A Result of Multiplicity of Nontrivial Solutions to an Elliptic Kirchhoff

Abstract

In this work, using Ljusternik–Schnirelman category theory, we study the existence and multiplicity of nontrivial solutions for the following class of elliptic Kirchhoff–Boussinesq type problems given by $\begin{aligned} ε^{4} Δ^{2} u \pm ε^{p} Δ_{p} u + V (x) u = f (u) + β | u |^{2_{* *} - 2} u in R^{N}, u \in H^{2} (R^{N}), \end{aligned}$ where $ε > 0$ , $2 < p < 2^{*} = 2 N / (N - 2)$ , $2_{* *} = 2 N / (N - 4)$ and $N \geq 5$ , $V : R^{N} \to R$ is a continuous function and $f : R \to R$ is a function of $C^{1}$ class. We consider the subcritical case, that is, $β = 0$ and critical case, that is, $β = 1$ .

Keywords

Kirchhoff–Boussinesq type problems subcritical or critical growth Ljusternik–Schnirelman category theory

1. Introduction

In a seminal paper (Rabinowitz, 1992), using the Mountain Pass Theorem (Ambrosetti & Rabinowitz, 1973) and the force of the parameter $ϵ$ , Rabinowitz showed existence of solution for a nonlinear Schrödinger equation given by

(R) where $V$ is a positive continuous potential. In Wang (1993), Wang showed that the solution found by Rabinowitz concentrates around a local minimum of the potential $V$ , when $ϵ$ converges to zero. In addition, Wang observed that the concentration of any family of solutions with energy uniformly bounded can only occur at a critical point of $V$ . The main tool used by Wang was the elliptic regularity theory. Nowadays, results of this type are called concentration results. In Cingolani and Lazzo (1997) Cingolani and Lazzo showed that problem ( R ) has multiplicity of solutions using Ljusternik–Schnirelman category theory. In the proof of the main result in Cingolani and Lazzo (1997) was important the uniqueness of solution of the autonomous problem associated with ( R ), that is, the problem when $V (x) =$ constant. The version of Rabinowitz (1992), Wang (1993) and Cingolani and Lazzo (1997) for the $p$ -Laplacian operator was presented by Alves and Figueiredo (2006), where the uniqueness of solution of the autonomous problem was removed and in the concentration result was used the method of Moser iteration. In Pimenta and Soares (2014) Pimenta and Soares found a family of solutions to a singularly perturbed biharmonic equation which has a concentration behavior. In order to show the concentration result, they used the arguments of regularity presented by Ramos (2009). After these articles addressing existence, concentration and multiplicity via Ljusternik–Schnirelman category theory, other articles appeared with versions for systems (Alves & Soares, 2005; Figueiredo, 2006; He, 2011; He & Zou, 2012), with exponential growth (Alves & Figueiredo, 2009), with discontinuous nonlinearity (Alves & Nascimento, 2013), with fractional Laplacian (Figueiredo & Siciliano, 2016) and so on.

In this article, we deal with the existence of a ground state solution for the problem
$\begin{aligned} {\begin{cases} ε^{4} Δ^{2} u \pm ε^{p} Δ_{p} u + V (x) u = f (u) + β | u |^{2_{* } - 2} u in R^{N}, \\ u \in H^{2} (R^{N}), \end{cases} \end{aligned}$
(P_ε)
where $ε > 0$ , $2 < p < 2^{} = 2 N / (N - 2)$ , $β \in {0, 1}$ , $2_{* } = 2 N / (N - 4)$ and $N \geq 5$ .

Along the article, we assume that $V : R^{N} \to R$ is a continuous function satisfying the following condition
$\begin{aligned} 0 < inf_{x \in R^{N}} V (x) = V_{0} < \underset{| x | \to \infty}{lim inf} V (x) = V_{\infty} \leq \infty \end{aligned}$
(V)
and the nonlinearity $f : R \to R$ fulfills the following hypotheses:

$(f_{1})$
We suppose that $f$ is $C^{1}$ class and
$\begin{aligned} lim_{| t | \to 0} \frac{f (t)}{t} = 0. \end{aligned}$

$(f_{2})$
There exists $q \in (p, 2_{ })$ such that
$\begin{aligned} lim_{| t | \to \infty} \frac{f (t)}{t^{q - 1}} = 0. \end{aligned}$

$(f_{3})$
The function $t \mapsto (f (t) / t^{p - 1})$ is increasing in $(0, \infty)$ and decreasing in $(- \infty, 0)$ .
$(f_{4})$
There are $p < r < 2_{ }$ and $σ > 0$ such that
$\begin{aligned} f (t) \geq σ | t |^{r - 2} t, \end{aligned}$
for all $t \geq 0$ .

Since we deal with the multiplicity of solutions of $(P_{ε})$ , we recall that if $Y$ is a given closed subset of a topological space $X$ , we denote by ${cat}_{X} (Y)$ the Ljusternik–Schnirelmann category of $Y$ in $X$ , that is the least number of closed and contractible sets in $X$ which cover $Y$ .

Let us denote by
$\begin{aligned} M = {x \in R^{N} : V (x) = V_{0}} and M_{δ} = {x \in R^{N} : dist (x, M) \leq δ}, for δ > 0. \end{aligned}$

Our main results are:
Theorem 1.1
Assume that conditions $(f_{1}) - (f_{4})$ and ( V ) hold with $β = 0$ . Then, for all $σ > 0$ and for any $δ > 0$ given, there exists $ε_{δ} > 0$ such that, for any $ε \in (0, ε_{δ})$ , problem ( Pε ) has at least ${cat}_{M_{δ}} (M)$ nontrivial solutions.
Theorem 1.2
Assume that conditions $(f_{1}) - (f_{4})$ and ( V ) hold with $β = 1$ . Then, for any $δ > 0$ given, there exist $ε_{δ} > 0$ and $σ^{} > 0$ such that, for any $ε \in (0, ε_{δ})$ and for all $σ > σ^{*}$ , problem ( Pε ) has at least ${cat}_{M_{δ}} (M)$ nontrivial solutions.

We emphasize that the operator studied in this article has a strong physical motivation. In an interesting article, Chueshov and Lasiecka (2011) consider the following nonlinear plate equation referred to as Kirchhoff–Boussinesq (K–B) model:
$\begin{aligned} w_{t t} + k w_{t} + Δ^{2} w = div (| \nabla w |^{p - 2} \nabla w) + σ Δ (w^{2}) - f (w) \end{aligned}$
(1.1)
defined on a bounded domain $Ω \subset R^{2}$ with a sufficiently smooth boundary and a suitable the initial data. The model (1.1) arises naturally, as shown in Chueshov and Lasiecka (2006), as the limit in Mindlin–Timoshenko equations which describe the dynamics of a plate that accounts for transverse shear effects (see, e.g., Lagnese, 1989 and Lagnese & Lions, 1988, Chapter 1) and the references therein. For details concerning the dynamics of Mindlin–Timoshenko plates we refer to Lagnese (1989) and Lagnese and Lions (1988).

More recently, some authors are focusing on stationary Kirchhoff–Boussinesq, that is, those in which there is only a single space variable $x \in Ω$ unlike in evolution problems, where the unknown also depends on the time variable $t \geq 0$ . For example, Sun, Liu, et al. (2017) are concerned with the following biharmonic equation with $p$ -Laplacian and Neumann boundary condition given by
$\begin{aligned} {\begin{cases} Δ^{2} u - λ Δ_{p} u = f (x, u) - \frac{μ}{| Ω |} \int_{Ω} f (y, u (y)) d y in Ω, \\ \frac{\partial Δ u}{\partial η} = \frac{\partial u}{\partial η} = 0 on \partial Ω . \end{cases} \end{aligned}$
Using the Fountain Theorem, the authors obtain the existence of infinitely many sign-changing high energy solutions.

The local case was studied by Carlos and Figueiredo (2024). More precisely these authors showed existence of solutions for the problem
$\begin{aligned} {\begin{cases} Δ^{2} u \pm Δ_{p} u = f (x, u) in Ω, \\ Δ u = u = 0 on \partial Ω, \end{cases} \end{aligned}$
considering the superlinear and sublinear cases.

Sun and Wu (2017) also study a class of biharmonic equations with $p$ -Laplacian and singular sign-changing potential as follows
$\begin{aligned} Δ^{2} u - β Δ_{p} u + V_{λ} (x) u = 0 in R^{N}, \end{aligned}$
where $N \geq 5$ , $β < 0$ with $V_{λ} (x) = λ a (x) - b (x)$ with $λ > 0$ . Under some suitable assumptions on $a$ and $b$ , the authors obtain the existence of nontrivial solutions for $λ$ large enough.

Sun, Chu, et al. (2017) also study problem
$\begin{aligned} Δ^{2} u - β Δ_{p} u + V_{λ} (x) u = f (x, u) in R^{N} \end{aligned}$
with the same arguments used in Sun and Wu (2017). Yang (2021) also obtain a nontrivial weak solution to a critical biharmonic system involving $p$ -Laplacian and Hardy potential via variational methods.

In Carlos et al. (2024), they studied the Kirchhoff–Boussinesq equation considering the nonlinearity of the Berestycki–Lions type and showed the existence of a solution in the zero mass case and in the positive mass case.

The present work is strongly influenced by the articles (Sun, Chu, et al., 2017; Sun, Liu, et al., 2017;; Sun & Wu, 2017; Yang, 2021). But, different from the results that can be found in these articles, here we consider two $Δ^{2} u \pm Δ_{p} u$ cases and we are considering subcritical and critical growth without parameters. Furthermore, we complete the study started in Alves and Figueiredo (2006), Cingolani and Lazzo (1997), Pimenta and Soares (2014), Rabinowitz (1992) because, possibly, this article presents the first multiplicity result involving category for this class of problems.

However, the presence of the $p$ -Laplacian operator in this class of problems implies that the arguments found in Alves and Figueiredo (2006), Cingolani and Lazzo (1997), Pimenta and Soares (2014), Rabinowitz (1992) cannot be adapted directly. For example, concentration results are open problems, as the presence of the biharmonic operator prevents the use of classical elliptical regularity theory or Moser’s iteration method. The presence of the $p$ -Laplacian operator prevents the use of the Ramos argument that can be found in Ramos (2009). Furthermore, the estimates are more delicate due to the presence of the $\pm$ sign in front of the $p$ -Laplacian operator.

The plan of the article is the following: In Section 2 we show the variational framework. In Section 3 we show the proof of Theorem 1.1, that is, we study the subcritical case. In Section 4 we show the proof of Theorem 1.2, that is, we study the critical case.
2. Variational Framework

For the proof of our results, we shall consider an equivalent problem to ( Pε ). Changing variables by $x \mapsto ε x$ , we can rewrite the problem ( Pε ) in to the following equivalent form

\begin{aligned} {\begin{cases} Δ^{2} u \pm Δ_{p} u + V (ε x) u = f (u) + β | u |^{2_{* *} - 2} u in R^{N}, \\ u \in H^{2} (R^{N}) . \end{cases} \end{aligned}

(P̃_ε)

If $u$ is a solution of problem $({\tilde{P}}_{ε})$ , then $\hat{u} (x) = u (x / ε)$ is a solution of problem ( Pε ). Thus, to study problem ( Pε ), it suffices to study problem $({\tilde{P}}_{ε})$ .

For $ε > 0$ , we define the rescaled weighted Sobolev space $X_{ε}$ by

\begin{aligned} X_{ε} = {u \in H^{2} (R^{N}) : \int_{R^{N}} V (ε x) | u |^{2} d x < \infty} \end{aligned}

endowed with the inner product

\begin{aligned} ⟨ u, v ⟩_{ε} = \int_{R^{N}} [Δ u Δ v + V (ε x) u v] d x \end{aligned}

and the norm

\begin{aligned} ‖ u ‖_{ε}^{2} = \int_{R^{N}} (| Δ u |^{2} + V (ε x) | u |^{2}) d x . \end{aligned}

An important result in this article is a Gagliardo–Nirenberg interpolation inequality:

Theorem 2.1

Suppose that $N, j, m$ are non-negative integers and that $1 \leq k_{1}, k_{2}, k_{3} \leq \infty$ and $θ \in [0, 1]$ are real numbers such that

\begin{aligned} \frac{1}{k_{1}} = \frac{j}{N} + (\frac{1}{k_{3}} - \frac{m}{N}) θ + \frac{1 - θ}{k_{2}} \end{aligned}

and

\begin{aligned} \frac{j}{m} \leq θ \leq 1. \end{aligned}

Then, there exist a constant

C > 0

independent of

u

such that

\begin{aligned} ‖ D^{j} u ‖_{k_{1}} \leq C ‖ D^{m} u ‖_{k_{3}}^{θ} ‖ u ‖_{k_{2}}^{1 - θ}, \forall u \in L^{k_{2}} (R^{N}) \cap W^{m, k_{3}} (R^{N}) . \end{aligned}

By using ( V ) and Sobolev embedding theorem we have

\begin{aligned} X_{ε} ↪ H^{2} (R^{N}) ↪ L^{s} (R^{N}), for 2 \leq s \leq 2_{* *} . \end{aligned}

(2.1)

Since $2 < p < 2^{*}$ , by using Theorem 2.1 for $j = 1$ , $m = 2$ , $1 / 2 < θ \leq 1$ , $k_{1} = p$ , $k_{2} = k_{3} = 2$ , we have the following continuous embedding

\begin{aligned} X_{ε} ↪ W^{1, p} (R^{N}) . \end{aligned}

(2.2)

From conditions $(f_{1})$ to $(f_{2})$ , given $Υ > 0$ , there exists a positive constant $C_{Υ} > 0$ such that

\begin{aligned} | f (t) | \leq Υ | t | + C_{Υ} | t |^{q - 1} . \end{aligned}

(2.3)

From condition $(f_{3})$ , we can show that

\begin{aligned} t f (t) - p F (t) is increasing for t > 0 and decreasing for t < 0 \end{aligned}

(2.4)

and

\begin{aligned} f^{'} (t) t^{2} - (p - 1) f (t) t > 0 for all t \neq 0, \end{aligned}

(2.5)

where

F (t) = \int_{0}^{t} f (s) d s

3. Subcritical Case

(β = 0)

In this section we study the problem

\begin{aligned} {\begin{cases} Δ^{2} u \pm Δ_{p} u + V (ε x) u = f (u) in R^{N}, \\ u \in H^{2} (R^{N}) . \end{cases} \end{aligned}

(S_ε)

From (2.1)–(2.3) we get that the functional associated to problem ( Sε ) given by

\begin{aligned} I_{ε} (u) = \frac{1}{2} \int_{R^{N}} (| Δ u |^{2} + V (ε x) | u |^{2}) d x \pm \frac{1}{p} \int_{R^{N}} | \nabla u |^{p} d x - \int_{R^{N}} F (u) d x \end{aligned}

is well defined for

u \in X_{ε}

. Moreover, using standard arguments, we can see that

I_{ε} \in C^{1} (X_{ε}, R)

with

\begin{aligned} I_{ε}^{'} (u) ϕ = \int_{R^{N}} (Δ u Δ ϕ + V (ε x) u ϕ) d x \pm \int_{R^{N}} | \nabla u |^{p - 2} \nabla u \nabla ϕ d x - \int_{R^{N}} f (u) ϕ d x \end{aligned}

for all

ϕ \in X_{ε}

. Then, the critical points of

I_{ε}

are weak solutions of ( Sε ).

The Nehari manifold associated to the functional $I_{ε}$ is given by

\begin{aligned} N_{ε} = {u \in X_{ε} ∖ {0} : I_{ε}^{'} (u) u = 0} . \end{aligned}

In order to use critical points theory we firstly derive results related to the Palais–Smale compactness condition for the functional $I_{ε}$ . A sequence $(u_{n}) \subset X_{ε}$ is a Palais–Smale sequence at level $c_{ε}$ for the functional $I_{ε}$ if

\begin{aligned} I_{ε} (u_{n}) \to c_{ε} \end{aligned}

and

\begin{aligned} I_{ε}^{'} (u_{n}) \to 0 in X_{ε}^{'}, \end{aligned}

where

\begin{aligned} c_{ε} = inf_{γ \in Γ_{ε}} max_{t \in [0, 1]} I_{ε} (γ (t)) > 0 \end{aligned}

and

\begin{aligned} Γ_{ε} = {γ \in C ([0, 1], X_{ε}) : γ (0) = 0, I_{ε} (γ (1)) < 0} . \end{aligned}

If every Palais–Smale sequence of

I_{ε}

has a strongly convergent subsequence then one says that

I_{ε}

satisfies the Palais–Smale condition (

(PS)

for short).

Lemma 3.1

The functional $I_{ε}$ satisfies the following conditions: (i)

There exist positive numbers $ρ$ and $α$ such that

\begin{aligned} I_{ε} (u) \geq α > 0 if ‖ u ‖_{ε} = ρ . \end{aligned}

(ii)

There exists $e \in X_{ε}$ with $I_{ε} (e) < 0$ and $‖ e ‖_{ε} > ρ$ .

Proof.

From (2.1) to (2.3), there exists $C_{1} > 0$ such that

\begin{aligned} I_{ε} (u) & \geq \frac{1}{2} ‖ u ‖_{ε}^{2} \pm \frac{1}{p} \int_{R^{N}} | \nabla u |^{p} d x - \frac{Υ}{2} ‖ u ‖_{L^{2} (R^{N})}^{2} - \frac{C_{Υ}}{q} ‖ u ‖_{L^{q} (R^{N})}^{q} \\ \geq \frac{1}{2} (1 - Υ C_{1}) ‖ u ‖_{ε}^{2} \pm \frac{1}{p} ‖ \nabla u ‖_{L^{p} (R^{N})}^{p} - C_{1} \frac{C_{Υ}}{q} ‖ u ‖_{ε}^{q} . \end{aligned}

In the case that the second term in the associated functional

I_{ε}

is positive, we get

\begin{aligned} I_{ε} (u) \geq \frac{1}{2} (1 - Υ C_{1}) ‖ u ‖_{ε}^{2} - \frac{1}{q} C_{Υ} C_{1} ‖ u ‖_{ε}^{q} \end{aligned}

and by taking

Υ > 0

sufficiently small, the proof of item

(i)

follows by choosing

ρ > 0

small enough.

In the case that the second term in the associated functional $I_{ε}$ is negative, we can use (2.2) to get $C_{2} > 0$ such that

\begin{aligned} I (u) \geq \frac{1}{2} (1 - Υ C_{1}) ‖ u ‖_{ε}^{2} - C_{2} ‖ u ‖_{ε}^{p} - \frac{1}{q} C_{Υ} C_{1} ‖ u ‖_{ε}^{q} . \end{aligned}

Since

2 < p < q

, then item

(i)

holds.

In order to prove $(i i)$ , fix $φ \in C_{0}^{\infty} (R^{N})$ . Now, from $(f_{4})$ we have

\begin{aligned} \frac{I_{ε} (t φ)}{t^{p}} & \leq \frac{1}{t^{p}} (\frac{t^{2}}{2} ‖ φ ‖_{ε}^{2} \pm \frac{t^{p}}{p} \int_{R^{N}} | \nabla φ |^{p} d x - \frac{σ}{r} t^{r} \int_{R^{N}} | φ |^{r} d x) \\ = \frac{1}{2 t^{p - 2}} ‖ φ ‖_{ε}^{2} \pm \frac{1}{p} \int_{R^{N}} | \nabla φ |^{p} d x - t^{r - p} \frac{σ}{r} \int_{R^{N}} | φ |^{r} d x \end{aligned}

for all

t > 0

. Since

2 < p < r

, there exists

t^{*} > 0

sufficiently large such that

e = t^{*} φ

satisfies

I_{ε} (e) < 0

and

‖ e ‖_{ε} > ρ

From Lemma 3.1, $I_{ε}$ has the mountain pass geometry. Hence, there exists a Palais–Smale sequence $(u_{n}) \subset X_{ε}$ at level $c_{ε}$ .

The next lemma is a key point in our arguments, because it establishes an important characterization involving the mountain pass for nonlocal elliptic problem.

To obtain a least energy solution, we need a characterization of the least energy. Define

\begin{aligned} c_{ε}^{*} = inf_{u \in N_{ε}} I_{ε} (u) and c_{ε}^{* *} = inf_{u \in X_{ε} ∖ {0}} max_{t \geq 0} I_{ε} (t u) . \end{aligned}

Lemma 3.2

For any $u \in X_{ε} ∖ {0}$ , we have $(i)$

There exists a unique $t_{0} = t_{0} (u)$ such that $t_{0} u \in N_{ε}$ . Moreover

\begin{aligned} I_{ε} (t_{0} u) = max_{t \geq 0} I_{ε} (t u) . \end{aligned}

(i i)

$c_{ε} = c_{ε}^{*} = c_{ε}^{* *} > 0$ .

Proof.

Given $u \in X_{ε}$ with $u \neq 0$ , let $h_{u} (t) = I_{ε} (t u)$ for $t > 0$ . Then, $t u \in N_{ε}$ if, and only if $h_{u}^{'} (t) = 0$ . Taking $Υ > 0$ sufficiently small in (2.3) and using (2.1), there exists $C_{3} > 0$ such that

\begin{aligned} h_{u} (t) \geq \frac{t^{2}}{2} (1 - Υ C_{3}) ‖ u ‖_{ε}^{2} \pm \frac{t^{p}}{p} \int_{R^{N}} | \nabla u |^{p} d x - C_{3} C_{Υ} \frac{t^{q}}{q} ‖ u ‖_{ε}^{q} . \end{aligned}

Thus, since

2 < p < q

, we have

h_{u} (t) > 0

for all

t > 0

sufficiently small.

Now, from $(f_{4})$ and using $2 < p < r$ , we have

\begin{aligned} \frac{h_{u} (t)}{t^{p}} \leq \frac{1}{2 t^{p - 2}} ‖ u ‖_{ε}^{2} \pm \frac{1}{p} \int_{R^{N}} | \nabla u |^{p} d x - σ \frac{t^{r - p}}{r} \int_{R^{N}} | u |^{r} d x . \end{aligned}

Hence,

lim_{t \to + \infty} h_{u} (t) = - \infty

. Then, there exists at least one

t_{u} > 0

such that

h_{u}^{'} (t_{u}) = 0

, that is,

t_{u} u \in N_{ε}

. Moreover, since

p > 2

, we get

\begin{aligned} h_{u}^{'} (t) & = t ‖ u ‖_{ε}^{2} \pm t^{p - 1} \int_{R^{N}} | \nabla u |^{p} d x - \int_{R^{N}} f (t u) u d x \\ = t^{p - 1} (\frac{1}{t^{p - 2}} ‖ u ‖_{ε}^{2} \pm \int_{R^{N}} | \nabla u |^{p} d x - \int_{R^{N}} \frac{f (t u)}{(t u)^{p - 1}} u^{p} d x) . \end{aligned}

From

(f_{3})

we conclude that

h_{u}^{'} (t) / t^{p - 1}

is decreasing. Then, it vanishes exactly once, and consequently there is not other

t > 0

such that

t u \in N_{ε}

. Note, in particular, that

t_{u}

is a global maximum point of

h_{u}

and

h_{u} (t_{u}) > 0

, that is,

I_{ε} (t_{u} u) > 0

. Since

t_{u} = 1

u \in N_{ε}

, we deduce that

I_{ε} (u) > 0

for every

u \in N_{ε}

. Now we can argue as in Willem (1996) in order to prove the item

(i i)

and to complete the proof.

The following result present an interesting property involving the $(P S)_{c_{ε}}$ sequence of $I_{ε}$ .

Lemma 3.3

Let $(u_{n}) \subset X_{ε}$ be a $(P S)_{c_{ε}}$ sequence for $I_{ε}$ . Then $(u_{n})$ is bounded in $X_{ε}$ .

Proof.

Let $(u_{n})$ be a $(P S)_{c_{ε}}$ sequence of $I_{ε}$ , that is

\begin{aligned} I_{ε} (u_{n}) \to c_{ε} and I_{ε}^{'} (u_{n}) \to 0 in X_{ε}^{'} as n \to \infty . \end{aligned}

Therefore, by using (2.4) we get

\begin{aligned} I_{ε} (u_{n}) - \frac{1}{p} I_{ε}^{'} (u_{n}) u_{n} & = (\frac{1}{2} - \frac{1}{p}) ‖ u_{n} ‖_{ε}^{2} + \frac{1}{p} \int_{R^{N}} (f (u_{n}) u_{n} - p F (u_{n})) d x \\ \geq (\frac{1}{2} - \frac{1}{p}) ‖ u_{n} ‖_{ε}^{2} . \end{aligned}

Then,

\begin{aligned} (\frac{1}{2} - \frac{1}{p}) ‖ u_{n} ‖_{ε}^{2} \leq I_{ε} (u_{n}) - \frac{1}{p} I_{ε}^{'} (u_{n}) u_{n} = c_{ε} + o (‖ u_{n} ‖_{ε}), \end{aligned}

where we conclude that

(u_{n})

is bounded in

X_{ε}

Lemma 3.4

Let $(u_{n}) \subset X_{ε}$ be a $(P S)_{c_{ε}}$ for $I_{ε}$ with $u_{n} ⇀ 0$ weakly in $X_{ε}$ . There is a sequence $(y_{n}) \subset R^{N}$ and constants $R, η > 0$ such that

\begin{aligned} \underset{n \to \infty}{lim inf} \int_{B_{R} (y_{n})} | u_{n} |^{2} d x \geq η > 0. \end{aligned}

Proof.

Suppose that the lemma does not hold. Since $(u_{n})$ is bounded in $X_{ε}$ , then, by Lions (1984, Lemma 1.1), we get

\begin{aligned} lim_{n \to \infty} \int_{R^{N}} | u_{n} |^{s} d x = 0 \end{aligned}

(3.1)

for all

s \in (2, 2_{* *})

. This limit combined with (2.3) yields

\begin{aligned} \int_{R^{N}} f (u_{n}) u_{n} d x \to 0. \end{aligned}

Thus, since

I_{ε}^{'} (u_{n}) u_{n} = o_{n} (1)

, we obtain

\begin{aligned} ‖ u_{n} ‖_{ε}^{2} \pm \int_{R^{N}} | \nabla u_{n} |^{p} d x = o_{n} (1) . \end{aligned}

From Theorem 2.1 for $j = 1$ , $k_{1} = p$ , $m = 2$ , $θ = 1 / 2$ , $k_{3} = 2$ and $k_{2} = 2 p / 4 - p$ there exist $C > 0$ such that

\begin{aligned} {(\int_{R^{N}} | \nabla u_{n} |^{p} d x)}^{1 / p} \leq C {(\int_{R^{N}} | Δ u_{n} |^{2} d x)}^{1 / 4} ‖ u_{n} ‖_{k_{2}}^{1 / 2} . \end{aligned}

Now, we can use the boundedness of

(u_{n})

X_{ε}

, (2.1) and (3.1) to get

\begin{aligned} \pm \int_{R^{N}} | \nabla u_{n} |^{p} d x \to 0. \end{aligned}

Thus,

‖ u_{n} ‖_{ε}^{2} \to 0

. Consequently

I_{ε} (u_{n}) \to 0

, which contradicts Lemma 3.2.

3.1. On the Autonomous Problem

In order to prove the main result of this section, we will need some basic results of the autonomous problem. Precisely, for any $ξ > 0$ we consider the following problem

\begin{aligned} {\begin{cases} Δ^{2} u \pm Δ_{p} u + ξ u = f (u) in R^{N}, \\ u \in H^{2} (R^{N}) . \end{cases} \end{aligned}

(S_ξ)

By (2.3), the above problem has a variational structure and the associated functional given by

\begin{aligned} I_{ξ} (u) = \frac{1}{2} \int_{R^{N}} (| Δ u |^{2} + ξ | u |^{2}) d x \pm \frac{1}{p} \int_{R^{N}} | \nabla u |^{p} d x - \int_{R^{N}} F (u) d x \end{aligned}

is well defined for

u \in X_{ξ} = H^{2} (R^{N})

. We denote the norm in

X_{ξ}

\begin{aligned} ‖ u ‖_{ξ}^{2} = \int_{R^{N}} (| Δ u |^{2} + ξ | u |^{2}) d x . \end{aligned}

We can show that

I_{ξ}

has the mountain pass geometry. Therefore, we can set the minimax level

c_{ξ}

in the following way

\begin{aligned} c_{ξ} = inf_{γ \in Γ_{ξ}} max_{t \in [0, 1]} I_{ξ} (γ (t)) > 0, \end{aligned}

where

Γ_{ξ} = {γ \in C ([0, 1], X_{ξ}) : γ (0) = 0, I_{ξ} (γ (1)) < 0}

. Moreover,

c_{ξ}

can be further characterized as

\begin{aligned} c_{ξ} = inf_{u \in M_{ξ}} I_{ξ} (u), \end{aligned}

where

M_{ξ}

is the Nehari manifold of

I_{ξ}

, that is,

\begin{aligned} M_{ξ} = {u \in X_{ξ} ∖ {0} : I_{ξ}^{'} (u) u = 0} . \end{aligned}

Proposition 3.5

Let $(u_{n}) \subset M_{ξ}$ be a sequence satisfying $I_{ξ} (u_{n}) \to c_{ξ}$ . Then, there exists a sequence $(y_{n}) \subset R^{N}$ such that, up to a subsequence, $v_{n} (x) = u_{n} (x + y_{n})$ converges strongly in $X_{ξ}$ . In particular, there exists a minimizer to $c_{ξ}$ .

Proof.

In similar arguments to Lemma 3.3, we can show that $(u_{n})$ is bounded in $X_{ξ}$ . Then, up to a subsequence, we may suppose that $u_{n} ⇀ u$ weakly in $X_{ξ}$ . By using a density argument, we can conclude that $u$ is a critical point of $I_{ξ}$ . Now, will divide our study in two cases.

Case 1. $u ≢ 0$ .

In this case $I_{ξ}^{'} (u) u = 0$ and therefore $u \in M_{ξ}$ . We are going to prove that

\begin{aligned} ‖ u ‖_{ξ} = lim_{n \to \infty} ‖ u_{n} ‖_{ξ} . \end{aligned}

(3.2)

Suppose, by contradiction, that (3.2) does not hold. Then by using (2.4) and Fatou’s lemma we get

\begin{aligned} c_{ξ} & \leq I_{ξ} (u) - \frac{1}{p} I_{ξ}^{'} (u) u \\ = (\frac{1}{2} - \frac{1}{p}) ‖ u ‖_{ξ}^{2} + \frac{1}{p} \int_{R^{N}} (f (u) u - p F (u)) d x \\ < \underset{n \to \infty}{lim inf} (\frac{1}{2} - \frac{1}{p}) ‖ u_{n} ‖_{ξ}^{2} + \underset{n \to \infty}{lim inf} \frac{1}{p} \int_{R^{N}} (f (u_{n}) u_{n} - p F (u_{n})) d x \\ \leq \underset{n \to \infty}{lim inf} [(\frac{1}{2} - \frac{1}{p}) ‖ u_{n} ‖_{ξ}^{2} + \frac{1}{p} \int_{R^{N}} (f (u_{n}) u_{n} - p F (u_{n})) d x] \\ = \underset{n \to \infty}{lim inf} [I_{ξ} (u_{n}) - \frac{1}{p} I_{ξ}^{'} (u_{n}) u_{n}] = c_{ξ}, \end{aligned}

which is a contradiction. Hence,

u_{n} \to u

X_{ξ}

. Consequently

I_{ξ} (u) = c_{ξ}

and

y_{n} = 0

for all

n \in N

Case 2. $u = 0$ .

In this case, by using similar arguments that Lemma 3.4, there exists $R, η > 0$ and $(y_{n}) \subset R^{N}$ such that

\begin{aligned} \underset{n \to \infty}{lim inf} \int_{B_{R} (y_{n})} | u_{n} |^{2} d x \geq η . \end{aligned}

Now, we define

\begin{aligned} v_{n} (x) = u_{n} (x + y_{n}) . \end{aligned}

Then, we can verify that

\begin{aligned} I_{ξ} (v_{n}) \to c_{ξ} and I_{ξ}^{'} (v_{n}) \to 0. \end{aligned}

It is clear that

(v_{n})

is bounded in

X_{ξ}

and there exists

v \in X_{ξ}

with

v \neq 0

such that

v_{n} ⇀ v

weakly in

X_{ξ}

. Repeating the same arguments used in Case 1, it follows that

v_{n} \to v

strongly in

X_{ξ}

and the proof of proposition is finished.

The following lemma describes a comparison between the least value of different parameters $ξ$ , which will play an important role in proving the existence result in the next subsection.

Lemma 3.6

Let $ξ_{1}, ξ_{2} > 0$ , with $ξ_{1} < ξ_{2}$ . Then $c_{ξ_{1}} < c_{ξ_{2}}$ .

Proof.

From Proposition 3.5, there exists $u \in M_{ξ_{2}}$ be such that

\begin{aligned} c_{ξ_{2}} = I_{ξ_{2}} (u) = max_{t \geq 0} I_{ξ_{2}} (t u) . \end{aligned}

Let

v = t_{0} u

be such that

\begin{aligned} I_{ξ_{1}} (v) = max_{t \geq 0} I_{ξ_{1}} (t u) . \end{aligned}

Thus

\begin{aligned} c_{ξ_{2}} & = max_{t \geq 0} I_{ξ_{2}} (t u) \\ \geq I_{ξ_{2}} (v) \\ = \frac{1}{2} \int_{R^{N}} (| Δ v |^{2} + ξ_{2} | v |^{2}) d x \pm \frac{1}{p} \int_{R^{N}} | \nabla v |^{p} d x - \int_{R^{N}} F (v) d x \\ > \frac{1}{2} \int_{R^{N}} (| Δ v |^{2} + ξ_{1} | v |^{2}) d x \pm \frac{1}{p} \int_{R^{N}} | \nabla v |^{p} d x - \int_{R^{N}} F (v) d x \\ = I_{ξ_{1}} (v) \geq c_{ξ_{1}} \end{aligned}

and the proof is over.

3.2. The Palais–Smale Condition

Lemma 3.7
Assume that $V_{\infty} < \infty$ and let $(u_{n}) \subset X_{ε}$ be a $(P S)_{d}$ sequence for $I_{ε}$ such that $u_{n} ⇀ 0$ weakly in $X_{ε}$ . If $u_{n} ↛ 0$ strongly in $X_{ε}$ , then $d \geq c_{V_{\infty}}$ , where $c_{V_{\infty}}$ is the infimum of $I_{V_{\infty}}$ over $M_{V_{\infty}}$ .
Proof.
We consider $t_{n} > 0$ such that $t_{n} u_{n} \in M_{V_{\infty}}$ . We start by proving that
$\begin{aligned} t_{0} = \underset{n \to \infty}{lim sup} t_{n} \leq 1. \end{aligned}$
Arguing by contradiction, we suppose that there exists $λ > 0$ and a subsequence, still denoted by $(t_{n})$ , such that
$\begin{aligned} t_{n} \geq 1 + λ for all n \in N . \end{aligned}$
(3.3)

From Lemma 3.3, $(u_{n})$ is bounded in $X_{ε}$ , then $I_{ε}^{'} (u_{n}) u_{n} = o_{n} (1)$ , or equivalently
$\begin{aligned} \int_{R^{N}} (| Δ u_{n} |^{2} + V (ε x) | u_{n} |^{2}) d x \pm \int_{R^{N}} | \nabla u_{n} |^{p} d x = \int_{R^{N}} f (u_{n}) u_{n} d x + o_{n} (1) . \end{aligned}$
(3.4)
Since $t_{n} u_{n} \in M_{V_{\infty}}$ , we also have that
$\begin{aligned} t_{n}^{2 - p} \int_{R^{N}} (| Δ u_{n} |^{2} + V_{\infty} | u_{n} |^{2}) d x \pm \int_{R^{N}} | \nabla u_{n} |^{p} d x = t_{n}^{1 - p} \int_{R^{N}} f (t_{n} u_{n}) u_{n} d x = \int_{R^{N}} \frac{f (t_{n} u_{n})}{(t_{n} u_{n})^{p - 1}} u_{n}^{p} d x . \end{aligned}$
(3.5)
Putting together (3.4) and (3.5), we get
$\begin{aligned} (t_{n}^{2 - p} - 1) \int_{R^{N}} | Δ u_{n} |^{2} d x + \int_{R^{N}} (t_{n}^{2 - p} V_{\infty} - V (ε x)) | u_{n} |^{2} d x = \int_{R^{N}} \frac{f (t_{n} u_{n})}{(t_{n} u_{n})^{p - 1}} u_{n}^{p} d x - \int_{R^{N}} f (u_{n}) u_{n} d x + o_{n} (1) . \end{aligned}$
By using (3.3) we obtain
$\begin{aligned} \int_{R^{N}} (V_{\infty} - V (ε x)) | u_{n} |^{2} d x \geq \int_{R^{N}} (\frac{f (t_{n} u_{n})}{(t_{n} u_{n})^{p - 1}} - \frac{f (u_{n})}{u_{n}^{p - 1}}) u_{n}^{p} d x + o_{n} (1) . \end{aligned}$
(3.6)

Now, using assumption ( V ) we can see that, given $ζ > 0$ , there exists $R = R (ζ) > 0$ such that
$\begin{aligned} V (ε x) \geq V_{\infty} - ζ for all | ε x | \geq R . \end{aligned}$
(3.7)
From this, taking into account that $u_{n} \to 0$ in $L^{2} (B_{R / ε} (0))$ and the boundedness of $(u_{n})$ in $X_{ε}$ , there exists $C_{4} > 0$ such that
$\begin{aligned} \int_{R^{N}} (V_{\infty} - V (ε x)) | u_{n} |^{2} d x & \leq V_{\infty} \int_{B_{R / ε} (0)} | u_{n} |^{2} d x + ζ \int_{R^{N} ∖ B_{R / ε} (0)} | u_{n} |^{2} d x \\ \leq o_{n} (1) + ζ C_{4} . \end{aligned}$
(3.8)
Combining (3.6) and (3.8), we have
$\begin{aligned} \int_{R^{N}} (\frac{f (t_{n} u_{n})}{(t_{n} u_{n})^{p - 1}} - \frac{f (u_{n})}{(u_{n})^{p - 1}}) u_{n}^{p} d x \leq o_{n} (1) + ζ C_{4} . \end{aligned}$
(3.9)
On the other hand, we can use the fact that $u_{n} ↛ 0$ to obtain $(y_{n}) \subset R^{N}$ and constants $\tilde{R}, η > 0$ such that
$\begin{aligned} \int_{B_{\tilde{R}} (y_{n})} | u_{n} |^{2} d x \geq η . \end{aligned}$
(3.10)
Let us consider $v_{n} (x) = u_{n} (x + y_{n})$ . Then we may assume that, up to a subsequence, $v_{n} ⇀ v$ in $X_{ε}$ . By (3.10) there exists $Ω \subset R^{N}$ with positive measure and such that $v > 0$ . From (3.3), $(f_{3})$ to (3.9), we can infer that
$\begin{aligned} 0 < \int_{Ω} (\frac{f ((1 + λ) v_{n})}{((1 + λ) v_{n})^{p - 1}} - \frac{f (v_{n})}{(v_{n})^{p - 1}}) v_{n}^{p} d x \leq o_{n} (1) + ζ C_{4} . \end{aligned}$
Taking the limit as $n \to \infty$ and applying Fatou’s lemma, we obtain
$\begin{aligned} 0 < \int_{Ω} (\frac{f ((1 + λ) v)}{((1 + λ) v)^{p - 1}} - \frac{f (v)}{{(v)}^{p - 1}}) v^{p} d x \leq ζ C_{4} . \end{aligned}$
Since $ζ > 0$ is arbitrary, we obtain a contradiction by taking $ζ \to 0$ . Therefore $t_{0} \leq 1$ .

Now, we divide the proof in two cases.

Case 1. $t_{0} < 1$ .

In this case we may suppose, without loss of generality, that $t_{n} < 1$ , for all $n \in N$ . Let us observe that
$\begin{aligned} d + o_{n} (1) & = I_{ε} (u_{n}) - \frac{1}{p} I_{ε}^{'} (u_{n}) u_{n} \\ = (\frac{1}{2} - \frac{1}{p}) ‖ u_{n} ‖_{ε}^{2} + \frac{1}{p} \int_{R^{N}} (f (u_{n}) u_{n} - p F (u_{n})) d x . \end{aligned}$
(3.11)
Recalling that $t_{n} u_{n} \in M_{V_{\infty}}$ and using (2.4) we obtain
$\begin{aligned} c_{V_{\infty}} & \leq I_{V_{\infty}} (t_{n} u_{n}) - \frac{1}{p} I_{V_{\infty}}^{'} (t_{n} u_{n}) (t_{n} u_{n}) \\ = t_{n}^{2} (\frac{1}{2} - \frac{1}{p}) ‖ u_{n} ‖_{V_{\infty}}^{2} + \frac{1}{p} \int_{R^{N}} (f (t_{n} u_{n}) (t_{n} u_{n}) - p F (t_{n} u_{n})) d x \\ \leq (\frac{1}{2} - \frac{1}{p}) ‖ u_{n} ‖_{V_{\infty}}^{2} + \frac{1}{p} \int_{R^{N}} (f (u_{n}) u_{n} - p F (u_{n})) d x . \end{aligned}$

On the other hand, we can use (3.8) in the above inequality to get
$\begin{aligned} c_{V_{\infty}} & \leq (\frac{1}{2} - \frac{1}{p}) \int_{R^{N}} (| Δ u_{n} |^{2} + V_{\infty} | u_{n} |^{2}) d x + \frac{1}{p} \int_{R^{N}} (f (u_{n}) u_{n} - p F (u_{n})) d x \\ \leq (\frac{1}{2} - \frac{1}{p}) ‖ u_{n} ‖_{ε}^{2} + \frac{1}{p} \int_{R^{N}} (f (u_{n}) u_{n} - p F (u_{n})) d x + ζ C_{4} (\frac{1}{2} - \frac{1}{p}) + o_{n} (1) . \end{aligned}$
Thus, using (3.11) we have
$\begin{aligned} c_{V_{\infty}} \leq d + o_{n} (1) + ζ C_{4} (\frac{1}{2} - \frac{1}{p}), \end{aligned}$
for any $ζ > 0$ . By taking $n \to \infty$ and from the arbitrariness of $ζ$ , we conclude that $c_{V_{\infty}} \leq d$ .

Case 2. $t_{0} = 1$ .

Up to a subsequence, we may suppose that $t_{n} \to 1$ . Taking into account that $I_{ε} (u_{n}) \to d$ , we have
$\begin{aligned} d + o_{n} (1) & = I_{ε} (u_{n}) \\ = I_{ε} (u_{n}) - I_{V_{\infty}} (t_{n} u_{n}) + I_{V_{\infty}} (t_{n} u_{n}) \\ \geq I_{ε} (u_{n}) - I_{V_{\infty}} (t_{n} u_{n}) + c_{V_{\infty}} . \end{aligned}$
Now, let us point out that
$\begin{aligned} I_{ε} (u_{n}) - I_{V_{\infty}} (t_{n} u_{n}) & = \frac{1}{2} (1 - t_{n}^{2}) \int_{R^{N}} | Δ u_{n} |^{2} d x + \frac{1}{2} \int_{R^{N}} (V (ε x) - t_{n}^{2} V_{\infty}) | u_{n} |^{2} d x \\ \pm \frac{1}{p} (1 - t_{n}^{p}) \int_{R^{N}} | \nabla u_{n} |^{p} d x + \int_{R^{N}} (F (t_{n} u_{n}) - F (u_{n})) d x \end{aligned}$
(3.12)
Note that, using (3.7) and the fact that $u_{n} \to 0$ in $L^{2} (B_{R / ε} (0))$ we get
$\begin{aligned} \int_{R^{N}} (V (ε x) - t_{n}^{2} V_{\infty}) | u_{n} |^{2} d x \\ = \int_{B_{R / ε} (0)} (V (ε x) - V_{\infty}) | u_{n} |^{2} d x + \int_{B_{R / ε}^{c} (0)} (V (ε x) - V_{\infty}) | u_{n} |^{2} d x + (1 - t_{n}^{2}) V_{\infty} \int_{R^{N}} | u_{n} |^{2} d x \\ \geq o_{n} (1) - ζ \int_{B_{R / ε}^{c} (0)} | u_{n} |^{2} d x + (1 - t_{n}^{2}) V_{\infty} \int_{R^{N}} | u_{n} |^{2} d x . \end{aligned}$
(3.13)

Applying the mean value theorem and (2.3), there exists $C_{5} > 0$ such that
$\begin{aligned} \int_{R^{N}} | F (t_{n} u_{n}) - F (u_{n}) | d x \leq C_{5} | t_{n} - 1 | \int_{R^{N}} | u_{n} |^{2} d x + C_{5} | t_{n} - 1 | \int_{R^{N}} | u_{n} |^{q} d x . \end{aligned}$
(3.14)
Thus, we can use the boundedness of $(u_{n})$ , (2.1), (2.2) and $t_{n} \to 1$ in (3.12)–(3.14) to get
$\begin{aligned} I_{ε} (u_{n}) - I_{V_{\infty}} (t_{n} u_{n}) \geq o_{n} (1) - ζ C_{6} \end{aligned}$
for some constant $C_{6} > 0$ . Therefore
$\begin{aligned} d + o_{n} (1) \geq o_{n} (1) - ζ C_{6} + c_{V_{\infty}} \end{aligned}$
and taking the limits as $n \to \infty$ and from the arbitrariness of $ζ$ we get $d \geq c_{V_{\infty}}$ .
Corollary 3.8
Suppose that $V_{\infty} = \infty$ . If $(u_{n}) \subset X_{ε}$ is a $(P S)_{d}$ sequence for $I_{ε}$ such that $u_{n} ⇀ 0$ weakly in $X_{ε}$ , then $u_{n} \to 0$ strongly in $X_{ε}$ .
Proof.
By using Proposition 3.5 we have that, for any $a > 0$ , the number $c_{a}$ is achieved by any $u \in X_{a}$ . Moreover, $lim_{a \to \infty} c_{a} = \infty$ .

Since $V_{\infty} = \infty$ we can take $a > 0$ in such way that $c_{a} > d$ and, for any given $ζ > 0$ , there exists $R > 0$ such that
$\begin{aligned} V (ε x) \geq a - ζ for all | ε x | \geq R . \end{aligned}$
(3.15)

Now, suppose by contradiction that $u_{n} ↛ 0$ in $X_{ε}$ . Then we can use (3.15) and argue as in the proof of Lemma 3.7 to conclude that $d \geq c_{a}$ , which does not make sense. Hence $u_{n} \to 0$ in $X_{ε}$ and the corollary is proved.
Proposition 3.9
For any fixed $ε > 0$ , $I_{ε}$ satisfies the Palais–Smale condition at level $c < c_{V_{\infty}}$ .
Proof.
Let $(u_{n}) \subset X_{ε}$ be such that
$\begin{aligned} I_{ε} (u_{n}) \to c and I_{ε}^{'} (u_{n}) \to 0 as n \to \infty . \end{aligned}$
From Lemma 3.3, $(u_{n})$ is bounded in $X_{ε}$ . Then, up to a subsequence we have that $u_{n} ⇀ u$ weakly in $X_{ε}$ with $u$ being a critical point of $I_{ε}$ . Moreover, we can show that $I_{ε}^{'} (u_{n} - u) \to 0$ and
$\begin{aligned} lim_{n \to \infty} I_{ε} (u_{n} - u) = c - I_{ε} (u) = d . \end{aligned}$
Recalling that $I_{ε}^{'} (u) = 0$ and by using (2.4) we get
$\begin{aligned} I_{ε} (u) = I_{ε} (u) - \frac{1}{p} I_{ε}^{'} (u) u = (\frac{1}{2} - \frac{1}{p}) ‖ u ‖_{ε}^{2} + \frac{1}{p} \int_{R^{N}} (f (u) u - p F (u)) d x \geq 0 \end{aligned}$
and therefore $d \leq c < c_{V_{\infty}}$ . If $V_{\infty} < \infty$ , it follows from Lemma 3.7 that $u_{n} \to u$ strongly in $X_{ε}$ . In view of Corollary 3.8, the same occurs in the complementary case $V_{\infty} = \infty$ . The proposition is proved.
Lemma 3.10
Let $J_{ε} : X_{ε} \to R$ be given by
$\begin{aligned} J_{ε} (u) = ‖ u_{n} ‖_{ε}^{2} \pm \int_{R^{N}} | \nabla u |^{p} d x - \int_{R^{N}} f (u) u d x . \end{aligned}$
Then, there exists $K > 0$ such that
$\begin{aligned} J_{ε}^{'} (u) u \leq - K for each u \in N_{ε} . \end{aligned}$
(3.16)
Proof.
Using (2.3), for any $u \in N_{ε}$ we have
$\begin{aligned} ‖ u ‖_{ε}^{2} \pm \int_{R^{N}} | \nabla u |^{p} d x = \int_{R^{N}} f (u) u d x \leq Υ ‖ u ‖_{L^{2} (R^{N})}^{2} + C_{Υ} ‖ u ‖_{L^{q} (R^{N})}^{q} \end{aligned}$
From (2.1), (2.2) and taking $Υ > 0$ sufficiently small we can deduce that
$\begin{aligned} 0 < (1 - Υ C_{8}) ‖ u ‖_{ε}^{2} \leq C_{7} ‖ u ‖_{ε}^{p} + C_{Υ} C_{9} ‖ u ‖_{ε}^{q} \end{aligned}$
for some constants $C_{7}, C_{8}, C_{9} > 0$ .

If $‖ u ‖ < 1$ , then $‖ u ‖_{ε}^{p} \geq ‖ u ‖_{ε}^{q}$ so we get
$\begin{aligned} {(\frac{1 - Υ C_{8}}{C_{7} + C_{Υ} C_{9}})}^{\frac{1}{p - 2}} \leq ‖ u ‖_{ε} . \end{aligned}$
Therefore, there exists $μ > 0$ such that
$\begin{aligned} μ \leq ‖ u ‖_{ε} for each u \in N_{ε} . \end{aligned}$
(3.17)
Then, for $u \in N_{ε}$ we have
$\begin{aligned} J_{ε}^{'} (u) u & = 2 ‖ u ‖_{ε}^{2} \pm p \int_{R^{N}} | \nabla u |^{p} d x - \int_{R^{N}} f (u) u d x - \int_{R^{N}} f^{'} (u) u^{2} d x \\ = 2 ‖ u ‖_{ε}^{2} - p ‖ u ‖_{ε}^{2} + p \int_{R^{N}} f (u) u d x - \int_{R^{N}} f (u) u d x - \int_{R^{N}} f^{'} (u) u^{2} d x \\ = - (p - 2) ‖ u ‖_{ε}^{2} - \int_{R^{N}} (f^{'} (u) u^{2} - (p - 1) f (u) u) d x . \end{aligned}$
Now, we can use (2.5) and (3.17) to get
$\begin{aligned} J_{ε}^{'} (u) u \leq - (p - 2) μ^{2} := - K . \end{aligned}$
The lemma is proved.
Proposition 3.11
The functional $I_{ε}$ constrained to $N_{ε}$ satisfies the $(P S)_{d}$ condition at any level $d < c_{V_{\infty}}$ .
Proof.
Let $(u_{n}) \subset N_{ε}$ be such that
$\begin{aligned} I_{ε} (u_{n}) \to d and ‖ I_{ε}^{'} (u_{n}) ‖_{*} \to 0. \end{aligned}$
Then there exists a sequence $(λ_{n}) \subset R$ such that
$\begin{aligned} I_{ε}^{'} (u_{n}) = λ_{n} J_{ε}^{'} (u_{n}) + o_{n} (1), \end{aligned}$
with $J_{ε}$ as in Lemma 3.10. Since $u_{n} \in N_{ε}$ we have that
$\begin{aligned} 0 = I_{ε}^{'} (u_{n}) u_{n} = λ_{n} J_{ε}^{'} (u_{n}) u_{n} + o_{n} (1) ‖ u_{n} ‖_{ε} . \end{aligned}$
Straightforward calculations show that $(u_{n})$ is bounded. Moreover, in view of Lemma 3.10, we may suppose that $J_{ε}^{'} (u_{n}) u_{n} \to l < 0$ . Hence, the above expression shows that $λ_{n} \to 0$ and therefore we conclude that $I_{ε}^{'} (u_{n}) \to 0$ in the dual space of $X_{ε}$ . It follows from Proposition 3.9 that $(u_{n})$ has a convergent subsequence.
Lemma 3.12
We have
$\begin{aligned} \underset{ε \to 0^{+}}{lim sup} c_{ε} \leq c_{V_{0}} . \end{aligned}$

Proof.
Let $ψ \in C_{0}^{\infty} (R^{N}, [0, 1])$ be such that $ψ (x) = 1$ , for $| x | < 1$ and $ψ (x) = 0$ , for $| x | \geq 2$ . For any $r > 0$ we define
$\begin{aligned} u_{r} (x) = ψ (\frac{x}{r}) w (x), \end{aligned}$
where $w$ is a ground state solution of the problem $(S_{V_{0}})$ given by Proposition 3.5.

Let $t_{ε, r} > 0$ be such that $t_{ε, r} u_{r} \in N_{ε}$ . Then
$\begin{aligned} c_{ε} \leq I_{ε} (t_{ε, r} u_{r}) = \frac{t_{ε, r}^{2}}{2} \int_{R^{N}} (| Δ u_{r} |^{2} + V (ε x) | u_{r} |^{2}) d x \pm \frac{t_{ε, r}^{p}}{p} \int_{R^{N}} | \nabla u_{r} |^{p} d x - \int_{R^{N}} F (t_{ε, r} u_{r}) d x . \end{aligned}$
It is easy to check that, for $r$ fixed, $t_{ε, r} \to t_{r} > 0$ as $ε \to 0^{+}$ . Moreover, without loos of generality, we may suppose that $V (0) = V_{0}$ . Hence, since $u_{r}$ has compact support, we can use Lebesgue’s theorem to get
$\begin{aligned} \underset{ε \to 0^{+}}{lim sup} c_{ε} & \leq \frac{t_{r}^{2}}{2} \int_{R^{N}} (| Δ u_{r} |^{2} + V_{0} | u_{r} |^{2}) d x \pm \frac{t_{r}^{p}}{p} \int_{R^{N}} | \nabla u_{r} |^{p} d x - \int_{R^{N}} F (t_{r} u_{r}) d x = I_{V_{0}} (t_{r} u_{r}) . \end{aligned}$
Since $w \in M_{V_{0}}$ and $u_{r} \to w$ in $H^{2} (R^{N})$ as $r \to \infty$ , we can check that $t_{r} \to 1$ as $r \to \infty$ . Thus, it follows from the above expression that
$\begin{aligned} \underset{ε \to 0^{+}}{lim sup} c_{ε} \leq lim_{r \to \infty} I_{V_{0}} (t_{r} u_{r}) = I_{V_{0}} (w) = c_{V_{0}} . \end{aligned}$
The lemma is proved.
Theorem 3.13
Assume that ( V ) and $(f_{1}) - (f_{4})$ hold. Then there exists $ε_{0} > 0$ such that, for any $ε \in (0, ε_{0})$ , problem ( Sε ) admits a ground state solution.
Proof.
From Lemma 3.12, we obtain $ε_{0} > 0$ such that $c_{ε} < c_{V_{0}}$ for any $ε \in (0, ε_{0})$ . For these values of $ε$ , since $I_{ε}$ has the mountain pass geometry, we can take a sequence $(u_{n}) \subset X_{ε}$ such that
$\begin{aligned} I_{ε} (u_{n}) \to c_{ε} and I_{ε}^{'} (u_{n}) \to 0. \end{aligned}$
Since $V_{0} < V_{\infty}$ , using Lemma 3.6, we guarantee that $c_{ε} < c_{V_{\infty}}$ . Thus, from Proposition 3.9 we get that, along a subsequence $u_{n} \to u_{ε}$ with $u_{ε}$ being a critical point of $I_{ε}$ and $I_{ε} (u_{ε}) = c_{ε}$ .
3.3. Multiple Solutions for ( Sε )

Proposition 3.14
Let $ε_{n} \to 0^{+}$ and $(u_{n}) \subset N_{ε_{n}}$ such that
$\begin{aligned} I_{ε_{n}} (u_{n}) \to c_{V_{0}} . \end{aligned}$
Then there exists a sequence $({\tilde{y}}_{n}) \subset R^{N}$ such that $v_{n} (x) = u_{n} (x + {\tilde{y}}_{n})$ has a convergent subsequence in $X_{V_{0}}$ . Moreover, up to a subsequence, $(y_{n}) = (ε_{n} {\tilde{y}}_{n})$ is such that $y_{n} \to y \in M$ .
Proof.
Since $I_{ε_{n}}^{'} (u_{n}) u_{n} = 0$ and $I_{ε_{n}} (u_{n}) \to c_{V_{0}}$ , we can proceed as in the proof of Lemma 3.3 to conclude that $(u_{n})$ is bounded in $X_{ε}$ . Moreover, since $c_{V_{0}} > 0$ , we cannot have $‖ u_{n} ‖_{ε_{n}} \to 0$ . Hence, arguing as in Lemma 3.4, we obtain a sequence $({\tilde{y}}_{n}) \subset R^{N}$ and constants $R, η > 0$ such that
$\begin{aligned} \underset{n \to \infty}{lim inf} \int_{B_{R} ({\tilde{y}}_{n})} | u_{n} |^{2} d x \geq η . \end{aligned}$
(3.18)
Let us define
$\begin{aligned} v_{n} (x) = u_{n} (x + {\tilde{y}}_{n}) . \end{aligned}$
In view of the boundedness of $(u_{n})$ in $X_{V_{0}}$ and (3.18), we may suppose, up to a subsequence that $v_{n} ⇀ v$ in $X_{V_{0}}$ for some $v \neq 0$ .

Let $t_{n} > 0$ be such that $w_{n} = t_{n} v_{n} \in M_{V_{0}}$ . Thus, by using the change of variables $z \mapsto x + {\tilde{y}}_{n}$ , $V (x) \geq V_{0}$ and the invariance by translation, we can see that
$\begin{aligned} c_{V_{0}} \leq I_{V_{0}} (w_{n}) \leq I_{ε_{n}} (t_{n} u_{n}) \leq I_{ε_{n}} (u_{n}) = c_{V_{0}} + o_{n} (1) . \end{aligned}$
which implies
$\begin{aligned} I_{V_{0}} (w_{n}) \to c_{V_{0}} . \end{aligned}$
Since $(v_{n})$ and $(w_{n})$ are bounded and $v_{n} ↛ 0$ , the sequence $(t_{n})$ is bounded. Thus, up to a subsequence, $t_{n} \to t_{0} \geq 0$ . Let us show that $t_{0} > 0$ . Otherwise, if $t_{0} = 0$ , by the boundedness of $(v_{n})$ , we get $w_{n} = t_{n} v_{n} \to 0$ in $X_{V_{0}}$ , that is $I_{V_{0}} (w_{n}) \to 0$ which contradicts with $c_{V_{0}} > 0$ . Thus $t_{0} > 0$ and, up to a subsequence, we may assume that $w_{n} ⇀ w$ in $X_{V_{0}}$ with $w = t_{0} v \neq 0$ . From Proposition 3.5, we conclude that $w_{n} \to w$ in $X_{V_{0}}$ , that is $v_{n} \to v$ in $X_{V_{0}}$ .

To conclude the proof of the proposition, we consider $y_{n} = ε_{n} {\tilde{y}}_{n}$ . Our goal is to show that $(y_{n})$ has a subsequence, still denoted by $(y_{n})$ , satisfying $y_{n} \to y$ for $y \in M$ . First of all, we claim that $(y_{n})$ is bounded. Assume by contradiction that $(y_{n})$ is not bounded, that is there exists a subsequence, still denoted by $(y_{n})$ , such that $| y_{n} | \to \infty$ . We will obtain a contradiction by considering two cases.

Case 1. $V_{\infty} = \infty$ .

Since $(u_{n}) \subset N_{ε_{n}}$ we have that
$\begin{aligned} \int_{R^{N}} (| Δ v_{n} |^{2} + V (ε_{n} x + y_{n}) | v_{n} |^{2}) d x \pm \int_{R^{N}} | \nabla v_{n} |^{p} d x = \int_{R^{N}} f (v_{n}) v_{n} d x . \end{aligned}$
In the case that the second term in the associated functional $I_{ε_{n}}$ is positive, we have
$\begin{aligned} \int_{R^{N}} V (ε_{n} x + y_{n}) | v_{n} |^{2} d x \leq \int_{R^{N}} f (v_{n}) v_{n} d x . \end{aligned}$
By applying Fatou’s lemma we obtain
$\begin{aligned} \infty \leq \underset{n \to \infty}{lim inf} \int_{R^{N}} f (v_{n}) v_{n} d x . \end{aligned}$
On the other hand, the boundedness of $(v_{n})$ , (2.3) and Sobolev embedding, imply that the right hand side in the above expression is bounded. Thus, we obtain a contradiction.

In the case that the second term in the associated functional $I_{ε_{n}}$ is negative, we can apply again Fatou’s lemma to get
$\begin{aligned} \infty \leq \underset{n \to \infty}{lim inf} (\int_{R^{N}} | \nabla v_{n} |^{p} d x + \int_{R^{N}} f (v_{n}) v_{n} d x) . \end{aligned}$
Since $X_{V_{0}} ↪ W^{1, p} (R^{N})$ , we can argue as in the previous case to get a contradiction.

Case 2. $V_{\infty} < \infty$ .

In this case, since $w_{n} \to w$ strongly in $X_{V_{0}}$ , $V_{0} < V_{\infty}$ and using the change of variable $z = x + {\tilde{y}}_{n}$ , we have
$\begin{aligned} c_{V_{0}} & = I_{V_{0}} (w) \\ < \frac{1}{2} \int_{R^{N}} (| Δ w |^{2} + V_{\infty} | w |^{2}) d x \pm \frac{1}{p} \int_{R^{N}} | \nabla w |^{p} d x - \int_{R^{N}} F (w) d x \\ \leq \underset{n \to \infty}{lim inf} (\frac{1}{2} \int_{R^{N}} (| Δ w_{n} |^{2} + V (ε_{n} x + y_{n}) | w_{n} |^{2}) d x \pm \frac{1}{p} \int_{R^{N}} | \nabla w_{n} |^{p} d x - \int_{R^{N}} F (w_{n}) d x) \\ = \underset{n \to \infty}{lim inf} (\frac{t_{n}^{2}}{2} \int_{R^{N}} (| Δ u_{n} |^{2} + V (ε_{n} z) | u_{n} |^{2}) d z \pm \frac{t_{n}^{p}}{p} \int_{R^{N}} | \nabla u_{n} |^{p} d z - \int_{R^{N}} F (t_{n} u_{n}) d z) \\ = \underset{n \to \infty}{lim inf} I_{ε_{n}} (t_{n} u_{n}) \leq \underset{n \to \infty}{lim inf} I_{ε_{n}} (u_{n}) = c_{V_{0}} \end{aligned}$
(3.19)
which is a contradiction.

Hence $(y_{n})$ is bounded and, up to a subsequence, we may assume that $y_{n} \to y$ . If $y \notin M$ then $V_{0} < V (y)$ and we have that
$\begin{aligned} c_{V_{0}} < \frac{1}{2} \int_{R^{N}} (| Δ w |^{2} + V (y) | w |^{2}) d x \pm \frac{1}{p} \int_{R^{N}} | \nabla w |^{p} d x - \int_{R^{N}} F (w) d x . \end{aligned}$
This inequality and the same kind of calculations performed in (3.19) provide a contradiction. Therefore, we can conclude that $y \in M$ .

Let $δ > 0$ be fixed and consider $ψ \in C^{\infty} (R^{+}, [0, 1])$ be a non-increasing function such that $ψ \equiv 1$ on $[0, δ / 2]$ and $ψ \equiv 0$ on $[δ, \infty)$ . For any $y \in M$ , we define
$\begin{aligned} Ψ_{ε, y} (x) = ψ (| ε x - y |) w (\frac{ε x - y}{ε}), \end{aligned}$
where $w \in X_{V_{0}}$ is a ground state solution of $(S_{V_{0}})$ given by Proposition 3.5.

If $t_{ε}$ denotes the unique positive number satisfying
$\begin{aligned} I_{ε} (t_{ε} Ψ_{ε, y}) = max_{t \geq 0} I_{ε} (t Ψ_{ε, y}) \end{aligned}$
we introduce the map $Φ_{ε} : M \to N_{ε}$ by setting
$\begin{aligned} Φ_{ε} (y) := t_{ε} Ψ_{ε, y} . \end{aligned}$

Lemma 3.15
Uniformly for $y \in M$ , we have
$\begin{aligned} lim_{ε \to 0^{+}} I_{ε} (Φ_{ε} (y)) = c_{V_{0}} . \end{aligned}$

Proof.
Assume by contradiction that there exists $δ_{0} > 0$ , $(y_{n}) \subset M$ and $ε_{n} \to 0^{+}$ such that
$\begin{aligned} | I_{ε_{n}} (Φ_{ε_{n}} (y_{n})) - c_{V_{0}} | \geq δ_{0} . \end{aligned}$
(3.20)
Since $I_{ε_{n}}^{'} (Φ_{ε_{n}} (y_{n})) Φ_{ε_{n}} (y_{n}) = 0$ , we can use the change of variable $z = ε_{n} x - y_{n} / ε_{n}$ to get
$\begin{aligned} t_{ε_{n}}^{2} \int_{R^{N}} (| Δ Ψ_{ε_{n}, y_{n}} (x) |^{2} + V (ε_{n} x) | Ψ_{ε_{n}, y_{n}} (x) |^{2}) d x \pm t_{ε_{n}}^{p} \int_{R^{N}} | \nabla Ψ_{ε_{n}, y_{n}} (x) |^{p} d x \\ = \int_{R^{N}} f (t_{ε_{n}} Ψ_{ε_{n}, y_{n}} (x)) t_{ε_{n}} Ψ_{ε_{n}, y_{n}} (x) d x \\ = \int_{R^{N}} f (t_{ε_{n}} ψ (| ε_{n} z |) w (z)) t_{ε_{n}} ψ (| ε_{n} z |) w (z) d z . \end{aligned}$
(3.21)
If $n_{0} \in N$ is such that $B_{δ / 2} (0) \subset B_{δ / 2 ε_{n}} (0)$ for all $n \geq n_{0}$ , we have
$\begin{aligned} t_{ε_{n}}^{2 - p} \int_{R^{N}} (| Δ Ψ_{ε_{n}, y_{n}} (x) |^{2} + V (ε_{n} x) | Ψ_{ε_{n}, y_{n}} (x) |^{2}) d x \pm \int_{R^{N}} | \nabla Ψ_{ε_{n}, y_{n}} (x) |^{p} d x \geq \int_{B_{δ / 2} (0)} \frac{f (t_{ε_{n}} w (z))}{(t_{ε_{n}} w (z))^{p - 1}} (w (z))^{p} d z . \end{aligned}$
(3.22)
On the other hand, by using Lebesgue’s theorem we may check that
$\begin{aligned} lim_{n \to \infty} \int_{R^{N}} (| Δ Ψ_{ε_{n}, y_{n}} (x) |^{2} + V (ε_{n} x) | Ψ_{ε_{n}, y_{n}} (x) |^{2}) d x = \int_{R^{N}} (| Δ w |^{2} + V_{0} | w |^{2}) d x \end{aligned}$
(3.23)
and
$\begin{aligned} lim_{n \to \infty} \int_{R^{N}} | \nabla Ψ_{ε_{n}, y_{n}} (x) |^{p} d x = \int_{R^{N}} | \nabla w |^{p} d x . \end{aligned}$
(3.24)

In the case that the second term on the left hand side in (3.22) is negative we have
$\begin{aligned} t_{ε_{n}}^{2 - p} \int_{R^{N}} (| Δ Ψ_{ε_{n}, y_{n}} |^{2} + V (ε_{n} x) | Ψ_{ε_{n}, y_{n}} |^{2}) d x \geq \int_{B_{δ / 2} (0)} \frac{f (t_{ε_{n}} w (z))}{(t_{ε_{n}} w (z))^{p - 1}} (w (z))^{p} d z . \end{aligned}$
If $t_{ε_{n}} \to \infty$ , we can use $(f_{3})$ to get $0 \geq \infty$ , which is absurd.

In the case that the second term on the left hand side in (3.22) is positive, we can observe that if $t_{ε_{n}} \to \infty$ , then $\int_{R^{N}} | \nabla Ψ_{ε_{n}, y_{n}} |^{p} d x \to + \infty$ , contradicting (3.24). Thus, up to a subsequence, $t_{ε_{n}} \to t_{0} \geq 0$ . If $t_{0} = 0$ , we can use (3.24), $p > 2$ and $(f_{1})$ in (3.21), to get
$\begin{aligned} \int_{R^{N}} (| Δ Ψ_{ε_{n}, y_{n}} |^{2} + V (ε_{n} x) | Ψ_{ε_{n}, y_{n}} |^{2}) d x \to 0, \end{aligned}$
contradicting (3.23). Hence, $t_{0} > 0$ . Now, we show that $t_{0} = 1$ . Letting $n \to \infty$ in (3.21), we can see that
$\begin{aligned} t_{0}^{2 - p} \int_{R^{N}} (| Δ w |^{2} + V_{0} | w |^{2}) d x \pm \int_{R^{N}} | \nabla w |^{p} d x = \int_{R^{N}} \frac{f (t_{0} w)}{(t_{0} w)^{p - 1}} w^{p} d x . \end{aligned}$
(3.25)
Since $w \in M_{V_{0}}$ we have
$\begin{aligned} \int_{R^{N}} (| Δ w |^{2} + V_{0} | w |^{2}) d x \pm \int_{R^{N}} | \nabla w |^{p} d x = \int_{R^{N}} f (w) w d x . \end{aligned}$
(3.26)
Putting together (3.25) and (3.26), we have
$\begin{aligned} (t_{0}^{2 - p} - 1) \int_{R^{N}} (| Δ w |^{2} + V_{0} | w |^{2}) d x = \int_{R^{N}} (\frac{f (t_{0} w)}{(t_{0} w)^{p - 1}} - \frac{f (w)}{w^{p - 1}}) w^{p} d x . \end{aligned}$
From $(f_{3})$ , we conclude that $t_{0} = 1$ . This fact and the Lebesgue’s theorem yield
$\begin{aligned} lim_{n \to \infty} \int_{R^{N}} F (t_{ε_{n}} Ψ_{ε_{n}, y_{n}}) d x = \int_{R^{N}} F (w) d x . \end{aligned}$
(3.27)
Thus, we can use (3.23), (3.24) and (3.27) to get
$\begin{aligned} lim_{n \to \infty} I_{ε_{n}} (Φ_{ε_{n}} (y_{n})) = I_{V_{0}} (w) = c_{V_{0}} \end{aligned}$
which contradicts (3.20). The lemma is proved.

Let us consider $ρ = ρ_{δ} > 0$ in such way that $M_{δ} \subset B_{ρ} (0)$ and define $Υ : R^{N} \to R^{N}$ by setting $Υ (x) = x$ for $| x | < ρ$ and $Υ (x) = ρ x / | x |$ for $| x | \geq ρ$ . We also consider the barycenter map $β_{ε} : N_{ε} \to R^{N}$ given by
$\begin{aligned} β_{ε} (u) = \frac{\int_{R^{N}} Υ (ε x) | u (x) |^{2} d x}{\int_{R^{N}} | u (x) |^{2} d x} . \end{aligned}$

Lemma 3.16
The function $β_{ε}$ satisfies
$\begin{aligned} lim_{ε \to 0^{+}} β_{ε} (Φ_{ε} (y)) = y uniformly for y \in M . \end{aligned}$

Proof.
Suppose, by contradiction, that the lemma is false. Then, there exists $δ_{0} > 0$ , $(y_{n}) \subset M$ and $ε_{n} \to 0^{+}$ such that
$\begin{aligned} | β_{ε_{n}} (Φ_{ε_{n}} (y_{n})) - y_{n} | \geq δ_{0} . \end{aligned}$
Using the definitions of $Φ_{ε_{n}} (y_{n})$ , $β_{ε_{n}}$ , $ψ$ and the change of variables $z = (ε_{n} x - y_{n}) / ε_{n}$ , we can write
$\begin{aligned} β_{ε_{n}} (Φ_{ε_{n}} (y_{n})) = y_{n} + \frac{\int_{R^{N}} (Υ (ε_{n} z + y_{n}) - y_{n}) | ψ (| ε_{n} z |) w (z) |^{2} d z}{\int_{R^{N}} | ψ (| ε_{n} z |) w (z) |^{2} d z} . \end{aligned}$
Taking into account $(y_{n}) \subset M \subset B_{ρ} (0)$ and $Υ |_{B_{ρ} (0)} = Id$ , we can use the above expression and the Lebesgue’s theorem to infer that
$\begin{aligned} | β_{ε_{n}} (Φ_{ε_{n}} (y_{n})) - y_{n} | = o_{n} (1), \end{aligned}$
which is a contradiction. The lemma is proved.

Following Cingolani and Lazzo (1997), we introduce the set
$\begin{aligned} Σ_{ε} = {u \in N_{ε} : I_{ε} (u) \leq c_{V_{0}} + h (ε)}, \end{aligned}$
where $h : R^{+} \to R^{+}$ is such that $h (ε) \to 0$ as $ε \to 0^{+}$ .

Given $y \in M$ , we can use Lemma 3.15 to conclude that $h (ε) = | I_{ε} (Φ_{ε} (y)) - c_{V_{0}} |$ satisfies $h (ε) \to 0$ as $ε \to 0^{+}$ . Thus, $Φ_{ε} (y) \in Σ_{ε}$ and therefore $Σ_{ε} \neq \emptyset$ , for any $ε > 0$ small.
Lemma 3.17
For any $δ > 0$ we have that
$\begin{aligned} lim_{ε \to 0^{+}} sup_{u \in Σ_{ε}} dist (β_{ε} (u), M_{δ}) = 0. \end{aligned}$

Proof.
Let $(ε_{n}) \subset R$ be such that $ε_{n} \to 0^{+}$ . For any $n \in N$ , there exists $(u_{n}) \subset Σ_{ε_{n}}$ such that
$\begin{aligned} dist (β_{ε_{n}} (u_{n}), M_{δ}) = sup_{u \in Σ_{ε_{n}}} dist (β_{ε_{n}} (u), M_{δ}) + o_{n} (1) . \end{aligned}$
Therefore, it suffices to prove that there exists $(y_{n}) \subset M_{δ}$ such that
$\begin{aligned} lim_{n \to \infty} | β_{ε_{n}} (u_{n}) - y_{n} | = 0. \end{aligned}$
(3.28)
Thus, recalling that $(u_{n}) \subset Σ_{ε_{n}} \subset N_{ε_{n}}$ , we obtain
$\begin{aligned} c_{V_{0}} & \leq max_{t \geq 0} I_{V_{0}} (t u_{n}) \\ \leq max_{t \geq 0} I_{ε_{n}} (t u_{n}) \\ = I_{ε_{n}} (u_{n}) \leq c_{V_{0}} + h (ε_{n}) \end{aligned}$
from which follows that $I_{ε_{n}} (u_{n}) \to c_{V_{0}}$ . Thus, we may invoke Proposition 3.14 to obtain a sequence $({\tilde{y}}_{n}) \subset R^{N}$ such that $y_{n} = ε_{n} {\tilde{y}}_{n} \in M_{δ}$ for $n$ sufficiently large. Hence
$\begin{aligned} β_{ε_{n}} (u_{n}) = \frac{\int_{R^{N}} Υ (ε_{n} x) | u_{n} (x) |^{2} d x}{\int_{R^{N}} | u_{n} (x) |^{2} d x} = y_{n} + \frac{\int_{R^{N}} (Υ (ε_{n} z + y_{n}) - y_{n}) | u_{n} (z + {\tilde{y}}_{n}) |^{2} d z}{\int_{R^{N}} | u_{n} (z + {\tilde{y}}_{n}) |^{2} d z} . \end{aligned}$
Since $ε_{n} z + y_{n} \to y \in M$ and from the strong convergence of $(u_{n} (\cdot + {\tilde{y}}_{n}))$ in $X_{V_{0}}$ , we have that $β_{ε_{n}} (u_{n}) = y_{n} + o_{n} (1)$ and therefore the sequence $(y_{n})$ satisfies (3.28). The lemma is proved.

Now we show that ( Sε ) admits at least ${cat}_{M_{δ}} (M)$ solutions. In order to achieve our aim, we recall the following result for critical points involving Ljusternik–Schnirelmann category. The proof of this abstract result can be seen in Ghoussoub (1993, Corollary 4.17).
Theorem 3.18
Let $I$ be a $C^{1}$ -functional defined on a $C^{1}$ -Finsler manifold $V$ . If I is bounded from below and satisfies the Palais–Smale condition, then $I$ has at least ${cat}_{V} (V)$ distinct critical points.

With a view to apply Theorem 3.18, the following abstract lemma provides a very useful tool since relates the topology of some sublevel of a functional to the topology of some subset of the space $R^{N}$ ; see Benci and Cerami (1994, Lemma 4.3).
Lemma 3.19
Let $Ω$ , $Ω_{1}$ and $Ω_{2}$ be closed sets with $Ω_{1} \subset Ω_{2}$ . Let $β : Ω \to Ω_{2}$ , $Φ : Ω_{1} \to Ω$ be two continuous maps such that $β \circ Φ$ is homotopically equivalent to the embedding $ι : Ω_{1} \to Ω_{2}$ . Then ${cat}_{Ω} (Ω) \geq {cat}_{Ω_{2}} (Ω_{1})$ .
Proof Proof of Theorem 1.1.

Given $δ > 0$ we can use Lemma 3.15, 3.16, 3.17, and argue as in Cingolani and Lazzo (1997, Section 6) to obtain $ε_{δ} > 0$ such that, for any $ε \in (0, ε_{δ})$ , the diagram

\begin{aligned} M \overset{Φ_{ε}}{⟶} Σ_{ε} \overset{β_{ε}}{⟶} M_{δ} \end{aligned}

is well defined and

β_{ε} \circ Φ_{ε}

is homotopically equivalent to the embedding

ι : M \to M_{δ}

. This fact and Lemma 3.19 imply that

{cat}_{Σ_{ε}} (Σ_{ε}) \geq {cat}_{M_{δ}} (M)

. Using the definition of

Σ_{ε}

, Proposition 3.11 and taking

ε_{δ}

small if necessary we guarantee that

I_{ε}

satisfies the Palais–Smale condition in

Σ_{ε}

. Hence, by using Theorem 3.18, we obtain that

I_{ε}

has at least

{cat}_{Σ_{ε}} (Σ_{ε})

critical points on

Σ_{ε}

. Then, we can infer that

I_{ε}

admits at least

{cat}_{M_{δ}} (M)

critical points. The theorem is proved.

4. Critical Case $(β = 1)$

In this section we present the proof of Theorem 1.2. Since many calculations are adaptions to that present in the early section, we will emphasis only the differences between the subcritical and critical case.

We first consider the critical version of problem $(S_{ξ})$ for $ξ = V_{0}$ , namely

\begin{aligned} {\begin{cases} Δ^{2} u \pm Δ_{p} u + V_{0} u = f (u) + | u |^{2_{* *} - 2} u in R^{N}, \\ u \in H^{2} (R^{N}), \end{cases} \end{aligned}

(C_{V
₀})

whose solutions are related with the critical points of

{\tilde{I}}_{V_{0}} : X_{V_{0}} \to R

defined as

\begin{aligned} {\tilde{I}}_{V_{0}} (u) = \frac{1}{2} \int_{R^{N}} (| Δ u |^{2} + V_{0} | u |^{2}) d x \pm \frac{1}{p} \int_{R^{N}} | \nabla u |^{p} d x - \int_{R^{N}} F (u) d x - \frac{1}{2_{* *}} \int_{R^{N}} | u |^{2_{* *}} d x . \end{aligned}

As in the subcritical case, we can show that ${\tilde{I}}_{V_{0}}$ has the mountain pass geometry and therefore we can set the minimax level ${\tilde{c}}_{V_{0}}$ in the following way

\begin{aligned} {\tilde{c}}_{V_{0}} := inf_{γ \in {\tilde{Γ}}_{V_{0}}} max_{t \in [0, 1]} {\tilde{I}}_{V_{0}} (γ (t)), \end{aligned}

where

{\tilde{Γ}}_{V_{0}} := {γ \in C ([0, 1], X_{V_{0}}) : γ (0) = 0, {\tilde{I}}_{V_{0}} (γ (1)) < 0}

Moreover, ${\tilde{c}}_{V_{0}}$ can be further characterized as

\begin{aligned} {\tilde{c}}_{V_{0}} = inf_{u \in {\tilde{M}}_{V_{0}}} {\tilde{I}}_{V_{0}} (u), \end{aligned}

where

{\tilde{M}}_{V_{0}}

is the Nehari manifold of

{\tilde{I}}_{V_{0}}

, that is,

\begin{aligned} {\tilde{M}}_{V_{0}} = {u \in X_{V_{0}} ∖ {0} : {\tilde{I}}_{V_{0}}^{'} (u) u = 0} . \end{aligned}

We denote by $S$ the best constant of the immersion, $D^{2, 2} (R^{N})$ into $L^{2_{* *}} (R^{N})$ , that is,

\begin{aligned} S = inf_{\begin{matrix} u \in D^{2, 2} (R^{N}) \\ u \neq 0 \end{matrix}} \frac{\int_{R^{N}} | Δ u |^{2} d x}{{(\int_{R^{N}} | u |^{2_{* *}} d x)}^{2 / 2_{* *}}} . \end{aligned}

The infimum

S > 0

is achieved by the functions

\begin{aligned} u_{δ, y} (x) = \frac{C_{N} δ^{\frac{N - 4}{4}}}{[δ + | x - y |^{2}]^{\frac{N - 4}{2}}}, \end{aligned}

for any

δ > 0

and

y \in R^{N}

(see Edmunds et al., 1990). Now, we define the following constant

\begin{aligned} \hat{S} = inf_{\begin{matrix} u \in H^{2} (R^{N}) \\ u \neq 0 \end{matrix}} \frac{\int_{R^{N}} | Δ u |^{2} d x + \int_{R^{N}} | \nabla u |^{p} d x}{{(\int_{R^{N}} | u |^{2_{* *}} d x)}^{2 / 2_{* *}}} . \end{aligned}

Note that

\begin{aligned} \frac{\int_{R^{N}} | Δ u |^{2} d x}{{(\int_{R^{N}} | u |^{2_{* *}} d x)}^{2 / 2_{* *}}} \leq \frac{\int_{R^{N}} | Δ u |^{2} d x + \int_{R^{N}} | \nabla u |^{p} d x}{{(\int_{R^{N}} | u |^{2_{* *}} d x)}^{2 / 2_{* *}}} \end{aligned}

implies

\begin{aligned} 0 < S < \hat{S} . \end{aligned}

From this we can deduce the following result:

Lemma 4.1

Let $(u_{n}) \subset {\tilde{M}}_{V_{0}}$ be a sequence such that ${\tilde{I}}_{V_{0}} (u_{n}) \to {\tilde{c}}_{V_{0}}$ with

\begin{aligned} {\tilde{c}}_{V_{0}} < min {(\frac{1}{p} - \frac{1}{2_{* *}}) {\hat{S}}^{N / 4}, (\frac{1}{2} - \frac{1}{p}) min {\frac{1}{2 C^{2}}, \frac{S}{2}}^{p / (p - 2)}, \frac{p - 2}{2 p}}, \end{aligned}

for some

C > 0

. Then we have either

(i)

$‖ u_{n} ‖_{V_{0}} \to 0$ , or

(ii)

there exists a sequence $(y_{n}) \subset R^{N}$ and constants $R, η > 0$ such that

\begin{aligned} \underset{n \to \infty}{lim inf} \int_{B_{R} (y_{n})} | u_{n} |^{2} d x \geq η . \end{aligned}

Proof.

Suppose that $(i i)$ does not hold. Since $(u_{n})$ is bounded in $X_{V_{0}}$ , then, by Lions (1984, Lemma I.1), we get

\begin{aligned} lim_{n \to \infty} \int_{R^{N}} | u_{n} |^{s} d x = 0 \end{aligned}

for all

s \in (2, 2_{* *})

. Given

Υ > 0

, we can use (2.3) to get

\begin{aligned} 0 \leq | \int_{R^{N}} f (u_{n}) u_{n} d x | \leq Υ \int_{R^{N}} | u_{n} |^{2} d x + C_{Υ} \int_{R^{N}} | u_{n} |^{q} d x \end{aligned}

for some constant

C_{Υ} > 0

. Since

(u_{n})

is bounded in

L^{2} (R^{N})

u_{n} \to 0

L^{q} (R^{N})

and

Υ

is arbitrary, we conclude that

\int_{R^{N}} f (u_{n}) u_{n} d x = o_{n} (1)

. Thus, from

{\tilde{I}}_{V_{0}}^{'} (u_{n}) u_{n} = 0

, we obtain

\begin{aligned} \int_{R^{N}} (| Δ u_{n} |^{2} + V_{0} | u_{n} |^{2}) d x \pm \int_{R^{N}} | \nabla u_{n} |^{p} d x = \int_{R^{N}} | u_{n} |^{2_{* *}} d x + o_{n} (1) . \end{aligned}

(4.1)

In the case that the second term on the left hand side in (4.1) is positive, we have

\begin{aligned} ‖ u_{n} ‖_{V_{0}}^{2} + \int_{R^{N}} | \nabla u_{n} |^{p} d x = \int_{R^{N}} | u_{n} |^{2_{* *}} d x + o_{n} (1) . \end{aligned}

Taking a subsequence, we obtain

l \geq 0

such that

\begin{aligned} ‖ u_{n} ‖_{V_{0}}^{2} + \int_{R^{N}} | \nabla u_{n} |^{p} d x \to l and \int_{R^{N}} | u_{n} |^{2_{* *}} d x \to l . \end{aligned}

(4.2)

From (2.3), we also have

\int_{R^{N}} F (u_{n}) d x \to 0

. Thus

\begin{aligned} {\tilde{c}}_{V_{0}} + o_{n} (1) = {\tilde{I}}_{V_{0}} (u_{n}) \geq \frac{1}{p} (‖ u_{n} ‖_{V_{0}}^{2} + \int_{R^{N}} | \nabla u_{n} |^{p} d x) - \frac{1}{2_{* *}} \int_{R^{N}} | u_{n} |^{2_{* *}} d x + o_{n} (1) . \end{aligned}

Letting

n \to \infty

and using (4.2), we get

\begin{aligned} {\tilde{c}}_{V_{0}} \geq (\frac{1}{p} - \frac{1}{2_{* *}}) l . \end{aligned}

(4.3)

Recalling the definition of

\hat{S}

we have

\begin{aligned} ‖ u_{n} ‖_{V_{0}}^{2} + \int_{R^{N}} | \nabla u_{n} |^{p} d x & \geq \int_{R^{N}} | Δ u_{n} |^{2} d x + \int_{R^{N}} | \nabla u_{n} |^{p} d x \\ \geq \hat{S} {(\int_{R^{N}} | u_{n} |^{2_{* *}} d x)}^{2 / 2_{* *}} . \end{aligned}

Taking the limit we conclude that

l \geq \hat{S} l^{2 / 2_{* *}}

. If

l > 0

we obtain from (4.3)

\begin{aligned} {\tilde{c}}_{V_{0}} \geq (\frac{1}{p} - \frac{1}{2_{* *}}) {\hat{S}}^{N / 4}, \end{aligned}

which is a contradiction. Hence

l = 0

and therefore

u_{n} \to 0

X_{V_{0}}

In the case that the second term on the left hand side in (4.1) is negative, we have

\begin{aligned} \int_{R^{N}} (| Δ u_{n} |^{2} + V_{0} | u_{n} |^{2}) d x = \int_{R^{N}} | \nabla u_{n} |^{p} d x + \int_{R^{N}} | u_{n} |^{2_{* *}} d x + o_{n} (1) . \end{aligned}

Taking a subsequence, we obtain

\hat{l} \geq 0

such that

\begin{aligned} ‖ u_{n} ‖_{V_{0}}^{2} \to \hat{l} and \int_{R^{N}} | \nabla u_{n} |^{p} d x + \int_{R^{N}} | u_{n} |^{2_{* *}} d x \to \hat{l} \end{aligned}

(4.4)

with

\hat{l} = l_{1} + l_{2}

\int_{R^{N}} | \nabla u_{n} |^{p} d x \to l_{1}

and

\int_{R^{N}} | u_{n} |^{2_{* *}} d x \to l_{2}

Since $\int_{R^{N}} F (u_{n}) d x \to 0$ we have

\begin{aligned} {\tilde{c}}_{V_{0}} + o_{n} (1) = {\tilde{I}}_{V_{0}} (u_{n}) \geq \frac{1}{2} ‖ u_{n} ‖_{V_{0}}^{2} - \frac{1}{p} (\int_{R^{N}} | \nabla u_{n} |^{p} d x + \int_{R^{N}} | u_{n} |^{2_{* *}} d x) + o_{n} (1) . \end{aligned}

Letting

n \to \infty

and using (4.4) we get

\begin{aligned} {\tilde{c}}_{V_{0}} \geq (\frac{1}{2} - \frac{1}{p}) \hat{l} . \end{aligned}

(4.5)

\hat{l} \geq 1

, then

\begin{aligned} {\tilde{c}}_{V_{0}} \geq \frac{p - 2}{2 p}, \end{aligned}

which is a contradiction by the hypotheses. Then, we have

0 \leq \hat{l} < 1

On the other hand, we can use the definition of $S$ to get

\begin{aligned} ‖ u_{n} ‖_{V_{0}}^{2} \geq \int_{R^{N}} | Δ u_{n} |^{2} d x \geq S {(\int_{R^{N}} | u_{n} |^{2_{* *}} d x)}^{2 / 2_{* *}} . \end{aligned}

(4.6)

Now, since $X_{V_{0}} ↪ W^{1, p} (R^{N})$ , there exists $C > 0$ such that

\begin{aligned} {(\int_{R^{N}} | \nabla u_{n} |^{p} d x)}^{1 / p} \leq C ‖ u_{n} ‖_{V_{0}} . \end{aligned}

Then,

\begin{aligned} ‖ u_{n} ‖_{V_{0}}^{2} \geq \frac{1}{C^{2}} {(\int_{R^{N}} | \nabla u_{n} |^{p} d x)}^{2 / p} . \end{aligned}

(4.7)

Using (4.6) and (4.7) we obtain

\begin{aligned} ‖ u_{n} ‖_{V_{0}}^{2} \geq \frac{1}{2} min {\frac{1}{C^{2}}, S} {{(\int_{R^{N}} | \nabla u_{n} |^{p} d x)}^{2 / p} + {(\int_{R^{N}} | u_{n} |^{2_{* *}} d x)}^{2 / 2_{* *}}} . \end{aligned}

Taking the limit and using (4.4), we have

\begin{aligned} \hat{l} \geq \frac{1}{2} min {\frac{1}{C^{2}}, S} (l_{1}^{2 / p} + l_{2}^{2 / 2_{* *}}) . \end{aligned}

Suppose that

0 < \hat{l} < 1

. In this case

0 < l_{1}, l_{2} < 1

. Then, since

2 < p < 2_{* *}

, we have

\begin{aligned} \hat{l} \geq \frac{1}{2} min {\frac{1}{C^{2}}, S} {\hat{l}}^{2 / p} . \end{aligned}

Since $\hat{l} > 0$ , we obtain

\begin{aligned} {\hat{l}}^{(p - 2) / p} \geq min {\frac{1}{2 C^{2}}, \frac{S}{2}} \end{aligned}

and from (4.5) we have that

\begin{aligned} {\tilde{c}}_{V_{0}} \geq (\frac{1}{2} - \frac{1}{p}) min {\frac{1}{2 C^{2}}, \frac{S}{2}}^{p / (p - 2)}, \end{aligned}

which is a contradiction. Hence

\hat{l} = 0

and therefore

(i)

holds.

Proposition 4.2

There exists $σ^{*} > 0$ such that for all $σ > σ^{*}$

\begin{aligned} {\tilde{c}}_{V_{0}} < min {(\frac{1}{p} - \frac{1}{2_{* *}}) {\hat{S}}^{N / 4}, (\frac{1}{2} - \frac{1}{p}) min {\frac{1}{2 C^{2}}, \frac{S}{2}}^{p / (p - 2)}, \frac{p - 2}{2 p}} . \end{aligned}

Proof.

Since ${\tilde{I}}_{V_{0}}$ has the mountain pass geometry, there exists $e \in X_{V_{0}}$ such that ${\tilde{I}}_{V_{0}} (e) < 0$ . We define $γ (t) = t e$ for $t \in [0, 1]$ . Then $γ \in {\tilde{Γ}}_{V_{0}}$ and by using $(f_{4})$ we get

\begin{aligned} 0 < {\tilde{c}}_{V_{0}} \leq max_{t \in [0, 1]} {\tilde{I}}_{V_{0}} (γ (t)) \leq max_{t \in [0, 1]} {\frac{t^{2}}{2} ‖ e ‖_{V_{0}}^{2} \pm \frac{t^{p}}{p} \int_{R^{N}} | \nabla e |^{p} d x - σ \frac{t^{r}}{r} \int_{R^{N}} | e |^{r} d x} . \end{aligned}

In the case that the second term in the associated functional

{\tilde{I}}_{V_{0}}

is positive, we have

\begin{aligned} 0 < {\tilde{c}}_{V_{0}} & \leq max_{t \in [0, 1]} {\frac{t^{2}}{2} ‖ e ‖_{V_{0}}^{2} + \frac{t^{p}}{p} \int_{R^{N}} | \nabla e |^{p} d x - σ \frac{t^{r}}{r} \int_{R^{N}} | e |^{r} d x} \\ \leq max_{t \geq 0} {\frac{t^{2}}{2} (‖ e ‖_{V_{0}}^{2} + ‖ \nabla e ‖_{L^{p} (R^{N})}^{p}) - σ \frac{t^{r}}{r} \int_{R^{N}} | e |^{r} d x} \\ = (\frac{1}{2} - \frac{1}{r}) \frac{1}{σ^{\frac{2}{r - 2}}} \frac{{(‖ e ‖_{V_{0}}^{2} + ‖ \nabla e ‖_{L^{p} (R^{N})}^{p})}^{\frac{r}{r - 2}}}{‖ e ‖_{L^{r} (R^{N})}^{\frac{2 r}{r - 2}}} . \end{aligned}

Thus,

{\tilde{c}}_{V_{0}} < (1 / p - 1 / 2_{* *}) {\hat{S}}^{N / 4}

, for all

σ > σ^{*}

, where

\begin{aligned} σ^{*} = {[\frac{\frac{1}{2} - \frac{1}{r}}{(\frac{1}{p} - \frac{1}{2_{* *}}) {\hat{S}}^{N / 4}}]}^{\frac{r - 2}{2}} \frac{{(‖ e ‖_{V_{0}}^{2} + ‖ \nabla e ‖_{L^{p} (R^{N})}^{p})}^{r / 2}}{‖ e ‖_{L^{r} (R^{N})}^{r}} . \end{aligned}

In the case that the second term in the associated functional

{\tilde{I}}_{V_{0}}

is negative, we have

\begin{aligned} 0 < {\tilde{c}}_{V_{0}} & \leq max_{t \geq 0} {\frac{t^{2}}{2} ‖ e ‖_{V_{0}}^{2} - σ \frac{t^{r}}{r} \int_{R^{N}} | e |^{r} d x} \\ = (\frac{1}{2} - \frac{1}{r}) \frac{1}{σ^{\frac{2}{r - 2}}} \frac{‖ e ‖_{V_{0}}^{\frac{2 r}{r - 2}}}{‖ e ‖_{L^{r} (R^{N})}^{\frac{2 r}{r - 2}}} . \end{aligned}

Thus,

{\tilde{c}}_{V_{0}} < min {(\frac{1}{2} - \frac{1}{p}) min {\frac{1}{2 C^{2}}, \frac{S}{2}}^{p / (p - 2)}, \frac{p - 2}{2 p}}

, for all

σ > σ^{*}

, where

\begin{aligned} σ^{*} = {[\frac{\frac{1}{2} - \frac{1}{r}}{min {(\frac{1}{2} - \frac{1}{p}) min {\frac{1}{2 C^{2}}, \frac{S}{2}}^{p / (p - 2)}, \frac{p - 2}{2 p}}}]}^{\frac{r - 2}{2}} \frac{‖ e ‖_{V_{0}}^{r}}{‖ e ‖_{L^{r} (R^{N})}^{r}} . \end{aligned}

The proof is complete.

The proof of the next result is in the same spirit of Proposition 3.5, but in this case we need to use Lemma 4.1.

Proposition 4.3

Let $(u_{n}) \subset {\tilde{M}}_{V_{0}}$ be a sequence satisfying ${\tilde{I}}_{V_{0}} (u_{n}) \to {\tilde{c}}_{V_{0}}$ . Then, there exists a sequence $(y_{n}) \subset R^{N}$ such that, up to a subsequence, $v_{n} (x) = u_{n} (x + y_{n})$ converges strongly in $X_{V_{0}}$ . In particular, there exists a minimizer to ${\tilde{c}}_{V_{0}}$ .

In order to obtain solutions for the critical version on problem ( Sε ) we will consider the problem (C_ε) and to look for critical points of the functional ${\tilde{I}}_{ε} : X_{ε} \to R$ given by

\begin{aligned} {\tilde{I}}_{ε} (u) = \frac{1}{2} ‖ u ‖_{ε}^{2} \pm \int_{R^{N}} | \nabla u |^{p} d x - \int_{R^{N}} F (u) d x - \frac{1}{2_{* *}} \int_{R^{N}} | u |^{2_{* *}} d x . \end{aligned}

The critical points of ${\tilde{I}}_{ε}$ belong to the Nehari manifold

\begin{aligned} {\tilde{N}}_{ε} = {u \in X_{ε} ∖ {0} : {\tilde{I}}_{ε}^{'} (u) u = 0} . \end{aligned}

and the ground state level is given by

\begin{aligned} {\tilde{c}}_{ε}^{*} = inf_{u \in {\tilde{N}}_{ε}} {\tilde{I}}_{ε} (u) . \end{aligned}

As in Lemma 3.1, the functional ${\tilde{I}}_{ε}$ satisfies the mountain pass geometry. Hence there exists a Palais–Smale sequence $(u_{n}) \subset X_{ε}$ at level ${\tilde{c}}_{ε}$ . The minimax level ${\tilde{c}}_{ε}$ is given by

\begin{aligned} {\tilde{c}}_{ε} = inf_{γ \in {\tilde{Γ}}_{ε}} max_{t \in [0, 1]} {\tilde{I}}_{ε} (γ (t)), \end{aligned}

where

{\tilde{Γ}}_{ε} = {γ \in C ([0, 1], X_{ε}) : γ (0) = 0, {\tilde{I}}_{ε} (γ (1)) < 0}

Also, using $(f_{3})$ we can show that

\begin{aligned} 0 < {\tilde{c}}_{ε} = {\tilde{c}}_{ε}^{*} = inf_{u \in X_{ε} ∖ {0}} max_{t \geq 0} {\tilde{I}}_{ε} (t u) = {\tilde{c}}_{ε}^{* *} . \end{aligned}

As before, the Palais–Smale condition for the functional ${\tilde{I}}_{ε}$ is related with $V_{\infty}$ . When this quantity is finite we define the limit functional ${\tilde{I}}_{V_{\infty}} : X_{V_{\infty}} \to R$ as being

\begin{aligned} {\tilde{I}}_{V_{\infty}} (u) = \frac{1}{2} ‖ u ‖_{V_{\infty}}^{2} \pm \int_{R^{N}} | \nabla u |^{p} d x - \int_{R^{N}} F (u) d x - \frac{1}{2_{* *}} \int_{R^{N}} | u |^{2_{* *}} d x . \end{aligned}

and its ground state level

\begin{aligned} {\tilde{c}}_{V_{\infty}} = inf_{u \in {\tilde{M}}_{V_{\infty}}} {\tilde{I}}_{V_{\infty}} (u), \end{aligned}

where

{\tilde{M}}_{V_{\infty}} = {u \in X_{V_{\infty}} ∖ {0} : {\tilde{I}}_{V_{\infty}}^{'} (u) u = 0} .

If $V_{\infty} = \infty$ , we set ${\tilde{c}}_{V_{\infty}} = \infty$ .

Now, we can argue as in the Section 3.2 to get a compactness result for the functional ${\tilde{I}}_{ε}$ . Hence, the following result holds.

Proposition 4.4

Let

\begin{aligned} c^{*} = min {{\tilde{c}}_{V_{\infty}}, (\frac{1}{p} - \frac{1}{2_{* *}}) {\hat{S}}^{N / 4}, (\frac{1}{2} - \frac{1}{p}) min {\frac{1}{2 C^{2}}, \frac{S}{2}}^{p / (p - 2)}, \frac{p - 2}{2 p}} . \end{aligned}

The functional

{\tilde{I}}_{ε}

constrained to

{\tilde{N}}_{ε}

satisfies the

(P S)_{c}

condition at any level

c < c^{*}

. Moreover, critical points of

{\tilde{I}}_{ε}

constrained to

N_{ε}

are critical points of

{\tilde{I}}_{ε}

X_{ε}

Since $V_{0} < V_{\infty}$ , we can argue as in Lemma 3.6 to get ${\tilde{c}}_{V_{0}} < {\tilde{c}}_{V_{\infty}}$ . Now, by using Proposition 4.2, we can conclude that ${\tilde{c}}_{V_{0}} < c^{*}$ .

Arguing as in Lemma 3.12 we can verify that

\begin{aligned} \underset{ε \to 0^{+}}{lim sup} {\tilde{c}}_{ε} \leq {\tilde{c}}_{V_{0}} . \end{aligned}

We are now ready to prove our second multiplicity result.

Proof Proof of Theorem 1.2.

Since the proof is very similar to that presented for Theorem 1.1, we only sketch it. Fix $δ$ and choose $\tilde{ψ} \in C^{\infty} (R^{+}, [0, 1])$ such that $\tilde{ψ} (s) = 1$ if $0 \leq s \leq δ / 2$ and $\tilde{ψ} (s) = 0$ if $s \geq δ$ . Let $\tilde{w} \in X_{V_{0}}$ be the solution of $(C_{V_{0}})$ given by Proposition 4.3 and define, for each $y \in M$ ,

\begin{aligned} {\tilde{Ψ}}_{ε, y} (x) = \tilde{ψ} (| ε x - y |) \tilde{w} (\frac{ε x - y}{ε}) . \end{aligned}

We introduce the map

{\tilde{Φ}}_{ε} : M \to {\tilde{N}}_{ε}

by setting

\begin{aligned} {\tilde{Φ}}_{ε} (y) := {\tilde{t}}_{ε} {\tilde{Ψ}}_{ε, y}, \end{aligned}

where

{\tilde{t}}_{ε}

is the unique positive number satisfying

\begin{aligned} {\tilde{I}}_{ε} ({\tilde{t}}_{ε} {\tilde{Ψ}}_{ε, y}) = max_{t \geq 0} {\tilde{I}}_{ε} (t {\tilde{Ψ}}_{ε, y}) . \end{aligned}

The following holds

\begin{aligned} lim_{ε \to 0^{+}} {\tilde{I}}_{ε} ({\tilde{Φ}}_{ε} (y)) = {\tilde{c}}_{V_{0}} uniformly for y \in M . \end{aligned}

Let $Υ : R^{N} \to R^{N}$ be the function defined in Subsection 3.3 and consider the barycenter map ${\tilde{β}}_{ε} : {\tilde{N}}_{ε} \to R^{N}$ given by

\begin{aligned} {\tilde{β}}_{ε} (u) = \frac{\int_{R^{N}} Υ (ε x) | u (x) |^{2} d x}{\int_{R^{N}} | u (x) |^{2} d x} . \end{aligned}

As before we can check that

\begin{aligned} lim_{ε \to 0^{+}} {\tilde{β}}_{ε} ({\tilde{Φ}}_{ε} (y)) = y uniformly for y \in M \end{aligned}

and

\begin{aligned} lim_{ε \to 0^{+}} sup_{u \in {\tilde{Σ}}_{ε}} dist ({\tilde{β}}_{ε} (u), M_{δ}) = 0, \end{aligned}

where

\begin{aligned} {\tilde{Σ}}_{ε} = {u \in {\tilde{N}}_{ε} : {\tilde{I}}_{ε} (u) \leq {\tilde{c}}_{V_{0}} + \tilde{h} (ε)}, \end{aligned}

and

\tilde{h} : [0, \infty) \to [0, \infty)

satisfies

\tilde{h} (ε) \to 0

ε \to 0^{+}

The above equations provide $ε_{δ} > 0$ such that, for any $ε \in (0, ε_{δ})$ , the diagram

\begin{aligned} M \overset{{\tilde{Φ}}_{ε}}{⟶} {\tilde{Σ}}_{ε} \overset{{\tilde{β}}_{ε}}{⟶} M_{δ} \end{aligned}

is well defined and

{\tilde{β}}_{ε} \circ {\tilde{Φ}}_{ε}

is homotopically equivalent to the embedding

ι : M \to M_{δ}

. Hence we conclude that

{cat}_{{\tilde{Σ}}_{ε}} ({\tilde{Σ}}_{ε}) \geq {cat}_{M_{δ}} (M)

In view of Proposition 4.4 and recalling that

{\tilde{c}}_{V_{0}} < c^{*}

we may suppose that

ε_{δ}

is small in such that

{\tilde{I}}_{ε}

satisfies the Palais–Smale condition in

{\tilde{Σ}}_{ε}

. The proof now follows from Ljusternik–Schnirelmann theory and the same arguments used in the subcritical case.

Footnotes

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Giovany M. Figueiredo was supported by CNPq, Capes and Fapdf — Brazil. Segundo Manuel. A. Salirrosas was supported by CNPq, Conselho Nacional de Desenvolvimento Científico e Tecnológico — Brazil (150632/2022-3).

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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A Result of Multiplicity of Nontrivial Solutions to an Elliptic Kirchhoff–Boussinesq Equation Type in R N

Abstract

Keywords

1. Introduction

4. Critical Case ( β = 1 )

Footnotes

Funding

Declaration of Conflicting Interests

References

4. Critical Case $(β = 1)$