Asymptotically vanishing nodal solutions for critical double phase problems

Abstract

We consider a Dirichlet double phase problem with unbalanced growth. In the reaction we have the combined effects of a critical term and of a locally defined Carathéodory perturbation. Using cut-off functions and truncation techniques we bypass the critical term and deal with a coercive problem. Using this auxillary problem, we show that the original Dirichlet equation has a whole sequence of nodal (sign-changing) solutions which converge to zero in the Musielak–Orlice–Sobolev space and in $L^{\infty}$ .

Keywords

Double phase integrand unbalanced growth Musielak–Orlice spaces critical term nodal solutions cut-off function

1. Introduction

Let $Ω \subseteq R^{N}$ be a bounded domain with a Lipschitz boundary $\partial Ω$ . In this paper we study the following critical Dirichlet double phase problem: $\begin{array}{l} (1) & \{\begin{matrix} - Δ_{p}^{a} u (z) - Δ_{q} u (z) = f (z, u (z)) + | u (z) |^{p^{*} - 2} u (z), & in Ω, \\ u |_{\partial Ω} = 0, & 1 < q < p < N . \end{matrix} \end{array}$

Given $a \in C^{0, 1} (\overline{Ω})$ with $a (z) > 0$ for all $z \in Ω$ and for $r \in (1, + \infty)$ by $Δ_{r}^{a}$ we denote the weighted r-Laplace differential operator defined by $\begin{matrix} Δ_{r}^{a} u = div (a (z) | D u |^{r - 2} D u) . \end{matrix}$ When $a \equiv 1$ , then we have the usual r-Laplace differential operator defined by $\begin{matrix} Δ_{r} u = div (| D u |^{r - 2} D u) . \end{matrix}$

In problem (1) we have the sum of two such operators with different exponents. For this differential operator the integrand in the energy functional is $\begin{matrix} ξ_{0} (z, x) = \frac{1}{p} a (z) x^{p} + \frac{1}{q} x^{q} for all z \in Ω, all x ⩾ 0 . \end{matrix}$

Since we do not assume that the weight function $a (\cdot)$ is bounded away from zero (that is, ${min}_{\overline{Ω}} a > 0$ ), the integrand $ξ_{0} (\cdot, \cdot)$ exhibits unbalanced growth, namely we have $\begin{matrix} \frac{1}{q} x^{q} ⩽ ξ_{0} (z, x) ⩽ c_{0} [1 + x^{p}] for all x \in Ω, all x ⩾ 0, some c_{0} > 0 . \end{matrix}$

Such integral functionals were first considered by Marcellini [16] and Zhikov [30,31] in connection with problems of nonlinear elasticity. Recntly, the interest for such problems was revived by Mingione and coworkers who produced local regularity results for local minimizers of such functionals (see Baroni–Colombo–Mingione [1], Colombo–Mingione [3,4]). However, a global regularity theory for double phase problems with unbalanced growth is still missing and this makes the study of such problems difficult.

In the reaction of (1) we have the effect of two distinct terms. One is the critical term $u \to | u |^{p^{*} - 2} u$ with $p^{*} = \frac{N p}{N - p}$ the critical exponent corresponding to $p > 1$ and the other is a Carathéodory perturbation $f (z, x)$ , with $f (z, \cdot)$ defined only locally, that is, $f : Ω \times [- η_{0}, η_{0}] \to R$ $(η_{0} > 0)$ is measurable in $z \in Ω$ and continuous in $x \in [- η_{0}, η_{0}]$ . No conditions on $f (z, \cdot)$ are imposed for $| x |$ big. The presence of the critical term leads to an energy functional which fails to satisfy the compactness condition needed to apply the results of critical point theory. In this paper using a symmetry condition on $f (z, \cdot)$ and cut-off techniques, we are able to bypass the critical term and deal with a coercive functional on which we can use variational tools. The main tool that we use is a generalized version of the symmetric mountain pass theorem due to Kajikiya [12]. Using that result, we are able to generate a whole sequence of nodal solutions ${u_{n}}_{n \in N}$ converging to zero in $L^{\infty} (Ω)$ and in the Musielak–Orlicz–Sobolev space on which the problem is defined.

Problems with a reaction which is only locally defined, were first considered by Wang [29], who studied equations driven by the Dirichlet p-Laplacian and a reaction of the form $u \to f (z, u) + λ | u |^{q - 2} u$ with $f \in C (Ω \times [- η_{0}, η_{0}])$ $λ > 0$ a parameter and $1 < q < p$ . Using different cut-off techniques, Wang [29] proved that for all $λ > 0$ small, the problem has a sequence of solutions ${u_{n}}_{n \in N} \subseteq W_{0}^{1, p} (Ω)$ converging to zero in $L^{\infty} (Ω)$ . However, he does not prove that these solutions are nodal. Later Li–Wang [14] considered a similar situation for semilinear Schrödinger equations and produced nodal solutions converging to zero in $L^{\infty} (Ω)$ . Recently there have been analogous results for more general parametric equations. We refer to the works of Gasiński–Papageorgiou [6,7], Leonardi–Papageorgiou [13], Papageorgiou–Rădulescu–Repovš [20,22]. None of the aforementioned works deals with double phase problems with unbalanced growth. Such problems were studied recently under different conditions on the data and using other methods, by Colasuonno–Squassina [2], Gasiński–Winkert [8], Ge–Lv–Lu [9], Liu–Dai [15], Papageorgiou–Rădulescu–Repovš [18], Papageorgiou–Scapellato [23], Papageorgiou–Vetro–Vetro [24,25], Papageorgiou–Zhang [27], and Ragusa–Tachikawa [28]. None of these works considers critical problems.

2. Mathematical background and hypotheses

The analysis of problem (1) requires the use of Musielak–Orlicz–Sobolev spaces. So, let $ξ : Ω \times R_{+} \to R_{+}$ $(R_{+} = [0, + \infty))$ be the Carathéodory function defined by $\begin{matrix} ξ (z, x) = a (z) x^{p} + x^{q} for all z \in Ω, all x ⩾ 0 . \end{matrix}$

(recall that $a \in C^{0, 1} (\overline{Ω})$ , that is, $a : \overline{Ω} \to R$ is Lipschitz continuous and $a (z) > 0$ for all $z \in Ω$ ). The function ξ is a generalized N-function (see Musielok [17, p. 82]) and satisfies the so-called ( $Δ_{2}$ )-condition (see Musielok [17, p. 52]), that is, $ξ (z, 2 x) ⩽ 2^{p} ξ (z, x)$ for all $z \in Ω$ , all $x ⩾ 0$ . Let $M (Ω)$ be the space of all functions $u : Ω \to R$ which are measurable. We identify two such functions which differ only on a Lebesgue-null set. Then we can define the Musielak–Orlicz–Lebesgue space $L^{ξ} (Ω)$ by $\begin{matrix} L^{ξ} (Ω) = {u \in M (Ω) : \int_{Ω} ξ (z, | u |) d z < \infty} . \end{matrix}$

We furnish this space with the so-called “Luxemburg norm” defined by $\begin{matrix} ‖ u ‖_{ξ} = inf {λ > 0 : \int_{Ω} ξ (z, \frac{| u |}{λ}) d z ⩽ 1} . \end{matrix}$

Using this space, we can define the corresponding Musielak–Orlicz–Sobolev space $W^{1, ξ} (Ω)$ by $\begin{matrix} W^{1, ξ} (Ω) = {u \in L^{ξ} (Ω) : | D u | \in L^{ξ} (Ω)} . \end{matrix}$

Here the gradient Du is understood in the weak sense. This space is equipped with the following norm $\begin{matrix} ‖ u ‖_{1, ξ} = ‖ u ‖_{ξ} + ‖ | D u | ‖_{ξ} . \end{matrix}$

In the sequel we set $‖ D u ‖_{ξ} = ‖ | D u | ‖_{ξ}$ .

Also,we set $\begin{matrix} W_{0}^{1, ξ} (Ω) = {\overline{C_{c}^{\infty} (Ω)}}^{‖ \cdot ‖_{1}, ξ} . \end{matrix}$

The spaces $L^{ξ} (Ω)$ , $W^{1, ξ} (Ω)$ , $W_{0}^{1, ξ} (Ω)$ are separable and uniformly convex (thus reflexive by the Milman–Pettis theorem, see Papageorgiou–Winkert [26, Theorem 3.4.28, p. 225]).

Suppose that $\frac{p}{q} < 1 + \frac{1}{N}$ , then from Proposition 2.18 of Colasuonno–Squassina [2], we know that there exists $c_{1} > 0$ such that $\begin{matrix} ‖ u ‖_{ξ} ⩽ c_{1} ‖ D u ‖_{ξ} for all u \in W_{0}^{1, ξ} (Ω) . \end{matrix}$

So the Poincaré inequality is valid for $W_{0}^{1, ξ} (Ω)$ and on account of this, on the space $W_{0}^{1, ξ} (Ω)$ we can use the following equivalent norm $\begin{matrix} ‖ u ‖ = ‖ D u ‖_{ξ} for all u \in W_{0}^{1, ξ} (Ω) . \end{matrix}$

The classical Sobolev embedding theorem exlends to these spaces as follows (recall that $1 < q < p < N$ ):

(a) $W_{0}^{1, ξ} (Ω) ↪ L^{r} (Ω)$ for all $1 ⩽ r ⩽ q^{*} = \frac{N q}{N - q}$ ; the embedding is continuous if $1 ⩽ r ⩽ q^{*}$ and compact if $1 ⩽ r < q^{*}$ ;

(b) $W_{0}^{1, ξ} (Ω) ↪ W_{0}^{1, q} (Ω)$ continuously.

In the study of these spaces (usually called Musielak–Orlicz spaces), important is the following modular function $\begin{matrix} ρ_{ξ} (u) = \int_{Ω} ξ (z, | u |) d z = \int_{Ω} [a (z) | u |^{p} + | u |^{q}] d z \end{matrix}$

This modular function is closely related to the Luxemburg norm.

Proposition 2.1.

For all $u \neq 0, ‖ u ‖_{ξ} = μ ⟺ ρ_{ξ} (\frac{u}{μ}) = 1$ .

$‖ u ‖_{ξ} < 1 (= 1, > 1) ⟺ ρ_{ξ} (u) < 1 (= 1, > 1)$ .

$‖ u ‖_{ξ} < 1 \Rightarrow ‖ u ‖_{ξ}^{p} ⩽ ρ_{ξ} (u) ⩽ ‖ u ‖_{ξ}^{q}$ .

$‖ u ‖_{ξ} > 1 \Rightarrow ‖ u ‖_{ξ}^{q} ⩽ ρ_{ξ} (u) ⩽ ‖ u ‖_{ξ}^{p}$ .

$‖ u ‖_{ξ} \to 0 ⟺ ρ_{ξ} (u) \to 0$ .

$‖ u ‖_{ξ} \to + \infty ⟺ ρ_{ξ} (u) \to + \infty$ .

In what follows by $⟨ \cdot, \cdot ⟩$ we denote the duality brackets for the pair $(W_{0}^{1, ξ} (Ω), W_{0}^{1, ξ} {(Ω)}^{})$ . Let $A_{p}^{a} : W_{0}^{1, ξ} (Ω) \to W_{0}^{1, ξ} {(Ω)}^{}$ be defined $\begin{matrix} ⟨ A_{p}^{a} (u), h ⟩ = \int_{Ω} a (z) | D u |^{p - 2} {(D u, D h)}_{R^{N}} d z for all u, h \in W_{0}^{1, ξ} (Ω) . \end{matrix}$

Also, if by ${⟨ \cdot, \cdot ⟩}_{1, q}$ we denote the duality brackets for the pair $(W_{0}^{1, q} (Ω), W^{- 1, q^{'}} (Ω) = W_{0}^{1, q} {(Ω)}^{})$ $(\frac{1}{q} + \frac{1}{q^{'}} = 1)$ , then $A_{q} : W_{0}^{1, q} (Ω) \to W^{- 1, q^{'}} (Ω)$ is defined by $\begin{matrix} {⟨ A_{q} (u), h ⟩}_{1, q} = \int_{Ω} | D u |^{q - 2} {(D u, D h)}_{R^{N}} d z for all u, h \in W_{0}^{1, q} (Ω) . \end{matrix}$

Recall that $W_{0}^{1, ξ} (Ω) ↪ W_{0}^{1, q} (Ω)$ and so we have $\begin{matrix} {⟨ A_{q} (u), h ⟩}_{1, q} = ⟨ A_{q} (u), h ⟩ for all u, h \in W_{0}^{1, ξ} (Ω) . \end{matrix}$

Both these operators are bounded (that is, map bounded sets to bounded sets), continuous, strictly monotone (thus maximal monotone too) and of type ${(S)}_{+}$ . Recall that if X is a reflexive Banach space and $A : X \to X^{}$ , then we say that $A (\cdot)$ is of type ${(S)}_{+}$ , if it has the following property (by ${⟨ \cdot, \cdot ⟩}_{X}$ we denote the duality brackets for the pair $(X, X^{})$ ): $\begin{array}{l} “ If u_{n} \overset{w}{\to} u in X and \underset{n \to \infty}{lim sup} {⟨ A (u_{n}), u_{n} - u ⟩}_{X} ⩽ 0, \\ then u_{n} \to u in X .^{″} \end{array}$

Also, if $φ \in C^{1} (X)$ , then we say that $φ (\cdot)$ satisfies the Palais–Smale condition (the PS-condition for short), if it has the following property: $\begin{array}{l} “ Every sequence {u_{n}}_{n \in N} \subseteq X such that \\ {φ (u_{n})}_{n \in N} \subseteq R is bounded \\ and φ^{'} (u_{n}) \to 0 in X^{} as n \to \infty, \\ admits a strongly convergent subsequence .^{″} \end{array}$

For every $u \in M (Ω)$ , we define $u^{\pm} = max {\pm u, 0}$ and have $\begin{array}{l} u = u^{+} - u^{-}, | u | = u^{+} + u^{-} . \\ If u \in W_{0}^{1, ξ} (Ω), then u^{\pm}, | u | \in W_{0}^{1, ξ} (Ω) . \end{array}$

A set $S \subseteq W_{0}^{1, ξ} (Ω)$ is said to be “downward directed” (resp.“upward directed”), if the following is true:

“If $u_{1}, u_{2} \in S$ , then we can find $u \in S$ such that $u ⩽ u_{1}, u ⩽ u_{2}$ (resp. if $v_{1}, v_{2} \in S$ , then we can find $v \in S$ such that $v_{1} ⩽ v, v_{2} ⩽ v$ ).”

Now we are ready to introduce our hypotheses on the data of (1).

$\underline{H_{0}} : a : \overline{Ω} \to R$ is Lipschitz continuous (that is, $a \in C^{0, 1} (\overline{Ω})$ ), $a (z) > 0$ for all $z \in Ω$ and $\frac{p}{q} < 1 + \frac{1}{N}$ . Remark 2.1.
The last condition implies that $p < q^{}$ .

$\underline{H_{1}} : f : Ω \times [- η_{0}, η_{0}] \to R (η_{0} > 0)$ is a Carathéodory function such that for a.a. $z \in Ω, f (z, 0) = 0, f (z, \cdot)$ is odd and
there exists $a_{0} \in L^{\infty} (Ω)$ such that $\begin{matrix} | f (z, x) | ⩽ a_{0} (z) for a.a. z \in Ω, all | x | ⩽ η_{0}; \end{matrix}$

there exists $τ \in (1, min {q, \frac{q^{2}}{N - q} + 1})$ such that $\begin{array}{l} lim_{x \to 0} \frac{f (z, x)}{| x |^{τ - 2} x} = 0 uniformly for a.a. z \in Ω and \\ lim_{x \to 0} \frac{f (z, x)}{| x |^{q - 2} x} = + \infty uniformly for a.a. z \in Ω; \end{array}$

if $F (z, x) = \int_{0}^{x} f (z, s) d s$ then $f (z, x) x ⩾ F (z, x)$ for a.a. $z \in Ω$ , all $| x | ⩽ η_{0}$ .

Remark 2.2.
We see that all conditions on $f (z, \cdot)$ are local (near zero).

Given any $η > 0$ , we can find $\hat{δ} = \hat{δ} (η) \in (0, min {\frac{η_{0}}{2}, 1})$ such that $\begin{matrix} (2) & f (z, x) x ⩾ η | x |^{q}, | f (z, x) | ⩽ | x |^{τ - 1}, | x |^{p^{}} ⩽ | x |^{τ} for a.a. z \in Ω, all | x | ⩽ \hat{δ} \end{matrix}$ (see hypothesis $H_{1} (\underline{ii})$ and recall that $\hat{δ} < 1 < τ < q < p$ ). We introduce an even cut-off function $θ \in C^{1} (R)$ such that $\begin{matrix} (3) & supp θ \subseteq [- η_{0}, η_{0}], θ ∣_{[\frac{- η_{0}}{2}, \frac{η_{0}}{2}]} \equiv 1 and 0 < θ ⩽ 1 on (- η_{0}, η_{0}) . \end{matrix}$

Then we define the Carathéodory function $k (z, x)$ by $\begin{matrix} (4) & k (z, x) = θ (x) [f (z, x) + | x |^{p^{*} - 2} x] + (1 - θ (x)) | x |^{τ - 2} x, for all (z, x) \in Ω \times R . \end{matrix}$

From (3) and hypothesis $H_{1} (\underline{i})$ , we see that for some $c_{2} > 0$ we have $\begin{matrix} (5) & | k (z, x) | ⩽ c_{2} [1 + | x |^{τ - 1}] for a.a. z \in Ω, all x \in R . \end{matrix}$

3. An auxiliary problem

In this section we deal with the following auxiliary Dirichlet double phase problem $\begin{matrix} (6) & - Δ_{p}^{a} u (z) - Δ_{q} u (z) = k (z, u (z)) in Ω, u ∣_{\partial Ω} = 0 . \end{matrix}$

Let $S_{+}$ (resp. $S_{-}$ ) denote the set of positive (resp.negative) solutions of problem (6).

Proposition 3.1.
If hypotheses $H_{0}$ , $H_{1}$ hold, then $\begin{matrix} \emptyset \neq S_{+}, S_{-} \subseteq W_{0}^{1, p} (Ω) \cap L^{\infty} (Ω) . \end{matrix}$
Proof.
First we show the nonemptiness of $S_{+}$ .

Let $K (z, x) = \int_{0}^{x} k (z, s) d s$ and consider the $C^{1}$ -functional $ψ_{+} : W_{0}^{1, ξ} (Ω) \to R$ defined by $\begin{matrix} ψ_{+} (u) = \frac{1}{p} \int_{Ω} a (z) | D u |^{p} d z + \frac{1}{q} ‖ D u ‖_{q}^{q} - \int_{Ω} K (z, u^{+}) d z for all u \in W_{0}^{1, ξ} (Ω) . \end{matrix}$

We have $\begin{matrix} ψ_{+} (u) ⩾ \frac{1}{p} ρ_{ξ} (D u) - \int_{Ω} K (z, u^{+}) d z . \end{matrix}$

On account of (5) since $τ < q$ , we see that $ψ_{+} (\cdot)$ is coercive. Also, by the Sobolev embedding theorem (see Section 2), $ψ_{+} (\cdot)$ is sequentially weakly lower semicontinuous. So by the Weierstrass–Tonelli theorem, we can find $u_{0} \in W_{0}^{1, ξ} (Ω)$ such that $\begin{matrix} (7) & ψ_{+} (u_{0}) = min [ψ_{+} (u) : u \in W_{0}^{1, ξ} (Ω)] . \end{matrix}$

Let ${\hat{u}}_{1} (q)$ be the positive, $L^{q}$ -normalized principal eigenfunction of the operator $(- Δ_{q}, W_{0}^{1, q} (Ω))$ . We know that ${\hat{u}}_{1} (q) \in W_{0}^{1, p} (Ω) \cap L^{\infty} (Ω)$ and ${\hat{u}}_{1} (q) (z) > 0$ for a.a. $z \in Ω$ . (see Gasiński–Papageorgiou [5, Theorem 6.2.6, p. 737]). Choose $t \in (0, 1)$ small so that $t {\hat{u}}_{1} (q) ⩽ \hat{δ}$ with $\hat{δ} \in (0, 1)$ as in (2). Then from (3) and (4) we have $\begin{matrix} ψ_{+} (t {\hat{u}}_{1} (q)) ⩽ \frac{t^{p}}{p} {‖ D {\hat{u}}_{1} (q) ‖}_{p}^{p} + \frac{t^{q}}{q} \int_{Ω} [\hat{λ_{1}} (q) - η] \hat{u_{1}} {(q)}^{q} d z . \end{matrix}$

Here $\hat{λ_{1}} (q) > 0$ is the principal eigenvalue of $(- Δ_{q}, W_{0}^{1, q} (Ω))$ and we have used the fact $‖ D {\hat{u}}_{1} ‖_{q}^{q} = \hat{λ_{1}} (q) ‖ {\hat{u}}_{1} (q) ‖_{q}^{q}$ . But $η > 0$ is arbitrary. So, choosing $η > \hat{λ_{1}} (q)$ , we obtain $\begin{array}{l} ψ_{+} (t {\hat{u}}_{1} (q)) ⩽ c_{3} t^{p} - c_{4} t^{q} for some c_{3}, c_{4} > 0, \\ \Rightarrow ψ_{+} (t {\hat{u}}_{1} (q)) < 0 (choosing t \in (0, 1) small, since q < p), \\ \Rightarrow ψ_{+} (u_{0}) < 0 = ψ_{+} (0) (see (7)), \\ \Rightarrow u_{0} \neq 0 . \end{array}$

From (7) we have $\begin{array}{l} ψ_{+}^{'} (u_{0}) = 0, \\ (8) & \Rightarrow ⟨ A_{p}^{a} (u_{0}), h ⟩ + ⟨ A_{q} (u_{0}), h ⟩ = \int_{Ω} k (z, u_{0}^{+}) h d z for all h \in W_{0}^{1, ξ} (Ω) . \end{array}$

In (8) we choose $h : - u_{0}^{-} \in W_{0}^{1, ξ} (Ω)$ and obtain $\begin{array}{l} ρ_{ξ} (D u_{0}^{-}) = 0, \\ \Rightarrow u_{0} ⩾ 0, u_{0} \neq 0 (by the Poincaré inequality and Proposition 2.1) . \end{array}$

Moreover, from Colasuonno–Squassina [2, p. 1934] we have that $u_{0} \in W_{0}^{1, p} (Ω) \cap L^{\infty} (Ω)$ .

Similarly for the set $S_{-}$ of negative solutions. In this case we work with the $C^{1}$ -functional $ψ_{-} : W_{0}^{1, ξ} (Ω) \to R$ defined by $\begin{matrix} ψ_{-} (u) = \frac{1}{p} \int_{Ω} a (z) | D u |^{p} d z + \frac{1}{q} ‖ D u ‖_{q}^{q} - \int_{Ω} K (z, - u^{-}) d z for all u \in W_{0}^{1, ξ} (Ω) . \end{matrix}$ □

In fact we can show the existence of extremal constant sign solutions, that is, we show that there exists a smallest positive solution $u_{} \in S_{+}$ and a biggest negative solution $v_{} \in S_{-}$ .
Proposition 3.2.
If hypotheses $H_{0}$ , $H_{1}$ hold, then there exists $u_{} \in S_{+}$ such that* $u_{} ⩽ u$ for all* $u \in S_{+}$ and there exists $v_{} \in S_{-}$ such that* $v ⩽ v_{}$ for all* $v \in S_{-}$ .
Proof.
From Papageorgiou–Rădulescu–Repovš [19] (proof of Proposition 7), we know that $S_{+}$ is downward directed. Then, Lemma 3.10, p. 178, of Hu–Papageorgiou [11], implies that we can find a decreasing sequence ${u_{n}}_{n \in N} \subseteq S_{+}$ such that $\begin{matrix} inf_{n \in N} u_{n} = inf S_{+} . \end{matrix}$

We have $\begin{array}{l} (9) & ⟨ A_{p}^{a} (u_{n}), h ⟩ + ⟨ A_{q} (u_{n}), h ⟩ = \int_{Ω} k (z, u_{n}) h d z for all h \in W_{0}^{1, ξ} (Ω), \\ (10) & 0 ⩽ u_{n} ⩽ u_{1} for all n \in N . \end{array}$

Choosing $h = u_{n} \in W_{0}^{1, ξ} (Ω)$ in (9) and using (10) and hypotheses $H_{1} (\underline{i})$ we obtain that $\begin{array}{l} ρ_{ξ} (D u_{n}) ⩽ c_{5} for some c_{5} > 0, all n \in N, \\ (11) & \Rightarrow {u_{n}}_{n \in N} \subseteq W_{0}^{1, ξ} (Ω) is bounded (see Proposition 2.1) . \end{array}$

The restriction on $τ > 1$ (see hypotheses $H_{1} (\underline{ii})$ ) implies that $\frac{N}{q} (τ - 1) < q^{}$ .We choose $s > \frac{N}{q}$ so that $s (τ - 1) < q^{}$ . Then on account of (11), we may assume that $\begin{matrix} (12) & u_{n} \overset{w}{⟶} u_{} in W_{0}^{1, ξ} (Ω) and u_{n} \to u_{} in L^{s (τ - 1)} (Ω) . \end{matrix}$

From (3), (4) we see that $\begin{matrix} (13) & | k (z, x) | ⩽ c_{6} | x |^{τ - 1} for a.a. z \in Ω, all x \in R, some c_{6} > 0 . \end{matrix}$

From (9) and Colasuonno–Squassina [2, p. 1934] (see also Guedda–Veron [10, Proposition 1.3]) since $s > \frac{N}{q}$ , we have $\begin{matrix} (14) & ‖ u_{n} ‖_{\infty} ⩽ c_{7} {‖ N_{k} (u_{n}) ‖}_{s}^{\frac{1}{q - 1}} \end{matrix}$ for some $c_{7} > 0$ , all $n \in N$ with $N_{k} (u) (\cdot) = k (\cdot, u (\cdot))$ for all $u \in M (Ω)$ (the Nemytskii (superposition) operator corresponding to k). From (13) and (14), we have $\begin{matrix} (15) & ‖ u_{n} ‖_{\infty} ⩽ c_{8} ‖ u_{n} ‖_{s (τ - 1)}^{\frac{τ - 1}{q - 1}} for some c_{8} > 0, all n \in N . \end{matrix}$

If $u_{} = 0$ , then from (15) and (12) if follows that $\begin{array}{l} ‖ u_{n} ‖_{\infty} \to 0 as n \to \infty, \\ (16) & \Rightarrow 0 ⩽ u_{n} (z) ⩽ \hat{δ} for a.a. z \in Ω, all n ⩾ n_{0}, \\ (17) & \Rightarrow k (z, u_{n} (z)) = f (z, u_{n} (z)) + u_{n} {(z)}^{p^{} - 1} for a.a. z \in Ω, all n ⩾ n_{0} . \end{array}$

Let $y_{n} = \frac{u_{n}}{‖ u_{n} ‖}$ $n \in N$ . Then $‖ y_{n} ‖ = 1, y_{n} ⩾ 0$ for all $n \in N$ . So we may assume that $\begin{matrix} (18) & y_{n} \overset{w}{⟶} y in W_{0}^{1, ξ} (Ω) and y_{n} \to y in L^{p} (Ω), y ⩾ 0 . \end{matrix}$

From (9) we have $\begin{array}{l} ‖ u_{n} ‖^{p - q} ⟨ A_{p}^{a} (y_{n}), h ⟩ + ⟨ A_{q} (y_{n}), h ⟩ = \int_{Ω} [\frac{f (z, u_{n})}{u_{n}^{q - 1}} + u_{n}^{p^{} - q}] y_{n}^{q - 1} h d z, \\ (19) & for all h \in W_{0}^{1, ξ} (Ω), all n \in N (see (17)) . \end{array}$

In (19) the left hand side is bounded for every $h \in W_{0}^{1, ξ} (Ω)$ . So, on account of hypothesis $H_{1} (\underline{ii})$ , we have $\begin{matrix} y = 0 and \frac{f (z, u_{n} (z))}{u_{n} {(z)}^{q - 1}} y_{n} {(z)}^{q - 1} \to 0 for a.a. z \in Ω . \end{matrix}$

Hence choosing $h = y_{n} \in W_{0}^{1, ξ} (Ω)$ in (19) and passing to the limit as $n \to \infty$ we obtain $\begin{array}{l} lim_{n \to \infty} ‖ D y_{n} ‖_{q} = 0, \\ \Rightarrow y_{n} \to 0 in W_{0}^{1, q} (Ω), \\ \Rightarrow D y_{n} (z) \to 0 in R^{N} for a.a. z \in Ω (at least for a subsequence), \\ \Rightarrow ξ (z, D y_{n} (z)) \to 0 in R^{N} for a.a. z \in Ω . \end{array}$

Since ${ξ (\cdot, D y_{n} (\cdot))}_{n \in N} \subseteq L^{1} (Ω)$ is uniformly integrable (see (18)), by Vitali’s convergence theorem, we have that $\begin{matrix} (20) & ρ_{ξ} (D y_{n}) \to 0 as n \to \infty . \end{matrix}$

From Proposition 2.1 we know that $\begin{array}{l} ‖ y_{n} ‖ = 1 if and only if ρ_{ξ} (D y_{n}) = 1 for all n \in N, \\ (21) & \Rightarrow ρ_{ξ} (D y_{n}) = 1 for all n \in N . \end{array}$

Comparing (20) and (21), we have a contradiction. Therefore $u_{} \neq 0$ and $S_{+} ∋ u_{} = inf S_{+}$ .

Similarly we produce $v_{} \in S_{-}$ such that $v_{} = sup S_{-}$ . Note that $S_{-}$ is upward directed □

The idea is to focus on the order interval $\begin{matrix} (22) & [v_{}, u_{}] = {u \in W_{0}^{1, ξ} (Ω) : v_{} (z) ⩽ u (z) ⩽ u_{} (z) for a.a. z \in Ω} . \end{matrix}$

On account of extremality of $u_{}$ and $v_{}$ , any nontrivial solution of (6) distinct from $u_{}$ and $v_{*}$ is necessarily nodal. We will use the Kajikiya [12] version of the symmetric mountain pass theorem, to generate a whole sequence of nodal solutions which converge to zero in $W_{0}^{1, ξ} (Ω)$ and $L^{\infty} (Ω)$ . Then on account of (3), (4), eventually these solutions are nodal solutions of (1). This strategy is implemented in the next section.
4. Nodal solutions

Using the two extremal constant sign solutions $u_{*} \in W_{0}^{1, ξ} (Ω) \cap L^{\infty} (Ω)$ and $v_{*} \in W_{0}^{1, ξ} (Ω) \cap L^{\infty} (Ω)$ from Proposition 3.2, we introduce the Carathéodory function $k_{*} (z, x)$ defined by $\begin{array}{l} (23) & k_{*} (z, x) = \{\begin{matrix} k (z, v_{*} (z)) & if x < v_{*} (z), \\ k (z, x) & if v_{*} (z) ⩽ x ⩽ u_{*} (z), \\ k (z, u_{*} (z)) & if u_{*} (z) < x . \end{matrix} \end{array}$

We set $K_{*} (z, x) = \int_{0}^{x} k_{*} (z, s) d s$ and consider the $C^{1}$ -functional $ψ_{*} : W_{0}^{1, ξ} (Ω) \to R$ defined by $\begin{array}{l} ψ_{*} (u) = \frac{1}{p} \int_{Ω} a (z) | D u |^{p} d z + \frac{1}{q} ‖ D u ‖_{q}^{q} - \int_{Ω} K_{*} (z, u) d z for all u \in W_{0}^{1, ξ} (Ω) . \end{array}$

Using (23) we can easily check that $\begin{matrix} (24) & K_{ψ_{*}} \subseteq [v_{*}, u_{*}] (see (22)) . \end{matrix}$

Proposition 4.1.

If hypotheses $H_{0}, H_{1}$ hold and $V \subseteq W_{0}^{1, ξ} (Ω)$ is a finite dimensional subspace, then we can find $r_{V} > 0$ such that $\begin{matrix} sup [ψ_{*} (u) : u \in V, ‖ u ‖ = r_{V}] < 0 . \end{matrix}$

Proof.

Since V is finite dimensional, all norms are equivalent (see Papageorgiou–Winkert [26, Proposition 3.1.17, p. 183]). Therefore we can find $r_{V} > 0$ such that $\begin{matrix} (25) & v \in V, ‖ v ‖ ⩽ r_{V} \Rightarrow | v (z) | ⩽ \hat{δ} for all a.a. z \in Ω, \end{matrix}$ with $\hat{δ} > 0$ as in (2). Recall that $\hat{δ} < \frac{η_{0}}{2}$ . It follows that $θ (v (z)) = 1$ for a.a. $z \in Ω$ (see (25) and (2)). Then using (4) and (23), we see that for $v \in V$ with $‖ v ‖ ⩽ r_{V}$ , then $\begin{array}{l} ψ_{*} (v) = & \frac{1}{p} \int_{Ω} a (z) | D v |^{p} d z + \frac{1}{q} ‖ D v ‖_{q}^{q} - \int_{Ω} K_{*} (z, v) d z - \int_{{v_{*} ⩽ v ⩽ u_{*}}} F (z, v) d z \\ - \int_{{u_{*} < v}} [F (z, v) - F (z, u_{*})] d z - \int_{{u_{*} < v}} f (z, u_{*}) u_{*} d z \\ - \int_{{v < v_{*}}} [F (z, v) - F (z, v_{*})] d z - \int_{{v < v_{*}}} f (z, v_{*}) v_{*} d z \\ = & \frac{1}{p} \int_{Ω} a (z) | D v |^{p} d z + \frac{1}{q} ‖ D v ‖_{q}^{q} - \int_{Ω} F (z, v) d z \\ - \int_{{u^{*} < v}} [f (z, u_{*}) u_{*} - F (z, u_{*}) d z - \int_{{v < v_{*}}} [f (z, v_{*}) v_{*} - F (z, v_{*})] d z \\ (26) & ⩽ & \frac{1}{p} \int_{Ω} a (z) | D v |^{p} d z + \frac{1}{q} ‖ D v ‖_{q}^{q} - \frac{η}{q} ‖ v ‖_{q}^{q} (see (2), (25) and hypotheses H_{1} (\underline{iii}) . \end{array}$

Since $η > 0$ is arbitrary and as we already mentioned all norms on V are equivalent, from (26) we obtain $\begin{matrix} ψ_{*} (v) ⩽ c_{9} ‖ v ‖^{p} - c_{10} ‖ v ‖^{q} for some c_{9}, c_{10} > 0, all v \in V with ‖ v ‖ = r_{V} . \end{matrix}$

But $1 < q < p$ . So, taking $r_{V} \in (0, 1)$ even smaller, we have $\begin{matrix} sup [ψ_{*} (v) : v \in V, ‖ v ‖ = r_{V}] < 0 . \end{matrix}$ □

Now we are ready for the main result of this paper. We will produce a whole sequence ${y_{n}}_{n \in N}$ of nodal solutions of problem (1) which converges to zero in $W_{0}^{1, ξ} (Ω)$ and in $L^{\infty} (Ω)$ .

Theorem 4.1.

If hypotheses $H_{0}, H_{1}$ hold, then problem ( 1 ) has a whole sequence ${y_{n}}_{n \in_{N}} W_{0}^{1, p} (Ω) \cap L^{\infty} (Ω)$ of nodal solutions such that $‖ y_{n} ‖, ‖ y_{n} ‖_{\infty} \to 0$ .

Proof.

Evidently the functional $ψ_{*} (\cdot)$ is even and also coercive (see (23)). So, $ψ_{*} (\cdot)$ is bounded below and satisfies the PS-condition (see Papageorgiou–Rădulescu–Repovš [21, Proposition 5.1.15, p. 369]). This fact and Proposition 4.1, permit the use of Theorem 1 of Kajikiya [12]. So, we can find ${y_{n}}_{n \in N} \subseteq W_{0}^{1, p} (Ω)$ such that $\begin{array}{l} (27) & y_{n} \in K_{ψ_{*}} \subseteq [v_{*}, u_{*}] (see (24)), y_{n} \neq 0, ψ_{*} (y_{n}) ⩽ 0, for all n \in N, \\ (28) & ‖ y_{n} ‖ \to 0 as n \to \infty . \end{array}$ From (27) and the extremality of $v_{*}$ , $u_{*}$ , we see that $\begin{matrix} {y_{n}}_{n \in N} \subseteq W_{0}^{1, ξ} (Ω) \cap L^{\infty} (Ω) are nodal solutions of (6) . \end{matrix}$

From the proof of Proposition 3.2 we know that $\begin{matrix} ‖ y_{n} ‖_{\infty} ⩽ c_{11} ‖ y_{n} ‖_{s (τ - 1)}^{\frac{τ - 1}{q - 1}} for all n \in N, with s > \frac{N}{q}, s (τ - 1) < q^{*} . \end{matrix}$ Then from (28), we have $‖ y_{n} ‖_{\infty} \to 0$ as $n \to \infty$ .

Therefore, we can find $n_{0} \in N$ such that $\begin{array}{l} | y_{n} (z) | ⩽ \frac{η_{0}}{2} for a.a. z \in Ω, all n ⩾ n_{0} \\ \Rightarrow θ (y_{n} (z)) = 1 for a.a. z \in Ω, all n ⩾ n_{0} (see (3)) \\ \Rightarrow {y_{n}}_{n ⩾ n_{0}} \subseteq W_{0}^{1, ξ} (Ω) \cap L^{\infty} (Ω) are nodal solutions of (1) (see (27), (23) and (4)) . \end{array}$ □

Footnotes

Acknowledgements

The work was supported by NNSF of China Grant No. 12071413, NSF of Guangxi Grant No. 2018GXNSFDA138002.

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