ}^{+} |}^{\frac{p - 2}{2}} A \nabla {\tilde{u}}_{ℓ}^{+} \cdot \nabla ({\tilde{ρ}}_{ℓ^{'}}^{p} {\tilde{u}}_{ℓ}^{+}) = {\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{+}) \int_{Ω_{ℓ}^{+}} {| {\tilde{u}}_{ℓ}^{+} |}^{p} {\tilde{ρ}}_{ℓ^{'}}^{p} . \end{matrix}$ Let $D_{ℓ^{'}}^{+} = Ω_{ℓ^{'} + 1}^{+} ∖ Ω_{ℓ^{'}}^{+}$ . An analogous calculation to derive (3.10) in the proof of Theorem 1.1 gives $\begin{matrix} (5.7) & (λ_{D}^{1} (Ω_{ℓ / 2}) - {\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{+})) \int_{Ω_{ℓ}^{+}} {| {\tilde{u}}_{ℓ}^{+} |}^{p} {\tilde{ρ}}_{ℓ^{'}}^{p} ⩽ p ‖ A ‖_{\infty}^{\frac{p}{2}} C^{'} \int_{D_{ℓ^{'}}^{+}} {| \nabla {\tilde{u}}_{ℓ}^{+} |}^{p} . \end{matrix}$ By (1.3), (4.7) and the assumption (5.1), there exists a ${\tilde{β}}^{'} > 0$ such that for $ℓ > ℓ_{0}$ we have $λ_{D}^{1} (Ω_{ℓ / 2}) - {\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{+}) ⩾ {\tilde{β}}^{'}$ , which implies $\begin{matrix} (5.8) & {\tilde{β}}^{'} \int_{Ω_{ℓ}^{+}} {| {\tilde{u}}_{ℓ}^{+} |}^{p} {\tilde{ρ}}_{ℓ^{'}}^{p} ⩽ p ‖ A ‖_{\infty}^{\frac{p}{2}} C^{'} \int_{D_{ℓ^{'}}^{+}} {| \nabla {\tilde{u}}_{ℓ}^{+} |}^{p} . \end{matrix}$ Again by an analogous computation as in derivation of the (3.13) in the proof of Theorem 1.1, and by the properties of ${\tilde{ρ}}_{ℓ^{'}}$ , we obtain $\begin{matrix} {\tilde{β}}^{'} λ^{\frac{p}{2}} \int_{Ω_{ℓ}^{+} ∖ Ω_{ℓ^{'} + 1}^{+}} {| \nabla {\tilde{u}}_{ℓ}^{+} |}^{p} ⩽ p ‖ A ‖_{\infty}^{\frac{p}{2}} (λ_{M}^{1} (Ω_{ℓ}) C^{'} + {\tilde{β}}^{'} C_{p}) \int_{D_{ℓ^{'}}^{+}} {| \nabla {\tilde{u}}_{ℓ}^{+} |}^{p}, \end{matrix}$ which gives $\begin{matrix} (5.9) & \tilde{β} \int_{Ω_{ℓ}^{+} ∖ Ω_{ℓ^{'} + 1}^{+}} {| \nabla {\tilde{u}}_{ℓ}^{+} |}^{p} ⩽ \tilde{C} \int_{Ω_{ℓ^{'} + 1}^{+} ∖ Ω_{ℓ^{'}}^{+}} {| \nabla {\tilde{u}}_{ℓ}^{+} |}^{p}, \end{matrix}$ where $\tilde{β} = {\tilde{β}}^{'} λ^{\frac{p}{2}}$ and $\tilde{C} = p ‖ A ‖_{\infty}^{\frac{p}{2}} (μ^{1} (ω) C^{'} + {\tilde{β}}^{'} C_{p})$ . From (5.9) we get $\begin{matrix} (\tilde{β} + \tilde{C}) \int_{Ω_{ℓ}^{+} ∖ Ω_{ℓ^{'} + 1}^{+}} {| \nabla {\tilde{u}}_{ℓ}^{+} |}^{p} ⩽ \tilde{C} \int_{Ω_{ℓ}^{+} ∖ Ω_{ℓ^{'}}^{+}} {| \nabla {\tilde{u}}_{ℓ}^{+} |}^{p}, \end{matrix}$ which implies $\begin{matrix} (5.10) & \int_{Ω_{ℓ}^{+} ∖ Ω_{ℓ^{'} + 1}^{+}} {| \nabla {\tilde{u}}_{ℓ}^{+} |}^{p} ⩽ \tilde{α} \int_{Ω_{ℓ}^{+} ∖ Ω_{ℓ^{'}}^{+}} {| \nabla {\tilde{u}}_{ℓ}^{+} |}^{p}, \end{matrix}$ where $\tilde{α} = \frac{\tilde{C}}{\tilde{β} + \tilde{C}} < 1$ . Fix $r > 1$ . Applying this procedure successively for $ℓ^{'} = r - 1, r - 2, \dots, r - [r]$ and then using (2.3), we obtain $\begin{array}{rcl} \int_{Ω_{ℓ}^{+} ∖ Ω_{r}^{+}} {| \nabla {\tilde{u}}_{ℓ}^{+} |}^{p} & ⩽ & {\tilde{α}}^{[r]} \int_{Ω_{ℓ}^{+}} {| \nabla {\tilde{u}}_{ℓ}^{+} |}^{p} \\ (5.11) & ⩽ & \frac{{\tilde{α}}^{[r]}}{λ^{\frac{p}{2}}} \int_{Ω_{ℓ}^{+}} {| A \nabla {\tilde{u}}_{ℓ}^{+} \cdot \nabla {\tilde{u}}_{ℓ}^{+} |}^{\frac{p}{2}} = \frac{{\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{+})}{λ^{\frac{p}{2}}} {\tilde{α}}^{[r]} ⩽ \frac{μ^{1} (ω)}{λ^{\frac{p}{2}}} {\tilde{α}}^{[r]} . \end{array}$ By Poincaré inequality (2.1) and (5.11), we have $\begin{matrix} \int_{Ω_{ℓ}^{+} ∖ Ω_{r}^{+}} {| {\tilde{u}}_{ℓ}^{+} |}^{p} ⩽ C_{p}^{p} \int_{Ω_{l}^{+} ∖ Ω_{r}^{+}} {| \nabla {\tilde{u}}_{ℓ}^{+} |}^{p} ⩽ \frac{μ^{1} (ω) C_{p}^{p}}{λ^{\frac{p}{2}}} {\tilde{α}}^{[r]} = C {\tilde{α}}^{[r]}, \forall ℓ > r, \end{matrix}$ which implies $\begin{matrix} (5.12) & \int_{Ω_{r}^{+}} {| {\tilde{u}}_{ℓ}^{+} |}^{p} ⩾ 1 - C {\tilde{α}}^{[r]} . \end{matrix}$ Since ${\tilde{u}}_{ℓ_{k}}^{+} \to {\tilde{u}}^{+}$ strongly in $L^{p} (Ω_{r}^{+})$ , we deduce from (5.12) that $\begin{matrix} \int_{Ω_{r}^{+}} {| {\tilde{u}}^{+} |}^{p} ⩾ 1 - C {\tilde{α}}^{[r]} . \end{matrix}$ Finally, limit $r \to \infty$ gives $\begin{matrix} (5.13) & \int_{Ω_{\infty}^{+}} {| {\tilde{u}}^{+} |}^{p} = 1 . \end{matrix}$ Since ${\tilde{u}}^{+} ≢ 0$ , we can conclude that it is a minimizer realizing $ν_{\infty}^{+}$ in (1.9). It is a standard fact that the minimizer is unique upto multiplication by a constant and ${\tilde{u}}^{+}$ has constant sign follows from the fact that ${\tilde{u}}_{ℓ_{k}}^{+}$ are non-negative.

To show that ${\tilde{u}}^{+}$ is a weak solution of (5.2), for any fixed $ϕ \in V (Ω_{\infty}^{+})$ and $t \in R$ we define the function $\begin{matrix} J_{ϕ} (t) = \frac{\int_{Ω_{\infty}^{+}} | A \nabla ({\tilde{u}}^{+} + t ϕ) \cdot \nabla ({\tilde{u}}^{+} + t ϕ) |^{\frac{p}{2}}}{\int_{Ω_{\infty}^{+}} | {\tilde{u}}^{+} + t ϕ |^{p}} . \end{matrix}$ The derivative of $J_{ϕ}$ with respect to t is $\begin{array}{rcl} J_{ϕ}^{'} (t) & = & \frac{\frac{p}{2} \int_{Ω_{\infty}^{+}} | A \nabla ({\tilde{u}}^{+} + t ϕ) \cdot \nabla ({\tilde{u}}^{+} + t ϕ) |^{\frac{p - 2}{2}} (2 A \nabla {\tilde{u}}^{+} \cdot \nabla ϕ + 2 t A \nabla ϕ \cdot \nabla ϕ)}{\int_{Ω_{\infty}^{+}} | {\tilde{u}}^{+} + t ϕ |^{p}} \\ - \frac{\frac{p}{2} (\int_{Ω_{\infty}^{+}} | A \nabla ({\tilde{u}}^{+} + t ϕ) \cdot \nabla ({\tilde{u}}^{+} + t ϕ) |^{\frac{p}{2}}) (\int_{Ω_{\infty}^{+}} | {\tilde{u}}^{+} + t ϕ |^{p - 2} (2 {\tilde{u}}^{+} ϕ + 2 t ϕ^{2}))}{{(\int_{Ω_{\infty}^{+}} | {\tilde{u}}^{+} + t ϕ |^{p})}^{2}} . \end{array}$ Since ${\tilde{u}}^{+}$ is a minimizer, we must have $J_{ϕ}^{'} (0) = 0$ . Substituting this and using (5.4) we get $\begin{matrix} (5.14) & \int_{Ω_{\infty}^{+}} {| A \nabla {\tilde{u}}^{+} \cdot \nabla {\tilde{u}}^{+} |}^{\frac{p - 2}{2}} A \nabla {\tilde{u}}^{+} \cdot \nabla ϕ = ν_{\infty}^{+} \int_{Ω_{\infty}^{+}} {| {\tilde{u}}^{+} |}^{p - 2} u ϕ . \end{matrix}$ This completes the proof. □
Proof of Theorem 1.3.
(i) Assume that $A_{12} \cdot \nabla_{X_{2}} W ≢ 0$ and (1.11) holds. For simplicity of notation, we use $\nabla W = \nabla_{X_{2}} W$ in this proof. For any $ℓ > 0$ we define the set $T_{ℓ} = {x \in ω : dist (x, \partial ω) ⩽ ℓ}$ . Let $β \in (0, 1)$ is fixed and $ρ_{ℓ}$ be an approximation of the characteristic function of ω, i.e. $ρ_{ℓ} \to 1$ pointwise as $ℓ \to 0$ and $ρ_{ℓ}$ satisfies the following properties: $\begin{matrix} (5.15) & ρ_{ℓ} \in C_{c}^{\infty} (ω), 0 ⩽ ρ_{ℓ} ⩽ 1, ρ_{ℓ} = 1 in ω ∖ T_{ℓ}, | \nabla ρ_{ℓ} | ⩽ \frac{1}{ℓ^{β}} in T_{ℓ} . \end{matrix}$ Define the function $\begin{matrix} v_{ℓ}^{ϵ} (x_{1}, X_{2}) = W (X_{2}) - x_{1} ρ_{ℓ} (X_{2}) G_{ϵ} (X_{2}) \end{matrix}$ on $Ω_{ℓ}$ , where ${G_{ϵ}} \subset C_{c}^{\infty} (ω)$ satisfies $\begin{matrix} lim_{ϵ \to 0} G_{ϵ} (X_{2}) = \frac{A_{12} (X_{2}) \cdot \nabla W (X_{2})}{a_{11} (X_{2})} in L^{p} (ω) . \end{matrix}$ We claim that $\begin{matrix} (5.16) & inf_{ϵ > 0} lim_{ℓ \to 0} \frac{\int_{Ω_{ℓ}^{-}} | A \nabla v_{ℓ}^{ϵ} \cdot \nabla v_{ℓ}^{ϵ} |^{\frac{p}{2}}}{\int_{Ω_{ℓ}^{-}} | v_{ℓ}^{ϵ} |^{p}} ⩽ \int_{ω} {(A_{22} \nabla W \cdot \nabla W - \frac{A_{12} \cdot \nabla W}{a_{11}})}^{\frac{p}{2}} . \end{matrix}$ Since $\begin{matrix} a_{11} G_{ϵ}^{2} - 2 (A_{12} \cdot \nabla W) G_{ϵ} + A_{22} \nabla W \cdot \nabla W \to A_{22} \nabla W \cdot \nabla W - \frac{A_{12} \cdot \nabla W}{a_{11}} \end{matrix}$ in $L^{p} (ω)$ as $ϵ \to 0$ , and L.H.S is always positive by uniform ellipticity of the matrix A, the function on the R.H.S of (5.16) is always non-negative. The following inequality holds for any two vectors a, b and $q ⩾ 1$ : $\begin{matrix} | b |^{q} ⩾ | a |^{q} + q ⟨ | a |^{q - 2} a, b - a ⟩ . \end{matrix}$ Using this we obtain $\begin{array}{rcl} \int_{Ω_{ℓ}^{-}} {| v_{ℓ}^{ϵ} |}^{p} & = & \int_{Ω_{ℓ}^{-}} {| W (X_{2}) - x_{1} ρ_{ℓ} (X_{2}) G_{ϵ} (X_{2}) |}^{p} \\ ⩾ & \int_{Ω_{ℓ}^{-}} {| W (X_{2}) |}^{p} - p \int_{Ω_{ℓ}^{-}} {| W (X_{2}) |}^{p - 1} x_{1} ρ_{ℓ} (X_{2}) G_{ϵ} (X_{2}) \\ (5.17) & = & ℓ + p \frac{ℓ^{2}}{2} \int_{ω} | W |^{p - 1} ρ_{ℓ} G_{ϵ} . \end{array}$ Now $\begin{array}{c} \int_{Ω_{ℓ}^{-}} {| A \nabla v_{ℓ}^{ϵ} \cdot \nabla v_{ℓ}^{ϵ} |}^{\frac{p}{2}} \\ = \int_{Ω_{ℓ}^{-}} {| a_{11} {(\partial_{x_{1}} v_{ℓ}^{ϵ})}^{2} + 2 (A_{12} \cdot \nabla_{X_{2}} v_{ℓ}^{ϵ}) \partial_{x_{1}} v_{ℓ}^{ϵ} + A_{22} \nabla_{X_{2}} v_{ℓ}^{ϵ} \cdot \nabla_{X_{2}} v_{ℓ}^{ϵ} |}^{\frac{p}{2}} \\ = \int_{Ω_{ℓ}^{-}} | a_{11} ρ_{ℓ}^{2} G_{ϵ}^{2} - 2 ρ_{ℓ} G_{ϵ} (A_{12} \cdot \nabla W - x_{1} (G_{ϵ} A_{12} \cdot \nabla ρ_{ℓ} + ρ_{ℓ} A_{12} \cdot \nabla G_{ϵ})) \\ (5.18) & + (A_{22} \nabla W - x_{1} (G_{ϵ} A_{22} \nabla ρ_{ℓ} + ρ_{ℓ} A_{22} \nabla G_{ϵ})) \cdot (\nabla W - x_{1} (G_{ϵ} \nabla ρ_{ℓ} + ρ_{ℓ} \nabla G_{ϵ})) |^{\frac{p}{2}} . \end{array}$ Applying Minkowski’s inequality in (5.18), we get $\begin{matrix} (5.19) & \int_{Ω_{ℓ}^{-}} {| A \nabla v_{ℓ}^{ϵ} \cdot \nabla v_{ℓ}^{ϵ} |}^{\frac{p}{2}} ⩽ {({I_{1}^{ϵ}}^{\frac{2}{p}} + {I_{2}^{ϵ}}^{\frac{2}{p}} + {I_{3}^{ϵ}}^{\frac{2}{p}})}^{\frac{p}{2}}, \end{matrix}$ where the integrals $\begin{array}{c} (5.20) & \begin{aligned} I_{1}^{ϵ} & = \int_{Ω_{ℓ}^{-}} {| a_{11} G_{ϵ}^{2} - 2 (A_{12} \cdot \nabla W) G_{ϵ} + A_{22} \nabla W \cdot \nabla W |}^{\frac{p}{2}} \\ = ℓ \int_{ω} {| a_{11} G_{ϵ}^{2} - 2 (A_{12} \cdot \nabla W) G_{ϵ} + A_{22} \nabla W \cdot \nabla W |}^{\frac{p}{2}}, \end{aligned} \\ (5.21) & I_{2}^{ϵ} = \int_{Ω_{ℓ}^{-}} {| a_{11} G_{ϵ}^{2} (ρ_{ℓ}^{2} - 1) + 2 (A_{12} \cdot \nabla W) G_{ϵ} (1 - ρ_{ℓ}) |}^{\frac{p}{2}} = ℓ K_{ℓ}^{ϵ}, \end{array}$ where $\begin{matrix} K_{ℓ}^{ϵ} = \int_{ω} {| a_{11} G_{ϵ}^{2} (ρ_{ℓ}^{2} - 1) + 2 (A_{12} \cdot \nabla W) G_{ϵ} (1 - ρ_{ℓ}) |}^{\frac{p}{2}} \end{matrix}$ and $\begin{matrix} (5.22) & I_{3}^{ϵ} = \int_{Ω_{ℓ}^{-}} {| 2 x_{1} H_{ℓ}^{ϵ} (X_{2}) + x_{1}^{2} F_{ℓ}^{ϵ} (X_{2}) |}^{\frac{p}{2}}, \end{matrix}$ where $\begin{matrix} H_{ℓ}^{ϵ} (X_{2}) = ρ_{ℓ} G_{ϵ}^{2} (A_{12} \cdot \nabla ρ_{ℓ}) + ρ_{ℓ}^{2} G_{ϵ} (A_{12} \cdot \nabla G_{ϵ}) - G_{ϵ} (A_{22} \nabla W \cdot \nabla ρ_{ℓ}) - ρ_{ℓ} (A_{22} \nabla W \cdot \nabla G_{ϵ}) \end{matrix}$ and $\begin{matrix} F_{ℓ}^{ϵ} (X_{2}) = (G_{ϵ} A_{22} \nabla ρ_{ℓ} + ρ_{ℓ} A_{22} \nabla G_{ϵ}) \cdot (G_{ϵ} \nabla ρ_{ℓ} + ρ_{ℓ} \nabla G_{ϵ}) . \end{matrix}$ Since $ρ_{ℓ} \to 1$ pointwise as $ℓ \to 0$ , $K_{ℓ}^{ϵ}$ converges to 0 as $ℓ \to 0$ by dominated convergence theorem. For any fix $ϵ > 0$ , (5.15) gives $\begin{matrix} (5.23) & | H_{ℓ}^{ϵ} | ⩽ C_{1} + \frac{C_{2}}{ℓ^{β}} and | F_{ℓ}^{ϵ} | ⩽ C_{3} + \frac{C_{4}}{ℓ^{2 β}}, \end{matrix}$ where the constants $C_{i} > 0$ ( $1 ⩽ i ⩽ 4$ ) are independent of ℓ. Applying Minkowski’s inequality in (5.22), we obtain $\begin{array}{rcl} I_{3}^{ϵ} & ⩽ & {({(\int_{Ω_{ℓ}^{-}} {| 2 x_{1} H_{ℓ}^{ϵ} (X_{2}) |}^{\frac{p}{2}})}^{\frac{2}{p}} + {(\int_{Ω_{ℓ}^{-}} {| x_{1}^{2} F_{ℓ}^{ϵ} (X_{2}) |}^{\frac{p}{2}})}^{\frac{2}{p}})}^{\frac{p}{2}} \\ ⩽ & {(2 ℓ {(\int_{Ω_{ℓ}^{-}} {| H_{ℓ}^{ϵ} (X_{2}) |}^{\frac{p}{2}})}^{\frac{2}{p}} + ℓ^{2} {(\int_{Ω_{ℓ}^{-}} {| F_{ℓ}^{ϵ} (X_{2}) |}^{\frac{p}{2}})}^{\frac{2}{p}})}^{\frac{p}{2}} \\ = & ℓ {(2 ℓ {(\int_{ω} {| H_{ℓ}^{ϵ} (X_{2}) |}^{\frac{p}{2}})}^{\frac{2}{p}} + ℓ^{2} {(\int_{ω} {| F_{ℓ}^{ϵ} (X_{2}) |}^{\frac{p}{2}})}^{\frac{2}{p}})}^{\frac{p}{2}} \end{array}$ By (5.23), we get $\begin{matrix} (5.24) & I_{3}^{ϵ} ⩽ ℓ {[C_{1} ℓ + C_{2} ℓ^{1 - β} + C_{3} ℓ^{2} + C_{4} ℓ^{2 - 2 β}]}^{\frac{p}{2}} = ℓ C {(ℓ)}^{\frac{p}{2}}, \end{matrix}$ where $C (ℓ) \to 0$ as $ℓ \to 0$ . Using the estimates (5.20), (5.21) and (5.24) in (5.19), we obtain $\begin{array}{c} \int_{Ω_{ℓ}^{-}} {| A \nabla v_{ℓ}^{ϵ} \cdot \nabla v_{ℓ}^{ϵ} |}^{\frac{p}{2}} \\ (5.25) & ⩽ ℓ {[{(\int_{ω} {| a_{11} G_{ϵ}^{2} - 2 (A_{12} \cdot \nabla W) G_{ϵ} + A_{22} \nabla W \cdot \nabla W |}^{\frac{p}{2}})}^{\frac{2}{p}} + {(K_{ℓ}^{ϵ})}^{\frac{2}{p}} + C (ℓ)]}^{\frac{p}{2}} . \end{array}$ Hence we have by (5.17) and (5.25), $\begin{array}{c} \frac{\int_{Ω_{ℓ}^{-}} | A \nabla v_{ℓ}^{ϵ} \cdot \nabla v_{ℓ}^{ϵ} |^{\frac{p}{2}}}{\int_{Ω_{ℓ}^{-}} | v_{ℓ}^{ϵ} |^{p}} \\ (5.26) & ⩽ \frac{{[{(\int_{ω} | a_{11} G_{ϵ}^{2} - 2 (A_{12} \cdot \nabla W) G_{ϵ} + A_{22} \nabla W \cdot \nabla W |^{\frac{p}{2}})}^{\frac{2}{p}} + {(K_{ℓ}^{ϵ})}^{\frac{2}{p}} + C (ℓ)]}^{\frac{p}{2}}}{1 + \frac{p}{2} ℓ \int_{ω} | W |^{p - 1} ρ_{ℓ} G_{ϵ}} \end{array}$ Sending the limits $ℓ \to 0$ and $ϵ \to 0$ in (5.26), we prove our claim. In particular we have $\begin{matrix} inf_{ϵ > 0} lim_{ℓ \to 0} \frac{\int_{Ω_{ℓ}^{-}} | A \nabla v_{ℓ}^{ϵ} \cdot \nabla v_{ℓ}^{ϵ} |^{\frac{p}{2}}}{\int_{Ω_{ℓ}^{-}} | v_{ℓ}^{ϵ} |^{p}} < μ^{1} (ω) . \end{matrix}$ Hence we can find some $ℓ_{1}$ and $ϵ_{1}$ for which $\begin{matrix} (5.27) & - γ_{1} = \int_{Ω_{ℓ_{1}}^{-}} {| A \nabla v_{ℓ_{1}}^{ϵ_{1}} \cdot \nabla v_{ℓ_{1}}^{ϵ_{1}} |}^{\frac{p}{2}} - μ^{1} (ω) \int_{Ω_{ℓ_{1}}^{-}} {| v_{ℓ_{1}}^{ϵ_{1}} |}^{p} < 0 . \end{matrix}$ For any $α > 0$ we define the following function in $V (Ω_{\infty}^{+})$ : $\begin{matrix} z_{α} (x_{1}, X_{2}) = \{\begin{array}{ll} v_{ℓ_{1}}^{ϵ_{1}} (x_{1} - ℓ_{1}, X_{2}) & x_{1} \in [0, ℓ_{1}) \\ W (X_{2}) e^{- α (x_{1} - ℓ_{1})} & x_{1} \in [ℓ_{1}, \infty) . \end{array} \end{matrix}$ We have $\begin{matrix} (5.28) & \int_{Ω_{\infty}^{+}} | z_{α} |^{p} = \int_{Ω_{ℓ_{1}}^{-}} {| v_{ℓ_{1}}^{ϵ_{1}} |}^{p} + (\int_{0}^{\infty} e^{- p α x_{1}}) \int_{ω} | W |^{p} = \int_{Ω_{ℓ_{1}}^{-}} {| v_{ℓ_{1}}^{ϵ_{1}} |}^{p} + \frac{1}{p α}, \end{matrix}$ and $\begin{array}{rcl} \int_{Ω_{\infty}^{+}} | A \nabla z_{α} \cdot \nabla z_{α} |^{\frac{p}{2}} & = & \int_{Ω_{ℓ_{1}}^{-}} {| A \nabla v_{ℓ_{1}}^{ϵ_{1}} \cdot \nabla v_{ℓ_{1}}^{ϵ_{1}} |}^{\frac{p}{2}} \\ (5.29) & + \frac{1}{p α} \int_{ω} {| α^{2} a_{11} W^{2} - 2 α (A_{12} \cdot \nabla W) W + A_{22} \nabla W \cdot \nabla W |}^{\frac{p}{2}} . \end{array}$ Using (1.9) and then (5.28) and (5.29), we obtain $\begin{aligned} ν_{\infty}^{+} - μ^{1} (ω) & ⩽ \frac{\int_{Ω_{ℓ_{1}}^{-}} | A \nabla v_{ℓ_{1}}^{ϵ_{1}} \cdot \nabla v_{ℓ_{1}}^{ϵ_{1}} |^{\frac{p}{2}} + \frac{1}{p α} \int_{ω} | α^{2} a_{11} W^{2} - 2 α (A_{12} \cdot \nabla W) W + A_{22} \nabla W \cdot \nabla W |^{\frac{p}{2}}}{\int_{Ω_{ℓ_{1}}^{-}} | v_{ℓ_{1}}^{ϵ_{1}} |^{p} + \frac{1}{p α}} \\ (5.30) & - μ^{1} (ω) . \end{aligned}$ Substituting the value $γ_{1}$ defined in (5.27), we get $\begin{matrix} (5.31) & ν_{\infty}^{+} - μ^{1} (ω) ⩽ \frac{- γ_{1} + \frac{1}{p α} \int_{ω} | α^{2} a_{11} W^{2} - 2 α (A_{12} \cdot \nabla W) W + A_{22} \nabla W \cdot \nabla W |^{\frac{p}{2}} - \frac{μ^{1} (ω)}{p α}}{\int_{Ω_{ℓ_{1}}^{-}} | v_{ℓ_{1}}^{ϵ_{1}} |^{p} + \frac{1}{p α}} . \end{matrix}$ Now we approximate the integral $\begin{matrix} I_{α} = \int_{ω} {| α^{2} a_{11} W^{2} - 2 α (A_{12} \cdot \nabla W) W + A_{22} \nabla W \cdot \nabla W |}^{\frac{p}{2}} . \end{matrix}$ Using the inequality ${(a + b)}^{q} ⩽ a^{q} + q a^{q - 1} b + C a^{q - 2} b^{2}$ where $| b | < a$ , $q ⩾ 1$ and C is some positive constant, for small α, we have $\begin{array}{rcl} I_{α} & ⩽ & \int_{ω} | A_{22} \nabla W \cdot \nabla W |^{\frac{p}{2}} + \frac{p}{2} \int_{ω} | A_{22} \nabla W \cdot \nabla W |^{\frac{p - 2}{2}} (α^{2} a_{11} W^{2} - 2 α (A_{12} \cdot \nabla W) W) \\ + C \int_{ω} | A_{22} \nabla W \cdot \nabla W |^{\frac{p - 4}{2}} {(α^{2} a_{11} W^{2} - 2 α (A_{12} \cdot \nabla W) W)}^{2} \\ (5.32) & = & μ^{1} (ω) + α^{2} \frac{p}{2} a_{11} I_{1} - α p I_{2} + α^{2} C I_{3}, \end{array}$ where $\begin{array}{c} I_{1} = \int_{ω} | A_{22} \nabla W \cdot \nabla W |^{\frac{p - 2}{2}} W^{2}, \\ I_{2} = \int_{ω} | A_{22} \nabla W \cdot \nabla W |^{\frac{p - 2}{2}} (A_{12} \cdot \nabla W) W \end{array}$ and $\begin{matrix} I_{3} = \int_{ω} | A_{22} \nabla W \cdot \nabla W |^{\frac{p - 4}{2}} {(α a_{11} W^{2} - 2 (A_{12} \cdot \nabla W) W)}^{2} . \end{matrix}$ Using (5.32) in (5.31), we get $\begin{matrix} (5.33) & ν_{\infty}^{+} - μ^{1} (ω) ⩽ \frac{- γ_{1} + \frac{α}{2} a_{11} I_{1} - I_{2} + \frac{α}{p} C I_{3}}{\int_{Ω_{ℓ_{1}}^{-}} | v_{ℓ_{1}}^{ϵ_{1}} |^{p} + \frac{1}{p α}} . \end{matrix}$ Since $γ_{1} > 0$ , and $I_{2} ⩾ 0$ by (1.11), choose α small enough so that RHS becomes negative and this completes the proof.

(ii) It is known that the space $V_{s} (Ω_{\infty}^{+})$ defined in (4.8) is dense in $V (Ω_{\infty}^{+})$ . The following subspace $\begin{matrix} V_{s}^{0} (Ω_{\infty}^{+}) = {u \in V_{s} (Ω_{\infty}^{+}) : \exists δ = δ (u) > 0 s.t u (x) = 0 for x with dist (x, γ_{\infty}^{+}) ⩽ δ} \end{matrix}$ is also dense in $V (Ω_{\infty}^{+})$ . By regularity of the eigenfunction, we know that W is continuous and positive in ω (see [26]). Picone’s identity says that for $u ⩾ 0$ and $v > 0$ , we have $\begin{matrix} (5.34) & R (u, v) = | A \nabla u \cdot \nabla u |^{\frac{p}{2}} - | A \nabla v \cdot \nabla v |^{\frac{p - 2}{2}} A \nabla v \cdot \nabla (\frac{u^{p}}{v^{p - 1}}) ⩾ 0 . \end{matrix}$ Let $u \in V_{s}^{0} (Ω_{\infty}^{+})$ be such that $u ⩾ 0$ and let $v = W$ . Integrating (5.34) over $Ω_{\infty}^{+}$ and applying integration by parts, we get $\begin{array}{rcl} 0 & ⩽ & \int_{Ω_{\infty}^{+}} | A \nabla u \cdot \nabla u |^{\frac{p}{2}} - \int_{Ω_{\infty}^{+}} | A \nabla W \cdot \nabla W |^{\frac{p - 2}{2}} A \nabla W \cdot \nabla (\frac{u^{p}}{W^{p - 1}}) \\ = & \int_{Ω_{\infty}^{+}} | A \nabla u \cdot \nabla u |^{\frac{p}{2}} + \int_{Ω_{\infty}^{+}} div (| A \nabla W \cdot \nabla W |^{\frac{p - 2}{2}} A \nabla W) \frac{u^{p}}{W^{p - 1}} \\ - \int_{{0} \times ω} (| A \nabla W \cdot \nabla W |^{\frac{p - 2}{2}} A \nabla W \cdot ν) \frac{u^{p}}{W^{p - 1}} \\ (5.35) & = & \int_{Ω_{\infty}^{+}} | A \nabla u \cdot \nabla u |^{\frac{p}{2}} - μ^{1} (ω) \int_{Ω_{\infty}^{+}} u^{p} + \int_{ω} | A_{22} \nabla W \cdot \nabla W |^{\frac{p - 2}{2}} (A_{12} \cdot \nabla W) \frac{u^{p} (0, X_{2})}{W^{p - 1} (0, X_{2})} . \end{array}$ Here we have used $- div (| A \nabla W \cdot \nabla W |^{\frac{p - 2}{2}} A \nabla W) = μ^{1} (ω) W^{p - 1}$ , which is true in view of (1.2). The last integral is less than or equal to 0 by the assumption (1.12), we have $\begin{matrix} (5.36) & 0 ⩽ \int_{Ω_{\infty}^{+}} | A \nabla u \cdot \nabla u |^{\frac{p}{2}} - μ^{1} (ω) \int_{Ω_{\infty}^{+}} u^{p} . \end{matrix}$ Now let $u \in V_{s}^{0} (Ω_{\infty}^{+})$ and $u = u^{+} - u^{-}$ , where $u^{+} (x) = max {u (x), 0}$ and $u^{-} (x) = min {0, u (x)}$ . Clearly $0 ⩽ u^{\pm} \in V_{s}^{0} (Ω_{\infty}^{+})$ and $| u | = u^{+} + u^{-}$ . Since $u^{+}$ and $u^{-}$ have disjoint supports, it is easy to check that $\begin{matrix} (5.37) & \int_{Ω_{\infty}^{+}} | A \nabla u \cdot \nabla u |^{\frac{p}{2}} = \int_{Ω_{\infty}^{+}} {| A \nabla u^{+} \cdot \nabla u^{+} |}^{\frac{p}{2}} + \int_{Ω_{\infty}^{+}} {| A \nabla u^{-} \cdot \nabla u^{-} |}^{\frac{p}{2}} \end{matrix}$ and $\begin{matrix} (5.38) & \int_{Ω_{\infty}^{+}} | u |^{p} = \int_{Ω_{\infty}^{+}} {(u^{+})}^{p} + \int_{Ω_{\infty}^{+}} {(u^{-})}^{p} . \end{matrix}$ Hence from (5.36), (5.37) and (5.38) we obtain, for every $u \in V_{s}^{0} (Ω_{\infty}^{+})$ $\begin{matrix} (5.39) & \int_{Ω_{\infty}^{+}} | A \nabla u \cdot \nabla u |^{\frac{p}{2}} ⩾ μ^{1} (ω) \int_{Ω_{\infty}^{+}} u^{p}, i.e. \frac{\int_{Ω_{\infty}^{+}} | A \nabla u \cdot \nabla u |^{\frac{p}{2}}}{\int_{Ω_{\infty}^{+}} u^{p}} ⩾ μ^{1} (ω) . \end{matrix}$ By the density of the $V_{s}^{0} (Ω_{\infty}^{+})$ in $V (Ω_{\infty}^{+})$ , inequality (5.39) holds for every $u \in V (Ω_{\infty}^{+})$ and (1.9) gives $ν_{\infty}^{+} ⩾ μ^{1} (ω)$ . The reverse inequality follows from Lemma 4.1 and we conclude $ν_{\infty}^{+} = μ^{1} (ω)$ .

Now, if u be a minimizer realizing $ν_{\infty}^{+}$ , from (5.35) we must have $\begin{matrix} \int_{Ω_{\infty}^{+}} | A \nabla u \cdot \nabla u |^{\frac{p}{2}} = \int_{Ω_{\infty}^{+}} | A \nabla W \cdot \nabla W |^{\frac{p - 2}{2}} A \nabla W \cdot \nabla (\frac{u^{p}}{W^{p - 1}}) . \end{matrix}$ This is true only when $u = c W$ for some constant c. But this is a contradiction since $W \notin V (Ω_{\infty}^{+})$ . This completes the proof. □
Remark 3.
It is clear from Remark 1 that Theorem 1.3 (i) and (ii) follows for $ν_{\infty}^{-}$ if we replace the inequalities (1.11) and (1.12) by $\int_{ω} | A_{22} \nabla W \cdot \nabla W |^{\frac{p - 2}{2}} (A_{12} \cdot \nabla W) W ⩽ 0$ and $A_{12} \cdot \nabla W ⩾ 0$ respectively. In particular, if $A_{12} \cdot \nabla W ≢ 0$ and $\int_{ω} | A_{22} \nabla W \cdot \nabla W |^{\frac{p - 2}{2}} (A_{12} \cdot \nabla W) W = 0$ , then $ν_{\infty}^{+} < μ^{1} (ω)$ and $ν_{\infty}^{-} < μ^{1} (ω)$ . Hence, if A satisfies the symmetry (S) defined in Definition 1 and $A_{12} \cdot \nabla W ≢ 0$ , then it follows that $ν_{\infty}^{\pm} < μ^{1} (ω)$ . Another consequence of Theorem 1.3 is that if $A_{12} \cdot \nabla W ≢ 0$ , at least one of the $ν_{\infty}^{+}$ and $ν_{\infty}^{-}$ is strictly less than $μ^{1} (ω)$ , in particular $min {ν_{\infty}^{+}, ν_{\infty}^{-}} < μ^{1} (ω)$ , which already follows from Theorem 2 (ii) and Theorem 1.2.

In the next theorem we describe the behaviour of the eigenfunctions ${u_{ℓ}}$ of mixed boundary value problem (1.4) at the ends of the cylinder. Here we have shown two possible cases may happen: concentration near one of the ends of the cylinder, or near both ends. Let ${\tilde{u}}^{\pm}$ be the unique positive normalized minimizer in (1.9), whenever it exists. For each $ℓ > 0$ we define the functions $\begin{array}{c} {\tilde{v}}_{ℓ}^{+} (x_{1}, X_{2}) = u_{ℓ} (x_{1} - ℓ, X_{2}) on Ω_{ℓ}^{+}, \\ {\tilde{v}}_{ℓ}^{-} (x_{1}, X_{2}) = u_{ℓ} (x_{1} + ℓ, X_{2}) on Ω_{ℓ}^{-} . \end{array}$ Theorem 5.2.
(i) If $ν_{\infty}^{+} < ν_{\infty}^{-}$ , then for every $r > 0$ , $\begin{matrix} {\tilde{v}}_{ℓ}^{+} \to {\tilde{u}}^{+} in W^{1, p} (Ω_{r}^{+}) and {\tilde{v}}_{ℓ}^{-} \to 0 in W^{1, p} (Ω_{r}^{-}) . \end{matrix}$ (ii) If A satisfies the symmetry (S) defined in Definition 1 and ( 5.2 ) holds, then we have

${\tilde{v}}_{ℓ}^{+} (x_{1}, X_{2}) = {\tilde{v}}_{ℓ}^{-} (- x_{1}, - X_{2})$ and for every $r > 0$ , $\begin{matrix} {\tilde{v}}_{ℓ}^{+} \to {\tilde{u}}^{+} in W^{1, p} (Ω_{r}^{+}) and {\tilde{v}}_{ℓ}^{-} \to {\tilde{u}}^{-} in W^{1, p} (Ω_{r}^{-}) . \end{matrix}$
Proof.
(i) The convergence of ${{\tilde{v}}_{ℓ}^{-}}$ to 0 in $W^{1, p} (Ω_{r}^{-})$ for any $r > 0$ follows from Proposition 4.1. It remains to prove the convergence for ${{\tilde{v}}_{ℓ}^{+}}$ . Let ${{\tilde{v}}_{ℓ_{k}}^{+}} \subset {{\tilde{v}}_{ℓ}^{+}}$ be any sequence. Since ${{\tilde{v}}_{ℓ_{k}}^{+}}$ is bounded in $W^{1, p} (Ω_{r}^{+})$ for any $r > 0$ , using a diagonal argument in ${{\tilde{v}}_{ℓ_{k}}^{+}}$ , we get a subsequence, we also denote it by ${{\tilde{v}}_{ℓ_{k}}^{+}}$ , which converges weakly in $W^{1, p} (Ω_{r}^{+})$ and strongly in $L^{p} (Ω_{r}^{+})$ to a function $v^{+} \in W^{1, p} (Ω_{\infty}^{+})$ . By (1.8) we have $\begin{matrix} (5.40) & \int_{Ω_{r}^{+}} {| {\tilde{v}}_{ℓ}^{+} |}^{p} = \int_{Ω_{ℓ}^{-} ∖ Ω_{ℓ - r}} | u_{ℓ} |^{p} = 1 - \int_{Ω_{ℓ - r}^{-}} | u_{ℓ} |^{p} - \int_{Ω_{ℓ} +} | u_{ℓ} |^{p} ⩾ 1 - C α^{[r]} - \int_{Ω_{ℓ} +} | u_{ℓ} |^{p} . \end{matrix}$ Clearly the integral on the right hand side goes to 0 as $ℓ \to \infty$ by Proposition 4.1. Setting $ℓ = ℓ_{k}$ in (5.40) and taking the limit $ℓ_{k} \to \infty$ , we get $\begin{matrix} (5.41) & \int_{Ω_{r}^{+}} {| v^{+} |}^{p} ⩾ 1 - C α^{[r]} \end{matrix}$ We conclude that $\int_{Ω_{\infty}^{+}} | v^{+} |^{p} = 1$ by taking r to infinity. Now by (1.10) and by Fatou’s lemma, we have $\begin{array}{rcl} ν_{\infty}^{+} & = & lim_{ℓ \to \infty} λ_{M}^{1} (Ω_{ℓ}) = lim_{ℓ \to \infty} \int_{Ω_{ℓ}} | A \nabla u_{ℓ} \cdot \nabla u_{ℓ} |^{\frac{p}{2}} \\ (5.42) & ⩾ & \underset{k \to \infty}{lim sup} \int_{Ω_{r}^{+}} {| A \nabla {\tilde{v}}_{ℓ_{k}}^{+} \cdot \nabla {\tilde{v}}_{ℓ_{k}}^{+} |}^{\frac{p}{2}} ⩾ \int_{Ω_{r}^{+}} {| A \nabla v^{+} \cdot \nabla v^{+} |}^{\frac{p}{2}} . \end{array}$ Hence from (5.41) and (5.42) we deduce that $\int_{Ω_{\infty}^{+}} | A \nabla v^{+} \cdot \nabla v^{+} |^{\frac{p}{2}} ⩽ ν_{\infty}^{+}$ . The reverse inequality follows from (1.9) and we conclude that $v^{+}$ is a normalised minimizer realizing $ν_{\infty}^{+}$ . Since the minimizer is unique upto multiply by a constant, it necessarily coincides with ${\tilde{u}}^{+}$ . Now we define the following function on $(0, \infty)$ , $\begin{matrix} f (r) = \underset{k \to \infty}{lim sup} \int_{Ω_{r}^{+}} {| A \nabla {\tilde{v}}_{ℓ_{k}}^{+} \cdot \nabla {\tilde{v}}_{ℓ_{k}}^{+} |}^{\frac{p}{2}} - \int_{Ω_{r}^{+}} {| A \nabla v^{+} \cdot \nabla v^{+} |}^{\frac{p}{2}} . \end{matrix}$ From (5.42) it follows that f is non-negative and non-decreasing. On the other hand, $f (r) \to 0$ as $r \to \infty$ . Hence f must be identically zero on $(0, \infty)$ . This implies ${\tilde{v}}_{ℓ_{k}}^{+}$ converges to ${\tilde{u}}^{+}$ strongly in $W^{1, p} (Ω_{r}^{+})$ for any $r > 0$ . The convergence for the whole family ${{\tilde{v}}_{ℓ}^{+}}$ follows from the uniqueness of the all possible limits.

(ii) If A satisfies the symmetry (S), it is easy to check that $u_{ℓ} (x_{1}, X_{2}) = u_{ℓ} (- x_{1}, - X_{2})$ . Using this, a similar computation gives the required result for this case. □

6. Convergence of eigenvalues using Krasnoselskii genus

In this section we prove the convergence results for higher eigenvalues (Theorem 1.4 and Theorem 1.5) using the Krasnoselskii genus. The following lemma shows that for $p = 2$ , the k-th Krasnoselskii eigenvalue $β_{k} (Ω_{ℓ})$ of problem (1.4) is same as the k-th eigenvalue $λ_{M}^{k} (Ω_{ℓ})$ . This fact may be known, but for the sake of completeness we include the proof here.

Lemma 6.1.

Let for any $k \in N$ , $β_{k} (Ω_{ℓ})$ be defined as in Theorem 2.2 with the closed and symmetric $C^{1, 1}$ -manifold $\begin{matrix} M_{ℓ} = {u \in V (Ω_{ℓ}) : \int_{Ω_{ℓ}} u^{2} = 1} \end{matrix}$ of the Banach space $V (Ω_{ℓ})$ and with the even functional $\begin{matrix} E_{ℓ} (u) = \int_{Ω_{ℓ}} A \nabla u \cdot \nabla u \end{matrix}$ on $M_{ℓ}$ . Then $β_{k} (Ω_{ℓ}) = λ_{M}^{k} (Ω_{ℓ})$ , where $λ_{M}^{k} (Ω_{ℓ})$ is the k-th eigenvalue of the problem ( 1.4 ) for $p = 2$ .

Proof.

Let ${u_{ℓ}^{1}, u_{ℓ}^{2}, \dots u_{ℓ}^{k}}$ be the first k normalised eigenfunctions of the problem (1.4). It is well known that $u_{ℓ}^{i}$ ’s are mutually orthogonal and $\begin{matrix} (6.1) & λ_{M}^{k} (Ω_{ℓ}) = max_{\begin{array}{c} u \in sp {u_{ℓ}^{1}, \dots u_{ℓ}^{k}} \\ ‖ u ‖_{L^{2}} = 1 \end{array}} E_{ℓ} (u) = min_{\begin{array}{c} V \subset V (Ω_{ℓ}) \\ dim V = k \end{array}} max_{\begin{array}{c} u \in V \\ ‖ u ‖_{L^{2}} = 1 \end{array}} E_{ℓ} (u) . \end{matrix}$ Consider the subspace $V_{0} = sp {u_{ℓ}^{1}, u_{ℓ}^{2}, \dots, u_{ℓ}^{k}}$ of $V (Ω_{ℓ})$ . Define a closed symmetric subset $A_{0}$ of $M_{ℓ}$ as $\begin{matrix} A_{0} = V_{0} \cap M_{ℓ} = {\sum_{i = 1}^{k} α_{i} u_{ℓ}^{i} : \int_{Ω_{ℓ}} {(\sum_{i = 1}^{k} α_{i} u_{ℓ}^{i})}^{2} = 1} = {\sum_{i = 1}^{k} α_{i} u_{ℓ}^{i} : \sum_{i = 1}^{k} α_{i}^{2} = 1} . \end{matrix}$ Clearly, $V_{0} ≅ R^{k}$ and $A_{0}$ is homeomorphic to the unit sphere in $R^{k}$ , and therefore Proposition 2.1 gives $γ (A_{0}) = k$ . Now, definition of $β_{k} (Ω_{ℓ})$ gives $\begin{matrix} β_{k} (Ω_{ℓ}) ⩽ sup_{u \in A_{0}} E_{ℓ} (u) = sup_{\begin{array}{c} u \in sp {u_{ℓ}^{1}, \dots u_{ℓ}^{k}} \\ ‖ u ‖_{L^{2}} = 1 \end{array}} E_{ℓ} (u) = λ_{M}^{k} (Ω_{ℓ}) . \end{matrix}$ For the reverse inequality, first we fix $ϵ > 0$ . ∃ $A_{ϵ} \in F_{k}$ such that ${sup}_{u \in A_{ϵ}} E_{ℓ} (u) < β_{k} (Ω_{ℓ}) + ϵ$ . Clearly by (2.2) and (2.3), ${(u, v)}_{A} = \int_{Ω_{ℓ}} A \nabla u \cdot \nabla v$ is an equivalent inner product on $V (Ω_{ℓ})$ . By Proposition 2.1, let ${w_{1}, w_{2}, \dots w_{k}}$ be a set of orthogonal (w.r.t ${(\cdot, \cdot)}_{A}$ ) vectors in $A_{ϵ}$ .

Let $V_{ϵ} = sp {w_{1}, w_{2}, \dots w_{k}}$ . For any $u = α_{1} w_{1} + \dots + α_{k} w_{k} \in V_{ϵ}$ with $| | u | |_{L^{2} (Ω_{ℓ})} = 1$ , we have $\begin{array}{rcl} E_{ℓ} (u) & = & \int_{Ω_{ℓ}} A \nabla u \cdot \nabla u = \int_{Ω_{ℓ}} A \nabla (\sum_{i = 1}^{k} α_{i} w_{i}) \cdot \nabla (\sum_{i = 1}^{k} α_{i} w_{i}) \\ (6.2) & = & \sum_{i = 1}^{k} α_{i}^{2} E_{ℓ} (w_{i}) ⩽ max_{1 ⩽ i ⩽ k} E_{ℓ} (w_{i}) . \end{array}$ Hence from (6.2) $\begin{matrix} (6.3) & λ_{M}^{k} (Ω_{ℓ}) ⩽ max_{\begin{array}{c} u \in V_{ϵ} \\ ‖ u ‖_{L^{2}} = 1 \end{array}} E_{ℓ} (u) ⩽ max_{1 ⩽ i ⩽ k} E_{ℓ} (w_{i}) ⩽ sup_{u \in A_{ϵ}} E_{ℓ} (u) < β_{k} (Ω_{ℓ}) + ϵ . \end{matrix}$ Since ϵ is arbitrary, this completes the proof. □

Corollary 2.

For $p > 2$ , we have $β_{1} (Ω_{ℓ}) = λ_{M}^{1} (Ω_{ℓ})$ .

Proof of Theorem 1.4.

First we consider the even case, i.e. the $2 k$ -th eigenvalue of the problem (1.4). Since A satisfies the symmetry (S), ${\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{+}) = {\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{-})$ , holds $\forall ℓ > 0$ and Remark 1, Lemma 4.2 and Theorem 1.2 together give $\begin{matrix} (6.4) & lim_{ℓ \to \infty} λ_{M}^{1} (Ω_{ℓ}) = lim_{ℓ \to \infty} {\tilde{λ}}_{M}^{1} (Ω_{ℓ}^{+}) . \end{matrix}$ Again we have $λ_{M}^{1} (Ω_{ℓ}) ⩽ {\tilde{λ}}_{M}^{1} (Ω_{ℓ_{1}}^{+})$ , $\forall ℓ_{1} ⩽ ℓ$ by considering the test function $u \in V (Ω_{ℓ})$ in (2.9) as $u (x_{1}, X_{2}) = {\tilde{u}}_{ℓ_{1}}^{+} (x_{1} + ℓ, X_{2})$ on $Ω_{ℓ}^{-} ∖ Ω_{ℓ - ℓ_{1}}^{-}$ and 0 on $Ω_{ℓ - ℓ_{1}}^{-} \cup Ω_{ℓ}^{+}$ , where ${\tilde{u}}_{ℓ_{1}}^{+}$ is the eigenfunction corresponding to ${\tilde{λ}}_{M}^{1} (Ω_{ℓ_{1}}^{+})$ (see Remark 2). Let $ϵ > 0$ . Then ∃ $ℓ_{0} > 0$ such that for $1 ⩽ i ⩽ k$ $\begin{matrix} (6.5) & {\tilde{λ}}_{M}^{1} (Ω_{\frac{i}{k} ℓ}^{+}) < λ_{M}^{1} (Ω_{ℓ}) + ϵ, \forall \frac{ℓ}{k} > ℓ_{0} . \end{matrix}$ For $1 ⩽ i ⩽ k$ and $ℓ ⩾ ℓ_{0}$ , we define the following functions on $Ω_{ℓ}$ : $\begin{array}{c} (6.6) & h_{i}^{-} (x_{1}, X_{2}) = \{\begin{array}{ll} {\tilde{u}}_{\frac{i}{k} ℓ}^{+} (x_{1} + ℓ, X_{2}) & on Ω_{ℓ}^{-} ∖ Ω_{(1 - \frac{i}{k}) ℓ}^{-} \\ 0 & otherwise, \end{array} \\ (6.7) & h_{i}^{+} (x_{1}, X_{2}) = \{\begin{array}{ll} {\tilde{u}}_{\frac{i}{k} ℓ}^{-} (x_{1} - ℓ, X_{2}) & on Ω_{ℓ}^{+} ∖ Ω_{(1 - \frac{i}{k}) ℓ}^{+} \\ 0 & otherwise, \end{array} \end{array}$ where ${\tilde{u}}_{\frac{i}{k} ℓ}^{\pm}$ are defined in Remark 2 for $p = 2$ . Let $u_{2 i - 1} = h_{i}^{-}$ and $u_{2 i} = h_{i}^{+}$ , where $1 ⩽ i ⩽ k$ and we define the subspace $V_{0} = sp {u_{1}, u_{2}, \dots, u_{2 k}}$ of $V (Ω_{ℓ})$ . Consider the closed symmetric subset $A_{0} = V_{0} \cap M_{ℓ}$ , where $M_{ℓ}$ is the $C^{1, 1}$ -manifold defined in Lemma 6.1. Clearly $γ (A_{0}) = 2 k$ . Now for any $u = \sum_{j = 1}^{2 k} α_{j} u_{j} \in A_{0}$ , $\begin{matrix} (6.8) & \int_{Ω_{ℓ}} A \nabla u \cdot \nabla u = \sum_{j = 1}^{2 k} α_{j}^{2} \int_{Ω_{ℓ}} A \nabla u_{j} \cdot \nabla u_{j} + 2 \sum_{1 ⩽ i < j ⩽ 2 k} α_{i} α_{j} \int_{Ω_{ℓ}} A \nabla u_{i} \cdot \nabla u_{j} . \end{matrix}$ For $1 ⩽ j ⩽ 2 k$ , we have $\begin{matrix} (6.9) & \int_{Ω_{ℓ}} A \nabla u_{j} \cdot \nabla u_{j} = \{\begin{array}{ll} {\tilde{λ}}_{M}^{1} (Ω_{\frac{j + 1}{2 k} ℓ}^{+}) \int_{Ω_{ℓ}} u_{j}^{2} & if j is odd, \\ {\tilde{λ}}_{M}^{1} (Ω_{\frac{j}{2 k} ℓ}^{+}) \int_{Ω_{ℓ}} u_{j}^{2} & if j is even, \end{array} \end{matrix}$ and for $1 ⩽ i < j ⩽ 2 k$ , $\begin{matrix} (6.10) & \int_{Ω_{ℓ}} A \nabla u_{i} \cdot \nabla u_{j} = \{\begin{array}{ll} {\tilde{λ}}_{M}^{1} (Ω_{\frac{j + 1}{2 k} ℓ}^{+}) \int_{Ω_{ℓ}} u_{i} u_{j} & if both i and j are odd, \\ {\tilde{λ}}_{M}^{1} (Ω_{\frac{j}{2 k} ℓ}^{+}) \int_{Ω_{ℓ}} u_{i} u_{j} & if both i and j are even, \\ 0 & otherwise. \end{array} \end{matrix}$ Applying (6.9), (6.10) and (6.5) in (6.8) we get $\begin{array}{rcl} \int_{Ω_{ℓ}} A \nabla u \cdot \nabla u & ⩽ & (λ_{M}^{1} (Ω_{ℓ}) + ϵ) [\sum_{j = 1}^{2 k} α_{j}^{2} \int_{Ω_{ℓ}} u_{j}^{2} + 2 \sum_{1 ⩽ i < j ⩽ 2 k} α_{i} α_{j} \int_{Ω_{ℓ}} u_{i} u_{j}] \\ (6.11) & = & (λ_{M}^{1} (Ω_{ℓ}) + ϵ) \int_{Ω_{ℓ}} u^{2} = λ_{M}^{1} (Ω_{ℓ}) + ϵ . \end{array}$ Finally, from the definition of $β_{2 k} (Ω_{ℓ})$ we obtain $\begin{matrix} λ_{M}^{1} (Ω_{ℓ}) ⩽ λ_{M}^{2 k} (Ω_{ℓ}) = β_{2 k} (Ω_{ℓ}) ⩽ sup_{u \in A_{0}} \int_{Ω_{ℓ}} A \nabla u \cdot \nabla u ⩽ λ_{M}^{1} (Ω_{ℓ}) + ϵ . \end{matrix}$ Since ϵ is arbitrary, (1.13) is true for k is even. Convergence of odd eigenvalues follows from the fact $\begin{matrix} λ_{M}^{2 k - 2} (Ω_{ℓ}) ⩽ λ_{M}^{2 k - 1} (Ω_{ℓ}) ⩽ λ_{M}^{2 k} (Ω_{ℓ}) . \end{matrix}$ □

Proof of Theorem 1.5.

Let $h_{1}^{\pm}$ is defined as in (6.6) and (6.7) with $k = 1$ . Define the closed and symmetric subset $A_{0} = V_{0} \cap M_{ℓ}$ of genus 2, where $V_{0} = sp {h_{1}^{-}, h_{1}^{+}}$ . Since $h_{1}^{-}$ and $h_{1}^{+}$ have disjoint support, for any $u = α h_{1}^{-} + β h_{1}^{+} \in A_{0}$ , we have $\begin{matrix} \int_{Ω_{ℓ}} | A \nabla u \cdot \nabla u |^{\frac{p}{2}} = α^{p} \int_{Ω_{ℓ}} {| A \nabla h_{1}^{-} \cdot \nabla h_{1}^{-} |}^{\frac{p}{2}} + β^{p} \int_{Ω_{ℓ}} {| A \nabla h_{1}^{+} \cdot \nabla h_{1}^{+} |}^{\frac{p}{2}} . \end{matrix}$ An analogous calculation as in the proof of Theorem 1.4 gives the rest of the proof. □

Footnotes

Acknowledgements

The third author would like to thank the facilities provided by IMSc. Part of this work was carried out when the third author was visiting IMSc. Research of third author is funded by Core Research grant under project number CRG/2022/007867. The third author would like to thank Prof. Itai Shafrir and Prof. Michel Chipot for several interesting discussion on the subject. In fact, the key ideas of the proof of Theorem 1.2 and Theorem are due to Prof. Shafrir.

Note: This work is a part of doctoral thesis work of the second author. He acknowledges the support provided by IIT Kanpur, India and MHRD, Government of India (GATE fellowship).

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