Multiple solutions for a class of quasilinear problems involving variable exponents

Abstract

In this paper we prove the multiplicity of solutions for a class of quasilinear problems in $R^{N}$ involving variable exponents. The main tool used in the proof are the variational methods, Ekeland’s variational principle and some properties related to Nehari manifold.

Keywords

existence multiplicity variable exponents variational methods

1. Introduction

In this paper, we consider the existence and multiplicity of solutions for the following class of quasilinear problems involving variable exponents $\begin{matrix} (P_{λ, k}) & \{\begin{matrix} - Δ_{p (x)} u + | u |^{p (x) - 2} u = λ g (k^{- 1} x) | u |^{q (x) - 2} u + f (k^{- 1} x) | u |^{r (x) - 2} u, R^{N}, \\ u \in W^{1, p (x)} (R^{N}), \end{matrix} \end{matrix}$ where λ and k are nonnegative parameters with $k \in N$ , the operator $Δ_{p (x)} u = div (| \nabla u |^{p (x) - 2} \nabla u)$ is called $p (x)$ -Laplacian, which is a natural extension of the p-Laplace operator, with p being a positive constant. We assume that $p, q, r : R^{N} \to R$ are positive Lipschitz continuous functions, which are $Z^{N}$ -periodic and verify $\begin{matrix} (p_{1}) & 1 < p_{-} ⩽ p (x) ⩽ p_{+} < q_{-} ⩽ q (x) ⩽ r (x) ≪ p^{*} (x) a.e. in R^{N}, \end{matrix}$ where $p_{+} = ess {sup}_{x \in R^{N}} p (x)$ , $p_{-} = ess {inf}_{x \in R^{N}} p (x)$ and $\begin{matrix} p^{*} (x) = \{\begin{matrix} N p (x) / (N - p (x)) & if p (x) < N, \\ + \infty & if p (x) ⩾ N . \end{matrix} \end{matrix}$ Moreover, we say that a measurable function $h : R^{N} \to R$ is $Z^{N}$ -periodic if $\begin{matrix} h (x + z) = h (x) \forall x \in R^{N} and \forall z \in Z^{N}, \end{matrix}$ and the notation $u ≪ v$ means that ${inf}_{x \in R^{N}} (v (x) - u (x)) > 0$ .

We would like point out that we are supposing that the exponents are Lipschitz continuous functions to guarantee that the embeddings from $W^{1, p (x)} (R^{N})$ into $L^{r (x)} (R^{N})$ , $L^{q (x)} (R^{N})$ and $L^{p^{*} (x)} (R^{N})$ are continuous, for more details see [18] and [17].

Related to functions f and g, we suppose that they are nonnegative continuous functions verifying the following conditions:

$\begin{matrix} lim_{| x | \to \infty} g (x) = 0; \end{matrix}$

There exist ℓ points $a_{1}, a_{2}, \dots, a_{ℓ}$ in $Z^{N}$ with $a_{1} = 0$ such that $\begin{matrix} 1 = f (a_{i}) = max_{R^{N}} f (x) for 1 ⩽ i ⩽ ℓ . \end{matrix}$ Moreover, $0 < f_{\infty} < f (x)$ for any $x \in R^{N}$ and $\begin{matrix} lim_{| x | \to \infty} f (x) = f_{\infty} . \end{matrix}$

The variable exponents problems appear in a lot of applications, such as, the motion of electrorheological fluids [29], image processing [12] and flow in porous media [8]. Motivated by this fact, such problems have attracted an increasing attention, we would like to mention [3,4,6,11,16,26,28], and also the survey papers [7,14,30] for the advances and references in this area.

The problem ( P λ , k ) has been considered in the literature for the case where the exponents are constants, see for example, Adachi and Tanaka [1], Cao and Noussair [9], Cao and Zhou [10], Hirano [20], Hirano and Shioji [21], Hu and Tang [23], Jeanjean [24], Lin [27], Hsu, Lin and Hu [22], Tarantello [31], Wu [33,34] and their references.

In Cao and Noussair [9], the authors have studied the existence and multiplicity of positive and nodal solutions for the following problem $\begin{matrix} (P_{1}) & \{\begin{matrix} - Δ u + u = f (ϵ x) | u |^{r - 2} u in R^{N}, \\ u \in H^{1, 2} (R^{N}), \end{matrix} \end{matrix}$ where ϵ is a positive real parameter, $r \in (2, 2^{*})$ and f verifies condition (H2). By using variational methods, the authors showed the existence of at least ℓ positive solutions and ℓ nodal solutions if ϵ is small enough. After, Wu in [33] considered the perturbed problem $\begin{matrix} (P_{2}) & \{\begin{matrix} - Δ u + u = f (ϵ x) | u |^{r - 2} u + λ g (ϵ x) | u |^{q - 2} u in R^{N}, \\ u \in H^{1, 2} (R^{N}), \end{matrix} \end{matrix}$ where λ is a positive parameter and $q \in (0, 1)$ . In [33], the authors showed the existence of at least ℓ positive solutions for ( P 2 ) when ϵ and λ are small enough.

In Hsu, Lin and Hu [22], the authors have considered the following class of quasilinear problems $\begin{matrix} (P_{3}) & \{\begin{matrix} - Δ_{p} u + | u |^{p - 2} u = f (ϵ x) | u |^{r - 2} u + λ g (ϵ x) in R^{N}, \\ u \in W^{1, p} (R^{N}) \end{matrix} \end{matrix}$ with $N ⩾ 3$ and $2 ⩽ p < N$ . In that paper, the authors have proved the same type of results found in [9] and [33].

Motivated by results proved in [9,22] and [33], we intend in the present paper to prove the existence of multiple solutions for problem ( P λ , k ), by using the same type of approach explored in those papers. However, since we are working with variable exponents, some estimates that hold for the constant case are not immediate for the variable case, and so, a careful analysis is necessary to get some estimates. Here, for example, we were able to prove our results by assuming that some exponents are periodic and $k \in N$ .

Our main result is the following:

Theorem 1.1.
Assume that ( p 1 ) and (H1)–(H2) are satisfied. Then there are $Λ^{} > 0$ and* $k^{} \in N$ such that problem (* P λ , k ) admits at least ℓ solutions for $0 ⩽ λ < Λ^{}$ and* $k ⩾ k^{*}$ .

This paper is organized in the following way: In Section 2, we collect some preliminaries on variable exponent spaces that will be used throughout the paper, which can be found in [4,5,13,17,18,25]. In Section 3, we show some technical results, and finally in Section 4 we prove Theorem 1.1. Notation.
The following notation will be used in the present work:
C and $c_{i}$ denote generic positive constants, which may vary from line to line.

We denote by $\int f$ the integral $\int_{R^{N}} f d x$ for any measurable function f.

$B_{R} (z)$ denotes the open ball with center at z and radius R in $R^{N}$ .

If h is a bounded mensurable function, we denote by $h_{+}$ and $h_{-}$ the ensuing real numbers $\begin{matrix} h_{+} = ess sup_{x \in R^{N}} h (x) and h_{-} = ess inf_{x \in R^{N}} h (x) . \end{matrix}$ Moreover, we also denote by $h^{'} (x)$ the conjugate exponent of $h (x)$ given by $h^{'} (x) = \frac{h (x)}{h (x) - 1}$ .

2. Preliminaries on Lebesgue and Sobolev spaces with variable exponent in $R^{N}$

In this section, we recall the definitions and some results involving the spaces $L^{h (x)} (R^{N})$ and $W^{1, h (x)} (R^{N})$ . We refer to [4,5,13,17,18,25] for the fundamental properties of these spaces.

Hereafter, let us denote by $L_{+}^{\infty} (R^{N})$ the set $\begin{matrix} L_{+}^{\infty} (R^{N}) = {u \in L^{\infty} (R^{N}) : ess inf_{x \in R^{N}} u ⩾ 1} \end{matrix}$ and we will assume that $h \in L_{+}^{\infty} (R^{N})$ .

The variable exponent Lebesgue space $L^{h (x)} (R^{N})$ is defined by $\begin{matrix} L^{h (x)} (R^{N}) = {u : R^{N} \to R is measurable : \int {| u (x) |}^{h (x)} < + \infty}, \end{matrix}$ which is endowed with the norm $\begin{matrix} ∥ u ∥_{h (x)} = inf {t > 0 : \int {| \frac{u (x)}{t} |}^{h (x)} ⩽ 1} . \end{matrix}$ On space $L^{h (x)} (R^{N})$ , we consider the modular function $ρ : L^{h (x)} (R^{N}) \to R$ given by $\begin{matrix} ρ (u) = \int {| u (x) |}^{h (x)} . \end{matrix}$

Proposition 2.1.
Let $u \in L^{h (x)} (R^{N})$ and ${u_{n}}_{n \in N} \subset L^{h (x)} (R^{N})$ . Then,
If $u \neq 0$ , $∥ u ∥_{h (x)} = a \Leftrightarrow ρ (\frac{u}{a}) = 1$ .

$∥ u ∥_{h (x)} < 1 (= 1; > 1) \Leftrightarrow ρ (u) < 1 (= 1; > 1)$ .

$∥ u ∥_{h (x)} > 1 \Rightarrow ∥ u ∥_{h (x)}^{h_{-}} ⩽ ρ (u) ⩽ ∥ u ∥_{h (x)}^{h_{+}}$ .

$∥ u ∥_{h (x)} < 1 \Rightarrow ∥ u ∥_{h (x)}^{h_{+}} ⩽ ρ (u) ⩽ ∥ u ∥_{h (x)}^{h_{-}}$ .

${lim}_{n \to + \infty} ∥ u_{n} ∥_{h (x)} = 0 \Leftrightarrow {lim}_{n \to \infty} ρ (u_{n}) = 0$ .

${lim}_{n \to + \infty} ∥ u_{n} ∥_{h (x)} = + \infty \Leftrightarrow {lim}_{n \to \infty} ρ (u_{n}) = + \infty$ .

We have the following Hölder inequality for Lebesgue spaces with variable exponents.
Proposition 2.2 (Hölder-type inequality).

Let $u \in L^{h (x)} (R^{N})$ and $v \in L^{h^{'} (x)} (R^{N})$ . Then, $u v \in L^{1} (R^{N})$ and $\begin{matrix} \int | u (x) v (x) | ⩽ (\frac{1}{h_{-}} + \frac{1}{h_{-}^{'}}) ∥ u ∥_{h (x)} ∥ v ∥_{h^{'} (x)} . \end{matrix}$

The next three results are important tools to study the properties of some energy functionals, and their proofs can be found in [5].

Lemma 2.3 (Brezis–Lieb’s lemma, first version).

Let $(η_{n}) \subset L^{h (x)} (R^{N}, R^{m})$ with $m \in N$ verifying

$η_{n} (x) \to η (x)$ , a.e. in $R^{N}$ ;

${sup}_{n \in N} | η_{n} |_{L^{h (x)} (R^{N}, R^{m})} < \infty$ .

Then,

η \in L^{h (x)} (R^{N}, R^{m})

and

\begin{matrix} \int (| η_{n} |^{h (x)} - | η_{n} - η |^{h (x)} - | η |^{h (x)}) = o_{n} (1) . \end{matrix}

Lemma 2.4 (Brezis–Lieb’s lemma, second version).

Let $(η_{n}) \subset L^{h (x)} (R^{N}, R^{m})$ with $m \in N$ verifying

$η_{n} (x) \to η (x)$ , a.e. in $R^{N}$ ;

${sup}_{n \in N} | η_{n} |_{L^{h (x)} (R^{N}, R^{m})} < \infty$ .

Then

\begin{matrix} η_{n} ⇀ η in L^{h (x)} (R^{N}, R^{m}) . \end{matrix}

The next proposition is a Brezis–Lieb-type result.

Lemma 2.5 (Brezis–Lieb lemma, third version).

Let $(η_{n}) \subset L^{h (x)} (R^{N}, R^{m})$ with $m \in N$ such that

$η_{n} (x) \to η (x)$ , a.e. in $R^{N}$ ;

${sup}_{n \in N} | η_{n} |_{L^{h (x)} (R^{N}, R^{m})} < \infty$ .

Then

\begin{matrix} \int {| | η_{n} |^{h (x) - 2} η_{n} - | η_{n} - η |^{h (x) - 2} (η_{n} - η) - | η |^{h (x) - 2} η |}^{h^{'} (x)} = o_{n} (1) . \end{matrix}

The variable exponent Sobolev space $W^{1, h (x)} (R^{N})$ is defined by $\begin{matrix} W^{1, h (x)} (R^{N}) = {u \in W_{loc}^{1, 1} (R^{N}) : u \in L^{h (x)} (R^{N}) and | \nabla u | \in L^{h (x)} (R^{N})} . \end{matrix}$ The corresponding norm for this space is $\begin{matrix} ∥ u ∥_{W^{1, h (x)} (R^{N})} = ∥ u ∥_{h (x)} + ∥ \nabla u ∥_{h (x)} . \end{matrix}$ The spaces $L^{h (x)} (R^{N})$ and $W^{1, h (x)} (R^{N})$ are separable and reflexive Banach spaces when $h_{-} > 1$ .

On space $W^{1, h (x)} (R^{N})$ , we consider the modular function $ρ_{1} : W^{1, h (x)} (R^{N}) \to R$ given by $\begin{matrix} ρ_{1} (u) = \int ({| \nabla u (x) |}^{h (x)} + {| u (x) |}^{h (x)}) . \end{matrix}$ If we define $\begin{matrix} ∥ u ∥ = inf {t > 0 : \int \frac{(| \nabla u |^{h (x)} + | u |^{h (x)})}{t^{h (x)}} ⩽ 1}, \end{matrix}$ then $∥ \cdot ∥_{W^{1, h (x)} (R^{N})}$ and $∥ \cdot ∥$ are equivalent norms on $W^{1, h (x)} (R^{N})$ .

Proposition 2.6.
Let $u \in W^{1, h (x)} (R^{N})$ and ${u_{n}} \subset W^{1, h (x)} (R^{N})$ . Then, the same conclusion of Proposition 2.1 occurs replacing $∥ \cdot ∥_{h (x)}$ and ρ by $∥ \cdot ∥$ and $ρ_{1}$ , respectively.

3. Technical lemmas

Associated with problem ( P λ , k ), we have the energy functional $J_{λ, k} : W^{1, p (x)} (R^{N}) \to R$ defined by $\begin{matrix} J_{λ, k} (u) = \int \frac{1}{p (x)} (| \nabla u |^{p (x)} + | u |^{p (x)}) - λ \int \frac{g (k^{- 1} x)}{q (x)} | u |^{q (x)} - \int \frac{f (k^{- 1} x)}{r (x)} | u |^{r (x)} . \end{matrix}$ It is easy to see that $J_{λ, k} \in C^{1} (W^{1, p (x)} (R^{N}), R)$ with $\begin{array}{l} J_{λ, k}^{'} (u) v = & \int (| \nabla u |^{p (x) - 2} \nabla u \nabla v + | u |^{p (x) - 2} u v) - λ \int g (k^{- 1} x) | u |^{q (x) - 2} u v \\ - \int f (k^{- 1} x) | u |^{r (x) - 2} u v \end{array}$ for any $u, v \in W^{1, p (x)} (R^{N})$ . Thus, the critical points of $J_{λ, k}$ are (weak) solutions of ( P λ , k ). Since the functional $J_{λ, k}$ is not bounded from below on $W^{1, p (x)} (R^{N})$ , we will work on Nehari manifold $M_{λ, k}$ associated with the functional $J_{λ, k}$ , given by $\begin{matrix} M_{λ, k} = {u \in W^{1, p (x)} (R^{N}) ∖ {0} : J_{λ, k}^{'} (u) u = 0} \end{matrix}$ and with the level $\begin{matrix} c_{λ, k} = inf_{u \in M_{λ, k}} J_{λ, k} (u) . \end{matrix}$

Using well-known arguments found in Willem [32], it follows that $c_{λ, k}$ is the mountain pass level of functional $J_{λ, k}$ .

For $f \equiv 1$ and $λ = 0$ , we consider the problem $\begin{matrix} (P_{\infty}) & \{\begin{matrix} - Δ_{p (x)} u + | u |^{p (x) - 2} u = | u |^{r (x) - 2} u, R^{N}, \\ u \in W^{1, p (x)} (R^{N}) . \end{matrix} \end{matrix}$ Associated with the problem ( P ∞ ), we have the energy functional $J_{\infty} : W^{1, p (x)} (R^{N}) \to R$ given by $\begin{matrix} J_{\infty} (u) = \int \frac{1}{p (x)} (| \nabla u |^{p (x)} + | u |^{p (x)}) - \int \frac{1}{r (x)} | u |^{r (x)}, \end{matrix}$ the level $\begin{matrix} c_{\infty} = inf_{u \in M_{\infty}} J_{\infty} (u), \end{matrix}$ and the Nehari manifold $\begin{matrix} M_{\infty} = {u \in W^{1, p (x)} (R^{N}) ∖ {0} : J_{\infty}^{'} (u) u = 0} . \end{matrix}$

For $f \equiv f_{\infty}$ and $λ = 0$ , we fix the problem $\begin{matrix} (P_{f_{\infty}}) & \{\begin{matrix} - Δ_{p (x)} u + | u |^{p (x) - 2} u = f_{\infty} | u |^{r (x) - 2} u, R^{N}, \\ u \in W^{1, p (x)} (R^{N}), \end{matrix} \end{matrix}$ and as above, we denote by $J_{f_{\infty}}$ , $c_{f_{\infty}}$ and $M_{f_{\infty}}$ the energy functional, the mountain pass level and Nehari manifold associated with ( P f ∞ ), respectively.

The following result concerns the behavior of $J_{λ, k}$ on $M_{λ, k}$ .

Lemma 3.1.
The functional $J_{λ, k}$ is bounded from below on $M_{λ, k}$ . Moreover, $J_{λ, k}$ is coercive on $M_{λ, k}$ .
Proof.
For each $u \in M_{λ, k}$ , $J_{λ, k}^{'} (u) u = 0$ . Hence, $\begin{matrix} λ \int g (k^{- 1} x) | u |^{q (x)} = \int (| \nabla u |^{p (x)} + | u |^{p (x)}) - \int f (k^{- 1} x) | u |^{r (x)} . \end{matrix}$ Note that $\begin{array}{l} J_{λ, k} (u) ⩾ & \frac{1}{p_{+}} \int (| \nabla u |^{p (x)} + | u |^{p (x)}) - \frac{λ}{q_{-}} \int g (k^{- 1} x) | u |^{q (x)} - \frac{1}{r_{-}} \int f (k^{- 1} x) | u |^{r (x)} \\ = & \frac{1}{p_{+}} \int (| \nabla u |^{p (x)} + | u |^{p (x)}) - \frac{1}{r_{-}} \int f (k^{- 1} x) | u |^{r (x)} \\ - \frac{1}{q_{-}} (\int (| \nabla u |^{p (x)} + | u |^{p (x)}) - \int f (k^{- 1} x) | u |^{r (x)}) . \end{array}$ Since $p_{+} < q_{-} ⩽ q (x) ⩽ r (x) ≪ p^{} (x)$ , $\begin{matrix} J_{λ, k} (u) ⩾ (\frac{1}{p_{+}} - \frac{1}{q_{-}}) \int (| \nabla u |^{p (x)} + | u |^{p (x)}), \end{matrix}$ showing that J is bounded from below and coercive on $M_{λ, k}$ . □

As an immediate consequence of the last lemma, we have:
Corollary 3.2.
Let* ${u_{n}}$ be a sequence in $M_{λ, k}$ and $J_{λ, k} (u_{n}) \to c_{λ, k}$ . Then ${u_{n}}$ is bounded in $W^{1, p (x)} (R^{N})$ .

The next lemma establishes that Nehari manifold $M_{λ, k}$ has a positive distance from the origin.
Lemma 3.3.
Given $Λ > 0$ there exists $η > 0$ such that $\begin{matrix} (3.1) & ρ_{1} (u) ⩾ η \forall (u, λ, k) \in M_{λ, k} \times [0, Λ] \times N . \end{matrix}$ Here, $ρ_{1} (u)$ is defined with $h (x)$ replaced by $p (x)$ . Moreover, the functional $E_{λ, k}$ defined by $E_{λ, k} (u) = J_{λ, k}^{'} (u) u$ is of class $C^{1} (W^{1, p (x)} (R^{N}), R)$ and $\begin{matrix} (3.2) & E_{λ, k}^{'} (u) u ⩽ - (q_{-} - p_{+}) η \forall (u, λ, k) \in M_{λ, k} \times [0, Λ] \times N . \end{matrix}$
Proof.
Suppose by contradiction that (3.1) does not hold. Then, there is ${u_{n}} \subset M_{λ, k}$ such that $\begin{matrix} ρ_{1} (u_{n}) \to 0 as n \to \infty, \end{matrix}$ or equivalently, by Proposition 2.6, $\begin{matrix} ∥ u_{n} ∥ \to 0 as n \to \infty . \end{matrix}$ Since ${u_{n}} \subset M_{λ, k}$ and $∥ f ∥_{\infty} ⩽ 1$ , we derive $\begin{matrix} \int (| \nabla u_{n} |^{p (x)} + | u_{n} |^{p (x)}) ⩽ λ ∥ g ∥_{\infty} \int | u_{n} |^{q (x)} + \int | u_{n} |^{r (x)} . \end{matrix}$ On the other hand, by using the fact that $∥ u_{n} ∥ < 1$ for n large enough, it follows from Propositions 2.1 and 2.6 $\begin{array}{l} ∥ u_{n} ∥^{p_{+}} & ⩽ \int (| \nabla u_{n} |^{p (x)} + | u_{n} |^{p (x)}) \\ ⩽ λ ∥ g ∥_{\infty} max {∥ u_{n} ∥_{q (x)}^{q_{-}}, ∥ u_{n} ∥_{q (x)}^{q_{+}}} + max {∥ u_{n} ∥_{r (x)}^{r_{-}}, ∥ u_{n} ∥_{r (x)}^{r_{+}}} . \end{array}$ By the Sobolev embedding theorem, there are positive constants $c_{1}$ and $c_{2}$ such that $\begin{matrix} ∥ u_{n} ∥^{p_{+}} ⩽ λ ∥ g ∥_{\infty} c_{1} max {∥ u_{n} ∥^{q_{-}}, ∥ u_{n} ∥^{q_{+}}} + c_{2} max {∥ u_{n} ∥^{r_{-}}, ∥ u_{n} ∥^{r_{+}}}, \end{matrix}$ and so, for n large enough, $\begin{matrix} ∥ u_{n} ∥^{p_{+}} ⩽ λ c_{1} ∥ g ∥_{\infty} ∥ u_{n} ∥^{q_{-}} + c_{2} ∥ u_{n} ∥^{r_{-}}, \end{matrix}$ obtaining an absurd, because $p_{+} < q_{-} ⩽ r_{-}$ . Therefore, (3.1) is proved.

Next, we will show that (3.2) occurs. For each $u \in M_{λ, k}$ , a simple calculus gives $\begin{matrix} \begin{matrix} E_{λ, k}^{'} (u) u & ⩽ p_{+} \int (| \nabla u |^{p (x)} + | u |^{p (x)}) - λ q_{-} \int g (k^{- 1} x) | u |^{q (x)} - r_{-} \int f (k^{- 1} x) | u |^{r (x)} \\ ⩽ (p_{+} - q_{-}) \int (| \nabla u |^{p (x)} + | u |^{p (x)}) + (q_{-} - r_{-}) \int f (k^{- 1} x) | u |^{r (x)} . \end{matrix} \end{matrix}$ As $p_{+} < q_{-} ⩽ r_{-}$ , it follows that $\begin{matrix} E_{λ, k}^{'} (u) u ⩽ - (q_{-} - p_{+}) ρ_{1} (u) ⩽ - (q_{-} - p_{+}) η, \end{matrix}$ finishing the proof. □

As by product of the last lemma, we are able to prove that critical points of $J_{λ, k}$ restrict to $M_{λ, k}$ are in fact critical point of $J_{λ, k}$ on $W^{1, p (x)} (R^{N})$ .
Lemma 3.4.
If $u_{0} \in M_{λ, k}$ is a critical point of $J_{λ, k}$ restricted to $M_{λ, k}$ , then $u_{0}$ is a critical point of $J_{λ, k}$ in $W^{1, p (x)} (R^{N})$ .
Proof.
As $u_{0}$ is a critical point of $J_{λ, k}$ restricted to $M_{λ, k}$ , there is $τ \in R^{N}$ such that $\begin{matrix} J_{λ, k}^{'} (u_{0}) = τ E_{λ, k}^{'} (u_{0}) . \end{matrix}$ By Lemma 3.3, we know that $E_{λ, k}^{'} (u_{0}) u_{0} < 0$ , then we must have $τ = 0$ . Thereby, $\begin{matrix} J_{λ, k}^{'} (u_{0}) = 0, \end{matrix}$ implying that $u_{0}$ is critical point of $J_{λ, k} (u_{0})$ in $W^{1, p (x)} (R^{N})$ . □

The next result is very important in our arguments, because it implies that the weak limit of a $(PS)$ sequence is a critical point for the energy functional. Theorem 3.5.
Let ${u_{n}}$ be a sequence in $W^{1, p (x)} (R^{N})$ such that $u_{n} ⇀ u$ in $W^{1, p (x)} (R^{N})$ and $J_{λ, k}^{'} (u_{n}) \to 0$ as $n \to \infty$ . Then, for some subsequence, $\nabla u_{n} (x) \to \nabla u (x)$ a.e. in $R^{N}$ as $n \to \infty$ and $J_{λ, k}^{'} (u) = 0$ .
Proof.
Let $R > 0$ and $ϕ \in C_{0}^{\infty} (R^{N})$ such that $\begin{matrix} ϕ = 0 if | x | ⩾ 2 R, ϕ = 1 if | x | ⩽ R and 0 ⩽ ϕ (x) ⩽ 1 \forall x \in R^{n} . \end{matrix}$ In what follows, let us denote by ${P_{n}}$ the following sequence $\begin{matrix} P_{n} (x) = ⟨ {| \nabla u_{n} (x) |}^{p (x) - 2} \nabla u_{n} (x) - {| \nabla u (x) |}^{p (x) - 2} \nabla u (x), \nabla u_{n} (x) - \nabla u (x) ⟩ . \end{matrix}$ From definition of ${P_{n}}$ , $\begin{matrix} \int_{B_{R} (0)} P_{n} ⩽ \int | \nabla u_{n} |^{p (x)} ϕ - \int | \nabla u_{n} |^{p (x) - 2} \nabla u_{n} \nabla u ϕ - \int | \nabla u |^{p (x) - 2} \nabla u \nabla (u_{n} - u) ϕ . \end{matrix}$ Recalling that $u_{n} ⇀ u$ in $W^{1, p (x)} (R^{N})$ , we have $\begin{matrix} (3.3) & \int_{B_{R} (0)} | \nabla u |^{p (x) - 2} \nabla u \nabla (u_{n} - u) ϕ \to 0 as n \to \infty, \end{matrix}$ and so, $\begin{matrix} \int_{B_{R} (0)} P_{n} ⩽ \int | \nabla u_{n} |^{p (x)} ϕ - \int | \nabla u_{n} |^{p (x) - 2} \nabla u_{n} \nabla u ϕ + o_{n} (1) . \end{matrix}$ On the other hand, from $J_{λ, k}^{'} (u_{n}) (ϕ u_{n}) = o_{n} (1)$ and $J_{λ, k}^{'} (u_{n}) (ϕ u) = o_{n} (1)$ , $\begin{array}{l} \int_{B_{R} (0)} P_{n} ⩽ & o_{n} (1) - \int | \nabla u_{n} |^{p (x) - 2} \nabla u_{n} \nabla ϕ (u_{n} - u) \\ - \int | u_{n} |^{p (x) - 2} u_{n} (u - u_{n}) ϕ + λ \int g (k^{- 1} x) | u_{n} |^{q (x) - 2} u_{n} (u_{n} - u) ϕ \\ + \int f (k^{- 1} x) | u_{n} |^{r (x) - 2} u_{n} (u_{n} - u) ϕ . \end{array}$ Thus, $\begin{array}{l} \int_{B_{R} (0)} P_{n} ⩽ & o_{n} (1) + c_{1} \int_{supt ϕ} | \nabla u_{n} |^{p (x) - 1} | u_{n} - u | \\ + c_{1} \int_{supt ϕ} | u_{n} |^{p (x) - 1} | u_{n} - u | + c_{1} λ ∥ g ∥_{\infty} \int_{supt ϕ} | u_{n} |^{q (x) - 1} | u_{n} - u | \\ + c_{1} \int_{supt ϕ} | u_{n} |^{r (x) - 1} | u_{n} - u | . \end{array}$ Combining Hölder’s inequality and the Sobolev embedding theorem, we deduce that $\begin{matrix} \int_{B_{R} (0)} P_{n} \to 0 as n \to \infty . \end{matrix}$ In what follows, let us consider the sets $\begin{matrix} B_{R}^{+} = {x \in B_{R} (0) / p (x) ⩾ 2} and B_{R}^{-} = {x \in B_{R} (0) / 1 < p (x) < 2} . \end{matrix}$ Since $\begin{matrix} P_{n} (x) ⩾ \{\begin{matrix} \frac{2^{3 - p_{+}}}{p_{+}} | \nabla u_{n} - \nabla u |^{p (x)} & if p (x) ⩾ 2, \\ (p_{-} - 1) \frac{| \nabla u_{n} - \nabla u |^{2}}{{(| \nabla u_{n} | + | \nabla u |)}^{2 - p (x)}} & if 1 < p (x) < 2, \end{matrix} \end{matrix}$ we have $\begin{matrix} (3.4) & \int_{B_{R}^{+}} | \nabla u_{n} - \nabla u |^{p (x)} d x \to 0 as n \to \infty . \end{matrix}$ Applying again Hölder’s inequality, $\begin{matrix} \int_{B_{R}^{-}} | \nabla u_{n} - \nabla u |^{p (x)} ⩽ C ∥ g_{n} ∥_{L^{\frac{2}{p (x)}} (B_{R}^{-})} ∥ h_{n} ∥_{L^{\frac{2}{2 - p (x)}} (B_{R}^{-})}, \end{matrix}$ where $\begin{matrix} g_{n} (x) = \frac{| \nabla u_{n} (x) - \nabla u (x) |^{p (x)}}{{(| \nabla u_{n} (x) | + | \nabla u (x) |)}^{\frac{p (x) (2 - p (x))}{2}}} \end{matrix}$ and $\begin{matrix} h_{n} (x) = {(| \nabla u_{n} (x) | + | \nabla u (x) |)}^{\frac{p (x) (2 - p (x))}{2}} . \end{matrix}$ By a direct computation, ${∥ h_{n} ∥_{L^{\frac{2}{2 - p (x)}} (B_{R}^{-})}}$ is a bounded sequence and $\begin{matrix} \int_{B_{R}^{-}} | g_{n} |^{\frac{2}{p (x)}} ⩽ C \int_{B_{R}^{-}} P_{n} . \end{matrix}$ Then, $\begin{matrix} (3.5) & \int_{B_{R}^{-}} | \nabla u_{n} - \nabla u |^{p (x)} \to 0 when n \to \infty . \end{matrix}$ From (3.4) and (3.5), $\nabla u_{n} \to \nabla u$ a.e. in $B_{R} (0)$ . Since R is arbitrary, it follows that for some subsequence $\begin{matrix} \nabla u_{n} (x) \to \nabla u (x) a.e. in R^{N} . \end{matrix}$ This combined with Lemma 2.4 gives $\begin{matrix} | \nabla u_{n} |^{p (x) - 2} \nabla u_{n} ⇀ | \nabla u |^{p (x) - 2} \nabla u in {(L^{p^{'} (x)} (R^{N}))}^{N} . \end{matrix}$ Now, using the fact that $J_{λ, k}^{'} (u_{n}) v = o_{n} (1)$ for all $v \in W^{1, p (x)} (R^{N})$ together with the last limit, we derive that $J_{λ, k}^{'} (u) v = 0$ for all $v \in W^{1, p (x)} (R^{N})$ , finishing the proof. □

3.1. A result of compactness

The next theorem is a version of a result compactness on Nehari manifolds due to Alves [2] for variable exponents. It establishes that problem ( P ∞ ) has a ground state solution.

Theorem 3.6.
Suppose that ( p 1 ) holds and let ${u_{n}} \subset M_{\infty}$ be a sequence with $J_{\infty} (u_{n}) \to c_{\infty}$ . Then,
$u_{n} \to u$ in $W^{1, p (x)} (R^{N})$ ; or

there is ${y_{n}} \subset Z^{N}$ with $| y_{n} | \to + \infty$ and $w \in W^{1, p (x)} (R^{N})$ such that $w_{n} = u_{n} (\cdot + y_{n}) \to w$ in $W^{1, p (x)} (R^{N})$ and $J_{\infty} (w) = c_{\infty}$ .

Proof.
Similarly to Corollary 3.2, we can assume that ${u_{n}}$ is a bounded sequence, and so, there is $u \in W^{1, p (x)} (R^{N})$ and a subsequence of ${u_{n}}$ , still denoted by itself, such that $u_{n} ⇀ u$ in $W^{1, p (x)} (R^{N})$ . Applying the Ekeland’s variational principle, there is a sequence ${w_{n}}$ in $M_{\infty}$ satisfying $\begin{matrix} w_{n} = u_{n} + o_{n} (1), J_{\infty} (w_{n}) \to c_{\infty} \end{matrix}$ and $\begin{matrix} (3.6) & J_{\infty}^{'} (w_{n}) - τ_{n} E_{\infty}^{'} (w_{n}) = o_{n} (1), \end{matrix}$ where $(τ_{n}) \subset R$ and $E_{\infty} (w) = J_{\infty}^{'} (w) w$ for any $w \in W^{1, p (x)} (R^{N})$ .

Since ${u_{n}} \subset M_{\infty}$ , (3.6) leads to $\begin{matrix} τ_{n} E_{\infty}^{'} (w_{n}) w_{n} = o_{n} (1) . \end{matrix}$ By the arguments of Lemma 3.3, there exists $δ > 0$ such that $\begin{matrix} | E_{\infty}^{'} (w_{n}) w_{n} | > δ \forall n \in N . \end{matrix}$ From this, $τ_{n} \to 0$ as $n \to \infty$ and we can claim that $\begin{matrix} J_{\infty} (u_{n}) \to c_{\infty} and J_{\infty}^{'} (u_{n}) \to 0 . \end{matrix}$

Next, we will study the following possibilities: $u \neq 0$ or $u = 0$ . Case 1 ( $u \neq 0$ ).

Similarly to Theorem 3.5, it follows that the below limits are valid for some subsequence:

$u_{n} (x) \to u (x)$ and $\nabla u_{n} (x) \to \nabla u (x)$ a.e. in $R^{N}$ ,

$\int | \nabla u_{n} (x) |^{p (x) - 2} \nabla u_{n} (x) \nabla v \to \int | \nabla u (x) |^{p (x) - 2} \nabla u (x) \nabla v$ ,

$\int | u_{n} |^{p (x) - 2} u_{n} v \to \int | u |^{p (x) - 2} u_{n} v$

and

$\int | u_{n} |^{r (x) - 2} u_{n} v \to \int | u |^{r (x) - 2} u v$

for any

v \in W^{1, p (x)} (R^{N})

. Consequently, u is critical point of

J_{\infty}

. By Fatou’s lemma, it is easy to check that

\begin{array}{l} c_{\infty} ⩽ & J_{\infty} (u) = J_{\infty} (u) - \frac{1}{r_{-}} J_{\infty}^{'} (u) u \\ = & \int (\frac{1}{p (x)} - \frac{1}{r_{-}}) (| \nabla u |^{p (x)} + | u |^{p (x)}) + \int (\frac{1}{r_{-}} - \frac{1}{r (x)}) | u |^{r (x)} \\ ⩽ & \underset{n \to \infty}{lim inf} {\int (\frac{1}{p (x)} - \frac{1}{r_{-}}) (| \nabla u_{n} |^{p (x)} + | u_{n} |^{p (x)}) \\ + \int (\frac{1}{r_{-}} - \frac{1}{r (x)}) | u_{n} |^{r (x)}} \\ = & \underset{n \to \infty}{lim inf} {J_{\infty} (u_{n}) - \frac{1}{r_{-}} J_{\infty}^{'} (u_{n}) u_{n}} = c_{\infty} . \end{array}

Hence,

\begin{matrix} lim_{n \to \infty} \int (| \nabla u_{n} |^{p (x)} + | u_{n} |^{p (x)}) = \int (| \nabla u |^{p (x)} + | u |^{p (x)}), \end{matrix}

implying that

u_{n} \to u

W^{1, p (x)} (R^{N})

Case 2 ( $u = 0$ ).

In this case, we claim that there are $R, ξ > 0$ and ${y_{n}} \subset R^{N}$ satisfying $\begin{matrix} (3.7) & \underset{n \to \infty}{lim sup} \int_{B_{R} (y_{n})} | u_{n} |^{p (x)} ⩾ ξ . \end{matrix}$ If the claim is false, we must have $\begin{matrix} \underset{n \to \infty}{lim sup} sup_{y \in R^{N}} \int_{B_{R} (y)} | u_{n} |^{p (x)} = 0 . \end{matrix}$ Thus, by a Lions-type result for variable exponent proved in [19, Lemma 3.1], $\begin{matrix} u_{n} \to 0 in L^{s (x)} (R^{N}) \end{matrix}$ for any $s \in C (R^{N})$ with $p ≪ s ≪ p^{*}$ .

Recalling $J_{\infty}^{'} (u_{n}) u_{n} = o_{n} (1)$ , the last limits yield $\begin{matrix} \int (| \nabla u_{n} |^{p (x)} + | u_{n} |^{p (x)}) = o_{n} (1), \end{matrix}$ or equivalently $\begin{matrix} u_{n} \to 0 in W^{1, p (x)} (R^{N}), \end{matrix}$ leading to $c_{\infty} = 0$ , which is absurd. This way, (3.7) is true. By a routine argument, we can assume that $y_{n} \in Z^{N}$ and $| y_{n} | \to \infty$ as $n \to \infty$ . Setting $\begin{matrix} w_{n} (x) = u_{n} (x + y_{n}), \end{matrix}$ and using the fact that p and r are $Z^{N}$ -periodic, a change of variable gives $\begin{matrix} J_{\infty} (w_{n}) = J_{\infty} (u_{n}) and ∥ J_{\infty}^{'} (w_{n}) ∥ = ∥ J_{\infty}^{'} (u_{n}) ∥, \end{matrix}$ showing that ${w_{n}}$ is a sequence ${(PS)}_{c_{\infty}}$ for $J_{\infty}$ . If $w \in W^{1, p (x)} (R^{N})$ denotes the weak limit of ${w_{n}}$ , it follows from (3.7), $\begin{matrix} \int_{B_{R} (0)} | w |^{p (x)} ⩾ ξ, \end{matrix}$ showing that $w \neq 0$ .

By repeating the same argument of the first case for the sequence ${w_{n}}$ , we deduce that $w_{n} \to w$ in $W^{1, p (x)} (R^{N})$ , $w \in M_{\infty}$ and $J_{\infty} (w) = c_{\infty}$ . □

3.2. Estimates involving the minimax levels

The main goal of this section is to prove some estimates involving the minimax levels $c_{λ, k}$ , $c_{0, k}$ and $c_{\infty}$ .

First of all, as $λ ⩾ 0$ we recall the inequalities $\begin{matrix} J_{λ, k} (u) ⩽ J_{0, k} (u) and J_{\infty} (u) ⩽ J_{0, k} (u) \forall u \in W^{1, p (x)} (R^{N}), \end{matrix}$ which imply $\begin{matrix} c_{λ, k} ⩽ c_{0, k} and c_{\infty} ⩽ c_{0, k} . \end{matrix}$

Lemma 3.7.
The minimax levels $c_{0, k}$ and $c_{f_{\infty}}$ satisfy the inequality $c_{0, k} < c_{f_{\infty}}$ . Hence, $c_{\infty} < c_{f_{\infty}}$ .
Proof.
In a manner analogous to Theorem 3.6, there is $U \in W^{1, p (x)} (R^{N})$ verifying $\begin{matrix} J_{f_{\infty}} (U) = c_{f_{\infty}} and J_{f_{\infty}}^{'} (U) = 0 . \end{matrix}$ From Lemma 3.6 in [15], there exists $t > 0$ such that $t U \in M_{0, k}$ . Thus, $\begin{matrix} c_{0, k} ⩽ J_{0, k} (t U) = \int \frac{t^{p (x)}}{p (x)} (| \nabla U |^{p (x)} + | U |^{p (x)}) - \int f (k^{- 1} x) \frac{t^{r (x)}}{r (x)} | U |^{r (x)} . \end{matrix}$ Since that by (H2), $f_{\infty} < f (x)$ for all $x \in R^{N}$ , we derive $\begin{matrix} c_{0, k} < J_{f_{\infty}} (t U) ⩽ max_{s ⩾ 0} J_{f_{\infty}} (s U) = J_{f_{\infty}} (U) = c_{f_{\infty}} . □ \end{matrix}$

Using the last lemma, we are able to prove that $J_{λ, k}$ verifies the ${(PS)}_{d}$ condition for some values of d.
Lemma 3.8.
The functional $J_{λ, k}$ satisfies the ${(PS)}_{d}$ condition for $d ⩽ c_{\infty} + ϱ$ , where $ϱ = \frac{1}{2} (c_{f_{\infty}} - c_{\infty}) > 0$ .
Proof.
Let ${v_{n}} \subset W^{1, p (x)} (R^{N})$ be a ${(PS)}_{d}$ sequence for functional $J_{λ, k}$ with $d ⩽ c_{\infty} + ϱ$ . Similarly to Corollary 3.2, ${v_{n}}$ is a bounded sequence in $W^{1, p (x)} (R^{N})$ , and so, for some subsequence, still denoted by ${v_{n}}$ , $\begin{matrix} v_{n} ⇀ v in W^{1, p (x)} (R^{N}) \end{matrix}$ for some $v \in W^{1, p (x)} (R^{N})$ . Now, we claim that $\begin{matrix} (3.8) & J_{λ, k} (v_{n}) - J_{0, k} (w_{n}) - J_{λ, k} (v) = o_{n} (1) \end{matrix}$ and $\begin{matrix} (3.9) & ∥ J_{λ, k}^{'} (v_{n}) - J_{0, k}^{'} (w_{n}) - J_{λ, k}^{'} (v) ∥ = o_{n} (1), \end{matrix}$ where $w_{n} = v_{n} - v$ .

Indeed, proceeding as in proof of Theorem 3.5, up to subsequences, we have the following convergences $\begin{matrix} \nabla v_{n} (x) \to \nabla v (x) and v_{n} (x) \to v (x) a.e. in R^{N} . \end{matrix}$

Applying Lemma 2.3, it follows that $\begin{matrix} J_{λ, k} (v_{n}) = J_{0, k} (w_{n}) + J_{λ, k} (v) + o_{n} (1), \end{matrix}$ showing (3.8). The equality (3.9) follows combining (H1) with Lemmas 2.4 and 2.5.

Since $J_{λ, k}^{'} (v) = 0$ and $J_{λ, k} (v) ⩾ 0$ , from (3.8)–(3.9), we have that $w_{n} = v_{n} - v$ is a ${(PS)}_{d^{}}$ sequence for $J_{0, k}$ with $d^{} = d - J_{λ, k} (v) ⩽ c_{\infty} + ϱ$ . Claim 3.9.
There is $R > 0$ such that $\begin{matrix} \underset{n \to \infty}{lim sup} sup_{y \in R^{N}} \int_{B_{R} (y)} | w_{n} |^{p (x)} = 0 . \end{matrix}$

If the claim is true, we have $\begin{matrix} \int | w_{n} |^{r (x)} \to 0 . \end{matrix}$ On the other hand, by (3.9), we know that $J_{0, k}^{'} (w_{n}) = o_{n} (1)$ , then $\begin{matrix} \int (| \nabla w_{n} |^{p (x)} + | w_{n} |^{p (x)}) = o_{n} (1), \end{matrix}$ showing that $w_{n} \to 0$ in $W^{1, p (x)} (R^{N})$ , and so, $v_{n} \to v$ in $W^{1, p (x)} (R^{N})$ . Proof of Claim 3.9.
If the claim is not true, for each $R > 0$ given, we find $ξ > 0$ and ${y_{n}} \subset Z^{N}$ verifying $\begin{matrix} \underset{n \to \infty}{lim sup} \int_{B_{R} (y_{n})} | w_{n} |^{p (x)} ⩾ ξ > 0 . \end{matrix}$ Using that $w_{n} ⇀ 0$ in $W^{1, p (x)} (R^{N})$ , it follows that ${y_{n}}$ is an unbounded sequence. Setting $\begin{matrix} {\tilde{w}}_{n} = w_{n} (\cdot + y_{n}), \end{matrix}$ we have that ${{\tilde{w}}_{n}}$ is also a ${(PS)}_{d^{}}$ sequence for $J_{0, k}$ , and so, it must be bounded. Then, there are $\tilde{w} \in W^{1, p (x)} (R^{N})$ and a subsequence of ${{\tilde{w}}_{n}}$ , still denoted by itself, such that $\begin{matrix} {\tilde{w}}_{n} ⇀ \tilde{w} \in W^{1, p (x)} (R^{N}) ∖ {0} . \end{matrix}$ Moreover, since $J_{0, k}^{'} (w_{n}) ϕ (\cdot - y_{n}) = o_{n} (1)$ for each $ϕ \in W^{1, p (x)} (R^{N})$ and $\nabla {\tilde{w}}_{n} (x) \to \nabla \tilde{w} (x)$ a.e. in $R^{N}$ , we obtain $\begin{matrix} \int (| \nabla \tilde{w} |^{p (x) - 2} \nabla \tilde{w} \nabla ϕ + | \tilde{w} |^{p (x) - 2} \tilde{w} ϕ) = \int f_{\infty} | \tilde{w} |^{r (x) - 2} \tilde{w} ϕ, \end{matrix}$ from where it follows that $\tilde{w}$ is a weak solution of the problem ( P f ∞ ). Consequently, after some routine calculations, we get $\begin{matrix} c_{f_{\infty}} ⩽ J_{f_{\infty}} (\tilde{w}) = J_{f_{\infty}} (\tilde{w}) - \frac{1}{r_{-}} J_{f_{\infty}}^{'} (\tilde{w}) \tilde{w} ⩽ \underset{n \to \infty}{lim inf} {J_{0, k} (w_{n}) - \frac{1}{r_{-}} J_{0, k}^{'} (w_{n}) w_{n}} = d^{} \end{matrix}$ implying that $c_{f_{\infty}} ⩽ c_{\infty} + ϱ$ , which is an absurd because $ϱ < c_{f_{\infty}} - c_{\infty}$ . Therefore, the Claim 3.9 is true. □

In what follows, let us fix $ρ_{0}, r_{0} > 0$ satisfying:
$\overline{B_{ρ_{0}} (a_{i})} \cap \overline{B_{ρ_{0}} (a_{j})} = \emptyset$ for $i \neq j$ and $i, j \in {1, \dots, ℓ}$ ;

$⋃_{i = 1}^{ℓ} B_{ρ_{0}} (a_{i}) \subset B_{r_{0}} (0)$ ;

$K_{\frac{ρ_{0}}{2}} = ⋃_{i = 1}^{ℓ} \overline{B_{\frac{ρ_{0}}{2}} (a_{i})}$ .
Besides this, we define the function $Q_{k} : W^{1, p (x)} (R^{N}) \to R$ by $\begin{matrix} Q_{k} (u) = \frac{\int χ (k^{- 1} x) | u |^{p_{+}}}{\int | u |^{p_{+}}}, \end{matrix}$ where $χ : R^{N} \to R^{N}$ is given by $\begin{matrix} χ (x) = \{\begin{matrix} x & if | x | ⩽ r_{0}, \\ r_{0} \frac{x}{| x |} & if | x | > r_{0} . \end{matrix} \end{matrix}$

The next two lemmas will be useful to get important $(PS)$ -sequences associated with $J_{λ, k}$ .
Lemma 3.10.
There are $δ_{0} > 0$ and $k_{1} \in N$ such that if $u \in M_{0, k}$ and $J_{0, k} (u) ⩽ c_{\infty} + δ_{0}$ , then $\begin{matrix} Q_{k} (u) \in K_{\frac{ρ_{0}}{2}} for k ⩾ k_{1} . \end{matrix}$
Proof.
If the lemma does not occur, there must be $δ_{n} \to 0$ , $k_{n} \to + \infty$ and $u_{n} \in M_{0, k_{n}}$ satisfying $\begin{matrix} J_{0, k_{n}} (u_{n}) ⩽ c_{\infty} + δ_{n} \end{matrix}$ and $\begin{matrix} Q_{k_{n}} (u_{n}) \notin K_{\frac{ρ_{0}}{2}} . \end{matrix}$ Fixing $s_{n} > 0$ such that $s_{n} u_{n} \in M_{\infty}$ , we have that $\begin{matrix} c_{\infty} ⩽ J_{\infty} (s_{n} u_{n}) ⩽ J_{0, k_{n}} (s_{n} u_{n}) ⩽ max_{t ⩾ 0} J_{0, k_{n}} (t u_{n}) = J_{0, k_{n}} (u_{n}) ⩽ c_{\infty} + δ_{n} \end{matrix}$ hence, $\begin{matrix} {s_{n} u_{n}} \subset M_{\infty} and J_{\infty} (s_{n} u_{n}) \to c_{\infty} . \end{matrix}$

Applying the variational principle of Ekeland, we can assume without loss of generality that ${s_{n} u_{n}} \subset M_{\infty}$ is a sequence ${(PS)}_{c_{\infty}}$ for $J_{\infty}$ , that is, $\begin{matrix} J_{\infty} (s_{n} u_{n}) \to c_{\infty} and J_{\infty}^{'} (s_{n} u_{n}) \to 0 . \end{matrix}$ According to Theorem 3.6, we must consider the ensuing cases:
$s_{n} u_{n} \to U \neq 0$ in $W^{1, p (x)} (R^{N})$ ;
or
there exists ${y_{n}} \subset Z^{N}$ with $| y_{n} | \to + \infty$ such that $v_{n} = s_{n} u (\cdot + y_{n})$ is convergent in $W^{1, p (x)} (R^{N})$ for some $V \in W^{1, p (x)} (R^{N}) ∖ {0}$ .

By a direct computation, we can suppose that $s_{n} \to s_{0}$ for some $s_{0} > 0$ . Therefore, without loss of generality, we can assume that $\begin{matrix} u_{n} \to U or v_{n} = u (\cdot + y_{n}) \to V in W^{1, p (x)} (R^{N}) . \end{matrix}$ Analysis of (i).
By Lebesgue’s dominated convergence theorem $\begin{matrix} Q_{k_{n}} (u_{n}) = \frac{\int χ (k_{n}^{- 1} x) | u_{n} |^{p_{+}}}{\int | u_{n} |^{p_{+}}} \to \frac{\int χ (0) | U |^{p_{+}}}{\int | U |^{p_{+}}} = 0 \in K_{\frac{ρ_{0}}{2}}, \end{matrix}$ implying $Q_{k_{n}} (u_{n}) \in K_{\frac{ρ_{0}}{2}}$ for n large, which is an absurd.
Analysis of (ii).
Using again the Ekeland’s variational principle, we can suppose that $J_{0, k_{n}}^{'} (u_{n}) = o_{n} (1)$ . Hence, $J_{0, k_{n}}^{'} (u_{n}) ϕ (\cdot - y_{n}) = o_{n} (1)$ for any $ϕ \in W^{1, p (x)} (R^{N})$ , and so, $\begin{matrix} (3.10) & o_{n} (1) = \int (| \nabla v_{n} |^{p (x) - 2} \nabla v_{n} \nabla ϕ + | v_{n} |^{p (x) - 2} v_{n} ϕ) - \int f (k_{n}^{- 1} (x + y_{n})) | v_{n} |^{r (x) - 2} v_{n} ϕ . \end{matrix}$ The last limit implies that for some subsequence, $\begin{matrix} \nabla v_{n} (x) \to \nabla V (x) and v_{n} (x) \to V (x) a.e. in R^{N} . \end{matrix}$ Now, we will study two cases:
$| k_{n}^{- 1} y_{n} | \to + \infty$
and
$k_{n}^{- 1} y_{n} \to y$ for some $y \in R^{N}$ .

If (I) holds, it follows that $\begin{matrix} \int (| \nabla V |^{p (x) - 2} \nabla V \nabla ϕ + | V |^{p (x) - 2} V ϕ) = \int f_{\infty} | V |^{r (x) - 2} V ϕ, \end{matrix}$ showing that V is a nontrivial weak solution of the problem ( P f ∞ ). Now, by Fatou’s lemma, $\begin{matrix} c_{f_{\infty}} ⩽ J_{f_{\infty}} (V) = J_{f_{\infty}} (V) - \frac{1}{r_{-}} J_{f_{\infty}}^{'} (V) V ⩽ \underset{n \to \infty}{lim inf} {J_{\infty} (u_{n}) - \frac{1}{r_{-}} J_{\infty}^{'} (u_{n}) u_{n}} = c_{\infty}, \end{matrix}$ or equivalently, $c_{f_{\infty}} ⩽ c_{\infty}$ , contradicting Lemma 3.7.

Now, if $k_{n}^{- 1} y_{n} \to y$ for some $y \in R^{N}$ , then V is a weak solution of the following problem $\begin{matrix} (P_{f (y)}) & \{\begin{matrix} - Δ_{p (x)} u + | u |^{p (x) - 2} u = f (y) | u |^{r (x) - 2} u, R^{N}, \\ u \in W^{1, p (x)} (R^{N}) . \end{matrix} \end{matrix}$ Repeating the previous argument, we deduce that $\begin{matrix} (3.11) & c_{f (y)} ⩽ c_{\infty}, \end{matrix}$ where $c_{f (y)}$ the mountain pass level of the functional $J_{f (y)} : W^{1, p (x)} (R^{N}) \to R$ given by $\begin{matrix} J_{f (y)} (u) = \int \frac{1}{p (x)} (| \nabla u |^{p (x)} + | u |^{p (x)}) - \int \frac{f (y)}{r (x)} | u |^{r (x)} . \end{matrix}$ Observe that $\begin{matrix} c_{f (y)} = inf_{u \in M_{f (y)}} J_{f (y)} (u), \end{matrix}$ where $\begin{matrix} M_{f (y)} = {u \in W^{1, p (x)} (R^{N}) ∖ {0} : J_{f (y)}^{'} (u) u = 0} . \end{matrix}$ If $f (y) < 1$ , a similar argument explored in the proof of Lemma 3.7 shows that $c_{f (y)} > c_{\infty}$ , contradicting the inequality (3.11). Thereby, $f (y) = 1$ and $y = a_{i}$ for some $i = 1, \dots, ℓ$ . Hence, $\begin{array}{l} Q_{k_{n}} (u_{n}) & = \frac{\int χ (k_{n}^{- 1} x) | u_{n} |^{p_{+}}}{\int | u_{n} |^{p_{+}}} \\ = \frac{\int χ (k_{n}^{- 1} x + k_{n}^{- 1} y_{n}) | v_{n} |^{p_{+}}}{\int | v_{n} |^{p_{+}}} \to \frac{\int χ (y) | V |^{p_{+}}}{\int | V |^{p_{+}}} = a_{i} \in K_{\frac{ρ_{0}}{2}}, \end{array}$ implying that $Q_{k_{n}} (u_{n}) \in K_{\frac{ρ_{0}}{2}}$ for n large, which is a contradiction, since by assumption $Q_{k_{n}} (u_{n}) \notin K_{\frac{ρ_{0}}{2}}$ . □
Lemma 3.11.
Let $δ_{0} > 0$ be as in Lemma 3.10 and $k_{2} = max {k_{1}, k_{2}}$ . Then, there is $Λ^{} > 0$ such that* $\begin{matrix} Q_{k} (u) \in K_{\frac{ρ_{0}}{2}} \forall (u, λ, k) \in A_{λ, k} \times [0, Λ_{}) \times ([k_{2}, + \infty) \cap N) . \end{matrix}$
Proof.
Observe that $\begin{matrix} J_{λ, k} (u) = J_{0, k} (u) - λ \int \frac{g (k^{- 1} x)}{q (x)} | u |^{q (x)} \forall u \in W^{1, p (x)} (R^{N}) . \end{matrix}$ In what follows, let $t_{u} > 0$ such that $t_{u} u \in M_{0, k}$ . Then, $\begin{array}{l} J_{0, k} (t_{u} u) & = J_{λ, k} (t_{u} u) + λ \int \frac{g (k^{- 1} x)}{q (x)} {(t_{u})}^{q (x)} | u |^{q (x)} \\ (3.12) & ⩽ max_{t ⩾ 0} J_{λ, k} (t u) + λ \int \frac{g (k^{- 1} x)}{q (x)} {(t_{u})}^{q (x)} | u |^{q (x)} . \end{array}$ Claim 3.12.

There is a constant* $R > 0$ such that $\begin{matrix} A_{λ, k} = {u \in M_{λ, k}; J_{λ, k} (u) < c_{\infty} + \frac{δ_{0}}{2}} \subset B_{R} (0) \end{matrix}$ for $k ⩾ k_{1}$ , that is, $A_{λ, k}$ is a bounded set, where $k_{1}$ was given in Lemma 3.10. Moreover, R is independent of λ and k.

Let $u \in A_{λ, k}$ and $t_{u} > 0$ such that $t_{u} u \in M_{0, k}$ . Then, given $Λ > 0$ , there are $C > 0$ and $k_{2} ⩾ k_{1}$ such that $\begin{matrix} 0 ⩽ t_{u} ⩽ C for all (u, λ, k) \in A_{λ, k} \times [0, Λ] \times ([k_{2}, + \infty) \cap N) . \end{matrix}$

Proof of (a).
Taking $k ⩾ k_{1}$ , let $u \in M_{λ, k}$ be such that $J_{λ, k} (u) < c_{\infty} + \frac{δ_{0}}{2}$ for $k ⩾ k_{1}$ . Then, $\begin{matrix} \int (| \nabla u |^{p (x)} + | u |^{p (x)}) - λ \int g (k^{- 1} x) | u |^{q (x)} - \int f (k^{- 1} x) | u |^{r (x)} = 0 \end{matrix}$ and $\begin{matrix} \int \frac{1}{p (x)} (| \nabla u |^{p (x)} + | u |^{p (x)}) - λ \int \frac{g (k^{- 1} x)}{q (x)} | u |^{q (x)} - \int \frac{f (k^{- 1} x)}{r (x)} | u |^{r (x)} < c_{\infty} + \frac{δ_{0}}{2} . \end{matrix}$ Combining the last two expressions, we obtain $\begin{matrix} (\frac{1}{p_{+}} - \frac{1}{q_{-}}) \int (| \nabla u |^{p (x)} + | u |^{p (x)}) + (\frac{1}{q_{-}} - \frac{1}{r_{-}}) \int f (k^{- 1} x) | u |^{r (x)} < c_{\infty} + \frac{δ_{0}}{2} . \end{matrix}$ Therefore, $\begin{matrix} \int (| \nabla u |^{p (x)} + | u |^{p (x)}) < (c_{\infty} + \frac{δ_{0}}{2}) {(\frac{1}{p_{+}} - \frac{1}{q_{-}})}^{- 1}, \end{matrix}$ proving (a).
Proof of (b).
Arguing by contradiction, suppose that the lemma does not hold. Then, there are ${u_{n}} \subset A_{λ_{n}, k_{n}}$ with $λ_{n} \in [0, Λ]$ and $k_{n} \to + \infty$ such that $t_{u_{n}} u_{n} \in M_{0, k_{n}}$ and $t_{u_{n}} \to \infty$ as $n \to \infty$ . Without loss of generality, we assume that $t_{u_{n}} ⩾ 1$ . As $t_{u_{n}} u_{n} \in M_{0, k_{n}}$ , we derive $\begin{matrix} {(t_{u_{n}})}^{p_{+}} \int (| \nabla u_{n} |^{p (x)} + | u_{n} |^{p (x)}) ⩾ f_{\infty} {(t_{u_{n}})}^{r_{-}} \int | u_{n} |^{r (x)}, \end{matrix}$ or equivalently, $\begin{matrix} (3.13) & \int (| \nabla u_{n} |^{p (x)} + | u_{n} |^{p (x)}) ⩾ f_{\infty} t_{u_{n}}^{r_{-} - p_{+}} \int | u_{n} |^{r (x)} . \end{matrix}$ Now, we claim that there is $η_{1} > 0$ such that $\begin{matrix} (3.14) & \int | u_{n} |^{r (x)} > η_{1} \forall n \in N . \end{matrix}$ Indeed, arguing by contradiction, if $\int | u_{n} |^{r (x)} \to 0$ , by interpolation it follows that $\int_{R^{N}} | u_{n} |^{q (x)} \to 0$ . Since $u_{n} \in M_{λ_{n}, k_{n}}$ , $\begin{matrix} \int (| \nabla u_{n} |^{p (x)} + | u_{n} |^{p (x)}) ⩽ λ_{n} ∥ g ∥_{\infty} \int | u_{n} |^{q (x)} + \int | u_{n} |^{r (x)} = o_{n} (1), \end{matrix}$ or equivalently, $\begin{matrix} u_{n} \to 0 in W^{1, p (x)} (R^{N}), \end{matrix}$ which contradicts Lemma 3.3, proving (3.14). Thereby, from inequality (3.13), $\begin{matrix} ρ_{1} (u_{n}) = \int (| \nabla u_{n} |^{p (x)} + | u_{n} |^{p (x)}) \to + \infty, \end{matrix}$ implying that ${u_{n}}$ is an unbounded sequence. However, this is impossible, because by item (a), ${u_{n}}$ is bounded, showing that (b) holds.

Now, combining Claim 3.12(b) with (3.12), we get $\begin{matrix} J_{0, k} (t_{u} u) ⩽ J_{λ, k} (u) + \frac{λ}{q_{-}} ∥ g ∥_{\infty} C^{q_{+}} \int | u |^{q (x)} . \end{matrix}$ As $u \in A_{λ, k}$ , we derive $\begin{matrix} J_{0, k} (t_{u} u) < c_{\infty} + \frac{δ_{0}}{2} + λ c_{2} \int | u |^{q (x)} . \end{matrix}$ Using the Sobolev embedding theorem combined with Claim 3.12(a), we obtain $\begin{matrix} J_{0, k} (t_{u} u) < c_{\infty} + \frac{δ_{0}}{2} + c_{3} λ \forall u \in A_{λ, k}, \end{matrix}$ where $c_{3}$ is a positive constant. Setting $Λ^{} : = δ_{0} / 2 c_{3}$ and $λ \in [0, Λ^{})$ , it follows that $\begin{matrix} t_{u} u \in M_{0, k} and J_{0, k} (t_{u} u) < c_{\infty} + δ_{0} . \end{matrix}$ Then, by Lemma 3.10, $\begin{matrix} Q_{k} (t_{u} u) \in K_{\frac{ρ_{0}}{2}} . \end{matrix}$ Now, it remains to note that $\begin{matrix} Q_{k} (u) = Q_{k} (t_{u} u), \end{matrix}$ to conclude the proof of lemma. □

From now on, we will use the ensuing notation
$θ_{λ, k}^{i} = {u \in M_{λ, k}; | Q_{k} (u) - a_{i} | < ρ_{0}}$ ,

$\partial θ_{λ, k}^{i} = {u \in M_{λ, k}; | Q_{k} (u) - a_{i} | = ρ_{0}}$ ,

$β_{λ, k}^{i} = {inf}_{u \in θ_{λ, k}^{i}} J_{λ, k} (u)$
and

${\tilde{β}}_{λ, k}^{i} = {inf}_{u \in \partial θ_{λ, k}^{i}} J_{λ, k} (u)$ .

The above numbers are very important in our approach, because we will prove that there is a $(PS)$ sequence of $J_{λ, k}$ associated with each $θ_{λ, k}^{i}$ for $i = 1, 2, \dots, ℓ$ . To this end, we need of the following technical result. Lemma 3.13.
There is $k^{} \in N$ such that* $\begin{matrix} β_{λ, k}^{i} < c_{\infty} + ϱ and β_{λ, k}^{i} < {\tilde{β}}_{λ, k}^{i} \end{matrix}$ for all $λ \in [0, Λ^{})$ and* $k ⩾ k^{}$ , where* $ϱ = \frac{1}{2} (c_{f_{\infty}} - c_{\infty}) > 0$ .
Proof.
From now on, $U \in W^{1, p (x)} (R^{N})$ is a ground state solution associated with ( P ∞ ), that is, $\begin{matrix} J_{\infty} (U) = c_{\infty} and J_{\infty}^{'} (U) = 0 (see Theorem 3.6) . \end{matrix}$ For $1 ⩽ i ⩽ ℓ$ and $k \in N$ , we define the function ${\hat{U}}_{k}^{i} : R^{N} \to R$ by $\begin{matrix} {\hat{U}}_{k}^{i} (x) = U (x - k a_{i}) . \end{matrix}$ Claim 3.14.
For all $i \in {1, \dots, ℓ}$ , we have that $\begin{matrix} \underset{k \to + \infty}{lim sup} (sup_{t ⩾ 0} J_{λ, k} (t {\hat{U}}_{k}^{i})) ⩽ c_{\infty} . \end{matrix}$

Indeed, since p, q and r are $Z^{N}$ -periodic, and $a_{i} \in Z^{N}$ , a change of variable gives $\begin{array}{l} J_{λ, k} (t {\hat{U}}_{k}^{i}) = & \int \frac{t^{p (x)}}{p (x)} (| \nabla U |^{p (x)} + | U |^{p (x)}) - λ \int g (k^{- 1} x + a_{i}) \frac{t^{q (x)}}{q (x)} | U |^{q (x)} \\ - \int f (k^{- 1} x + a_{i}) \frac{t^{r (x)}}{r (x)} | U |^{r (x)} . \end{array}$ Moreover, we know that there exists $s = s (k) > 0$ such that $\begin{matrix} max_{t ⩾ 0} J_{λ, k} (t {\hat{U}}_{k}^{i}) = J_{λ, k} (s {\hat{U}}_{k}^{i}) . \end{matrix}$ By a direct computation, it follows that $s (k) ↛ 0$ and $s (k) ↛ \infty$ as $k \to \infty$ . Thus, without loss of generality, we can assume $s (k) \to s_{0} > 0$ as $k \to \infty$ . Thereby, $\begin{array}{l} lim_{k \to \infty} (max_{t ⩾ 0} J_{λ, k} (t {\hat{U}}_{k}^{i})) = & \int \frac{s_{0}^{p (x)}}{p (x)} (| \nabla U |^{p (x)} + | U |^{p (x)}) - λ \int g (a_{i}) \frac{s_{0}^{q (x)}}{q (x)} | U |^{q (x)} \\ - \int f (a_{i}) \frac{s_{0}^{r (x)}}{r (x)} | U |^{r (x)} \\ ⩽ & J_{\infty} (s_{0} U) ⩽ max_{s ⩾ 0} J_{\infty} (s U) = J_{\infty} (U) = c_{\infty} . \end{array}$ Consequently, $\begin{matrix} \underset{k \to + \infty}{lim sup} (sup_{t ⩾ 0} J_{λ, k} (t {\hat{U}}_{k}^{i})) ⩽ c_{\infty} for i \in {1, \dots, ℓ} . \end{matrix}$

Since $Q_{k} ({\hat{U}}_{k}^{i}) \to a_{i}$ as $k \to \infty$ , then ${\hat{U}}_{k}^{i} \in θ_{λ, k}^{i}$ for all k large enough. On the other hand, by Claim 3.14, $J_{λ, k} ({\hat{U}}_{k}^{i}) < c_{\infty} + \frac{δ_{0}}{4}$ holds also for k large enough and $λ \in [0, Λ^{})$ . This way, there exists $k_{3} \in N$ such that $\begin{matrix} β_{λ, k}^{i} < c_{\infty} + \frac{δ_{0}}{4} \forall λ \in [0, Λ^{}) and k ⩾ k_{3} . \end{matrix}$ Thus, decreasing $δ_{0}$ if necessary, we can assume that $\begin{matrix} β_{λ, k}^{i} < c_{\infty} + ϱ \forall λ \in [0, Λ^{}) and k ⩾ k_{3} . \end{matrix}$ In order to prove the other inequality, we observe that Lemma 3.11 yields $J_{λ, k} (u) ⩾ c_{\infty} + \frac{δ_{0}}{2}$ for all $u \in \partial θ_{λ, k}^{i}$ , if $λ \in [0, Λ^{})$ and $k ⩾ k_{2}$ . Therefore, $\begin{matrix} {\tilde{β}}_{λ, k}^{i} ⩾ c_{\infty} + \frac{δ_{0}}{2} for λ \in [0, Λ^{}) and k ⩾ k_{2} . \end{matrix}$ Fixing $k^{} = max {k_{2}, k_{3}}$ , we derive that $\begin{matrix} β_{λ, k}^{i} < {\tilde{β}}_{λ, k}^{i}, \end{matrix}$ for $λ \in [0, Λ_{})$ and $k ⩾ k^{}$ . □
Lemma 3.15.
For each $1 ⩽ i ⩽ ℓ$ , there exists a ${(PS)}_{β_{λ, k}^{i}}$ sequence, ${u_{n}^{i}} \subset θ_{λ, k}^{i}$ for functional $J_{λ, k}$ .
Proof.
By Lemma 3.13, we know that $β_{λ, k}^{i} < {\tilde{β}}_{λ, k}^{i}$ . Then, the lemma follows adapting the same ideas explored in [27]. □

4. Proof of Theorem 1.1

Let ${u_{n}^{i}} \subset θ_{λ, k}^{i}$ be a ${(PS)}_{β_{λ, k}^{i}}$ sequence for functional $J_{λ, k}$ given by Lemma 3.15. Since $β_{λ, k}^{i} < c_{\infty} + ϱ$ , by Lemma 3.8 there is $u^{i}$ such that $u_{n}^{i} \to u^{i}$ in $W^{1, p (x)} (R^{N})$ . Thus, $\begin{matrix} u^{i} \in θ_{λ, k}^{i}, J_{λ, k} (u^{i}) = β_{λ}^{i} and J_{λ, k}^{'} (u^{i}) = 0 . \end{matrix}$ Now, we infer that $u^{i} \neq u^{j}$ for $i \neq j$ as $1 ⩽ i, j ⩽ ℓ$ . To see why, it remains to observe that $\begin{matrix} Q_{k} (u^{i}) \in \overline{B_{ρ_{0}} (a_{i})} and Q_{k} (u^{j}) \in \overline{B_{ρ_{0}} (a_{j})} . \end{matrix}$ Once that $\begin{matrix} \overline{B_{ρ_{0}} (a_{i})} \cap \overline{B_{ρ_{0}} (a_{j})} = \emptyset for i \neq j, \end{matrix}$ it follows that $u^{i} \neq u^{j}$ for $i \neq j$ . From this, $J_{λ, k}$ has at least ℓ nontrivial critical points for $λ \in [0, Λ^{*})$ and $k ⩾ k^{*}$ , proving the theorem. □

Footnotes

Acknowledgements

This article was partially supported by INCT-MAT and PROCAD. C.O. Alves was partially supported by CNPq/Brazil 304036/2013-7.

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Multiple solutions for a class of quasilinear problems involving variable exponents

Abstract

Keywords

1. Introduction

Lemma 2.3 (Brezis–Lieb’s lemma, first version).

Lemma 2.4 (Brezis–Lieb’s lemma, second version).

Lemma 2.5 (Brezis–Lieb lemma, third version).

Proposition 2.6. Let u ∈ W 1 , h ( x ) ( R N ) and { u n } ⊂ W 1 , h ( x ) ( R N ) . Then, the same conclusion of Proposition 2.1 occurs replacing ∥ · ∥ h ( x ) and ρ by ∥ · ∥ and ρ 1 , respectively. 3. Technical lemmas

Case 2 ( u = 0 ).

3.2. Estimates involving the minimax levels

Footnotes

Acknowledgements

References

Proposition 2.6.
Let $u \in W^{1, h (x)} (R^{N})$ and ${u_{n}} \subset W^{1, h (x)} (R^{N})$ . Then, the same conclusion of Proposition 2.1 occurs replacing $∥ \cdot ∥_{h (x)}$ and ρ by $∥ \cdot ∥$ and $ρ_{1}$ , respectively.

3. Technical lemmas

Case 2 ( $u = 0$ ).