Multiplicity of solutions for a nonlocal nonhomogeneous Neumann boundary problem with two critical exponents

Abstract

This article concerns the multiplicity of solutions for a class of nonlocal and nonhomogeneous Neumann boundary value problems involving the $p (x)$ -Laplacian, in which both nonlinear terms assume critical growth. We use variational method, exploring an important truncation argument and properties of the genus.

Keywords

Nonlocal problem Neumann boundary conditions Sobolev spaces with variable exponent critical exponent truncation argument

1. Introduction

Let us consider the following problem $\begin{array}{l} \{\begin{matrix} - Δ_{p (x)} u + | u |^{p (x) - 2} u + ξ_{1} | u |^{h (x) - 2} u = λ | u |^{v (x) - 2} u + ξ_{2} | u |^{r (x) - 2} u {[\int_{Ω} \frac{1}{r (x)} | u |^{r (x)} d x]}^{α} & in Ω, \\ | \nabla u |^{p (x) - 2} \frac{\partial u}{\partial ν} = γ | u |^{q (x) - 2} u {[\int_{\partial Ω} \frac{1}{q (x)} | u |^{q (x)} d Γ]}^{β} & on \partial Ω, \end{matrix} \end{array}$ where $Ω \subset R^{N}$ is a bounded smooth domain of $R^{N}$ , $N ⩾ 2$ , and $p, h, v, r \in C (\overline{Ω})$ , $q \in C (\partial Ω)$ , $\frac{\partial u}{\partial ν}$ is the outer unit normal derivative, $d Γ$ denotes the boundary measure, $λ, γ, α, β, ξ_{1}, ξ_{2}$ nonnegative parameters and $Δ_{p (x)}$ is the $p (x)$ -Laplace operator, that is, $\begin{array}{l} Δ_{p (x)} u = \sum_{i = 1}^{N} \frac{\partial}{\partial x_{i}} (| \nabla u |^{p (x) - 2} \frac{\partial u}{\partial x_{i}}), 1 < p (x) < N . \end{array}$ Recalling the following critical Sobolev exponents $\begin{array}{l} p^{*} (x) = \frac{N p (x)}{N - p (x)} and p_{*} (x) = \frac{(N - 1) p (x)}{N - p (x)}, \end{array}$ where $p_{*}$ is the critical exponent from the point of view of the trace.

Note that when $ξ_{1} = ξ_{2} = β = 0$ , Corrêa and Costa [9] proved existence of nontrivial solution for the problem with critical growth on $\partial Ω$ , and subcritical growth in Ω. Furthermore, they showed existence of nontrivial solution for the problem with critical growth in Ω, and subcritical growth on $\partial Ω$ with $ξ_{1} = ξ_{2} = 0$ . When $ξ_{1} = 1$ , $ξ_{2} = β = 0$ , Corrêa and Costa [10] showed multiplicity of solutions for the problem, with critical growth on $\partial Ω$ , and subcritical growth in Ω. Considering $ξ_{1} = λ = 0$ and $ξ_{2}, α, β > 0$ , Costa, Ferreira and Tavares [11] proved existence of nontrivial solution for the problem, with critical growth simultaneously in Ω, and on $\partial Ω$ .

In this paper, we study questions of multiplicity of solutions to the following problem $\begin{array}{l} (1) & \{\begin{matrix} - Δ_{p (x)} u + | u |^{p (x) - 2} u + | u |^{h (x) - 2} u = λ | u |^{v (x) - 2} u + | u |^{r (x) - 2} u {[\int_{Ω} \frac{1}{r (x)} | u |^{r (x)} d x]}^{α} & in Ω, \\ | \nabla u |^{p (x) - 2} \frac{\partial u}{\partial ν} = | u |^{q (x) - 2} u {[\int_{\partial Ω} \frac{1}{q (x)} | u |^{q (x)} d Γ]}^{β} & on \partial Ω, \end{matrix} \end{array}$ where $Ω \subset R^{N}$ is a bounded smooth domain of $R^{N}$ , $N ⩾ 2$ , and $p, h, v, r \in C (\overline{Ω})$ , $q \in C (\partial Ω)$ , $\frac{\partial u}{\partial ν}$ is the outer unit normal derivative, $λ, α, β$ are positive parameters, $Δ_{p (x)}$ is the $p (x)$ -Laplace operator, and ${x \in \partial Ω; q (x) = p_{*} (x)}$ and ${x \in \overline{Ω}; r (x) = p^{*} (x)}$ are nonempty sets. Note that, $ξ_{1} = ξ_{2} = γ = 1$ .

It is important to emphasize, that this class of problems has several interesting motivations from both physical and mathematical point of view. They arise, for instance, in the so-called model of motion of electrorheological fluids (see [1,22,27,29]), characterized by their capability to change in drastic way the mechanical properties when influenced by an exterior electromagnetic field. See, for example, Mihailescu and Radulescu [25]. Another application of such a kind of equation is related to image processing. See Chen, Levine, and Rao [6], Ru˙z˘ic˘ka [28] and the references therein.

Problems in the form (1) are associated with the energy functional $\begin{array}{l} J_{λ} (u) = & \int_{Ω} \frac{1}{p (x)} (| \nabla u |^{p (x)} + | u |^{p (x)}) d x + \int_{Ω} \frac{1}{h (x)} | u |^{h (x)} d x - λ \int_{Ω} \frac{1}{v (x)} | u |^{v (x)} d x \\ - \frac{1}{α + 1} {[\int_{Ω} \frac{1}{r (x)} | u |^{r (x)} d x]}^{α + 1} - \frac{1}{β + 1} {[\int_{\partial Ω} \frac{1}{q (x)} | u |^{q (x)} d Γ]}^{β + 1}, \end{array}$ for all $u \in W^{1, p (x)} (Ω)$ , where $W^{1, p (x)} (Ω)$ is the generalized Lebesgue–Sobolev space.

This functional is differentiable and its Fréchet-derivative is given by $\begin{array}{l} J_{λ}^{'} (u) w = & \int_{Ω} | \nabla u |^{p (x) - 2} \nabla u \nabla w d x + \int_{Ω} | u |^{p (x) - 2} u w d x + \int_{Ω} | u |^{h (x) - 2} u w d x \\ - λ \int_{Ω} | u |^{v (x) - 2} u w d x - {[\int_{Ω} \frac{1}{r (x)} | u |^{r (x)} d x]}^{α} \int_{Ω} | u |^{r (x) - 2} u w d x \\ - {[\int_{\partial Ω} \frac{1}{q (x)} | u |^{q (x)} d Γ]}^{β} \int_{\partial Ω} | u |^{q (x) - 2} u w d Γ, \end{array}$ for all $u, w \in W^{1, p (x)} (Ω)$ . So $u \in W^{1, p (x)} (Ω)$ is a weak solution of problem (1) if, and only if, u is a critical point of $J_{λ}$ .

In the sequel, we set $\begin{array}{l} C_{+} (\overline{Ω}) : = {h \in C (\overline{Ω}); h (x) > 1 for all x \in \overline{Ω}} \end{array}$ and for each $h \in C_{+} (\overline{Ω})$ we define $\begin{array}{l} h^{+} : = max_{\overline{Ω}} h (x) and h^{-} : = min_{\overline{Ω}} h (x) . \end{array}$

We use a truncation argument, the concentration-compactness principle of Lions extended by Bonder and Silva [4], the same principle to the variable exponent spaces from the point of view of the trace, extended by Bonder and Silva [3], Fu [20] and an appropriate mini-max class of critical points via the classical concept and properties of the genus to prove our main result which reads as follows:

Theorem 1.
We denote $A_{1} : = {x \in \partial Ω; q (x) = p_{} (x)}$ and* $A_{2} : = {x \in \overline{Ω}; r (x) = p^{} (x)}$ , nonempty and disjoint sets. Moreover assume the existence of functions* $p (x), h (x), v (x), r (x) \in C_{+} (\overline{Ω})$ and $q (x) \in C_{+} (\partial Ω)$ there holds $\begin{array}{l} (2) & v^{\pm} < p^{-} ⩽ p^{+} < h^{+} < min {\frac{(α + 1) {(r^{-})}^{α + 1}}{{(r^{+})}^{α}}, \frac{(β + 1) {(q^{-})}^{β + 1}}{{(q^{+})}^{β}}} . \end{array}$ Then there exists $\bar{λ} > 0$ such that for all $0 < λ < \bar{λ}$ there are infinitely many solutions to ( 1 ) in $W^{1, p (x)} (Ω)$ .

2. Preliminaries on variable exponent spaces

We denote by $M (Ω)$ the set of real measurable functions defined on Ω.

For each $h \in C_{+} (\overline{Ω})$ , we define the generalized Lebesgue space as $\begin{array}{l} L^{h (x)} (Ω) : = {u \in M (Ω); \int_{Ω} {| u (x) |}^{h (x)} d x < \infty} . \end{array}$ We consider $L^{h (x)} (Ω)$ equipped with the Luxemburg norm $\begin{array}{l} | u |_{h (x)} : = inf {μ > 0; \int_{Ω} {| \frac{u (x)}{μ} |}^{h (x)} d x ⩽ 1} . \end{array}$ The dual space of $L^{h (x)} (Ω)$ is $L^{h^{'} (x)} (Ω)$ where $\begin{array}{l} \frac{1}{h (x)} + \frac{1}{h^{'} (x)} = 1, for all x \in \overline{Ω} . \end{array}$

The proof of the following propositions can be found in [13–16,23], Bonder and Silva [4], Bonder, Saintier and Silva [3], Fan, Shen and Zhao [17], Fan and Zhang [18] and Fan and Zhao [19].

Proposition 1 (Hölder Inequality, Fan and Zhang [18] and Kovácik and Jirí [23]).

For $u \in L^{h (x)} (Ω)$ and $w \in L^{h^{'} (x)} (Ω)$ , there holds $u w \in L^{1} (Ω)$ and $\begin{array}{l} | \int_{Ω} u w d x | ⩽ (\frac{1}{h^{-}} + \frac{1}{h^{' -}}) | u |_{h (x)} | w |_{h^{'} (x)} . \end{array}$

Denoting by $\begin{array}{l} ρ (u) : = \int_{Ω} {| u (x) |}^{h (x)} d x, u \in L^{h (x)} (Ω), \end{array}$ we derive the following proposition:

Proposition 2 (Fan and Zhang [18], Fan and Zhao [19] and David and Jirí [16]).

For all $u, u_{j} \in L^{h (x)} (Ω)$ , there holds:

For $u \neq 0, | u |_{h (x)} = λ \Leftrightarrow ρ (\frac{u}{λ}) = 1$ ;

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

If $| u |_{h (x)} > 1$ , then $| u |_{h (x)}^{h^{-}} ⩽ ρ (u) ⩽ | u |_{h (x)}^{h^{+}}$ ;

If $| u |_{h (x)} < 1$ , then $| u |_{h (x)}^{h^{+}} ⩽ ρ (u) ⩽ | u |_{h (x)}^{h^{-}}$ ;

${lim}_{j \to + \infty} | u_{j} |_{h (x)} = 0 \Leftrightarrow {lim}_{j \to + \infty} ρ (u_{j}) = 0$ ;

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

Proposition 3 (Lars Diening et al. [14]).

For any $u \in L^{h (x)} (Ω)$ and $w \subset L^{h (x)} (Ω)$ are valid:

$min {| u |_{h (x)}^{h^{-}}, | u |_{h (x)}^{h^{+}}} ⩽ ρ (u) ⩽ max {| u |_{h (x)}^{h^{-}}, | u |_{h (x)}^{h^{+}}}$ ;

$min {ρ {(u)}^{\frac{1}{h^{-}}}, ρ {(u)}^{\frac{1}{h^{+}}}} ⩽ | u |_{h (x)} ⩽ max {ρ {(u)}^{\frac{1}{h^{-}}}, ρ {(u)}^{\frac{1}{h^{+}}}}$ ;

$W \subset L^{h (x)} (Ω)$ is bounded in $L^{h (x)} (Ω)$ if and only if $ρ (W)$ is bounded in $R$ .

The generalized Lebesgue–Sobolev space $W^{1, h (x)} (Ω)$ is defined as $\begin{array}{l} W^{1, h (x)} (Ω) : = {u \in L^{h (x)} (Ω); | \nabla u | \in L^{h (x)} (Ω)} \end{array}$ with the norm $\begin{array}{l} ‖ u ‖ : = | u |_{h (x)} + | \nabla u |_{h (x)} . \end{array}$

The spaces $L^{h (x)} (Ω)$ and $W^{1, h (x)} (Ω)$ are separable and reflexive Banach spaces.

Denoting by $\begin{array}{l} ρ_{1, h (x)} : = \int_{Ω} (| u |^{h (x)} + | \nabla u |^{h (x)}) d x, u \in W^{1, h (x)} (Ω), \end{array}$ we derive the following proposition:

Proposition 4 (Fan and Zhang [18], Fan and Zhao [19] and David and Jirí [16]).

For all $u, u_{j} \in W^{1, h (x)} (Ω)$ , there holds:

$‖ u ‖ < 1 (= 1; > 1) \Leftrightarrow ρ_{1, h (x)} (u) < 1 (= 1; > 1)$ ;

If $‖ u ‖ > 1$ , then $‖ u ‖^{h^{-}} ⩽ ρ_{1, h (x)} (u) ⩽ ‖ u ‖^{h^{+}}$ ;

If $‖ u ‖ < 1$ , then $‖ u ‖^{h^{+}} ⩽ ρ_{1, h (x)} (u) ⩽ ‖ u ‖^{h^{-}}$ ;

${lim}_{j \to + \infty} ‖ u_{j} ‖ = 0 \Leftrightarrow {lim}_{j \to + \infty} ρ_{1, h (x)} (u_{j}) = 0$ ;

${lim}_{j \to + \infty} ‖ u_{j} ‖ = + \infty \Leftrightarrow {lim}_{j \to + \infty} ρ_{1, h (x)} (u_{j}) = + \infty$ .

Proposition 5 (Lars Diening et al. [14]).

Let $m \in L^{\infty} (Ω)$ with $m_{-} > 0$ . Then, given $u \in W^{1, h (x)} (Ω)$ , we have $\begin{array}{l} max {‖ u ‖^{m^{-}}, ‖ u ‖^{m^{+}}} ⩽ max {ρ_{1, h (x)} {(u)}^{\frac{m^{-}}{h^{+}}}, ρ_{1, h (x)} {(u)}^{\frac{m^{+}}{h^{-}}}} . \end{array}$

Proposition 6 (David Jirí [15] and Fan et al. [17]).

Suppose that Ω is a bounded smooth domain in $R^{N}$ and $h, h_{1} \in C_{+} (\overline{Ω})$ with $N > h^{+}$ . If $h_{1} (x) ⩽ h^{*} (x) (respectively h_{1} (x) < h^{*} (x))$ , for all $x \in \overline{Ω}$ , then the embedding $\begin{array}{l} W^{1, h (x)} (Ω) ↪ L^{h_{1} (x)} (Ω), \end{array}$ is continuous (respectively compact).

Let $\begin{array}{l} q \in C_{+} (\partial Ω) : = {h \in C (\partial Ω); h (x) > 1 for all x \in \partial Ω} . \end{array}$

The Lebesgue space $L^{q (x)} (\partial Ω)$ is defined as $\begin{array}{l} L^{q (x)} (\partial Ω) : = {u; u : \partial Ω \to R is measurable and \int_{\partial Ω} | u |^{q (x)} d Γ < \infty}, \end{array}$ and the corresponding (Luxemburg) norm is given by $\begin{array}{l} ‖ u ‖_{L^{q (x)} (\partial Ω)} : = ‖ u ‖_{q (x), \partial Ω} : = inf {λ > 0; \int_{\partial Ω} {| \frac{u (x)}{λ} |}^{q (x)} d Γ ⩽ 1} . \end{array}$

Proposition 7 (Diening [13] and Fan et al. [17]).

Suppose that Ω is a bounded smooth domain in $R^{N}$ , $q \in C_{+} (\partial Ω)$ and $p \in C_{+} (\overline{Ω})$ with $N > p^{+}$ . Then, if $q (x) < p_{*} (x)$ , for all $x \in \partial Ω$ , the embedding $\begin{array}{l} W^{1, p (x)} (Ω) ↪ L^{q (x)} (\partial Ω), \end{array}$ is compact.

Proposition 8 (Fan and Zhang [18]).

Let $L_{p (x)} : W^{1, p (x)} (Ω) \to {(W^{1, p (x)} (Ω))}^{'}$ be such that $\begin{array}{l} L_{p (x)} (u) (v) = \int_{Ω} | \nabla u |^{p (x) - 2} \nabla u \cdot \nabla v d x, \forall u, v \in W^{1, p (x)} (Ω) . \end{array}$ Then

$L_{p (x)} : W^{1, p (x)} (Ω) \to {(W^{1, p (x)} (Ω))}^{'}$ is a continuous, bounded and strictly monotone operator;

$L_{p (x)}$ is a mapping of type $S_{+}$ , i.e., if $u_{n} ⇀ u$ in $W^{1, p (x)} (Ω)$ and $\begin{array}{l} lim sup (L_{p (x)} (u_{n}) - L_{p (x)} (u), u_{n} - u) ⩽ 0, \end{array}$ then $u_{n} \to u$ in $W^{1, p (x)} (Ω)$ ;

$L_{p (x)} : W^{1, p (x)} (Ω) \to {(W^{1, p (x)} (Ω))}^{'}$ is a homeomorphism.

Proposition 9 (Bonder, Saintier and Silva [3]).

Let $Ω \subset R^{N}$ be a bounded smooth domain, $h \in C_{+} (\overline{Ω})$ , $l \in C_{+} (\partial Ω)$ with $\begin{array}{l} l (x) ⩽ h^{*} (x), \forall x \in \partial Ω, \end{array}$ and $(u_{j})$ a sequence in $W^{1, h (x)} (Ω)$ such that $\begin{array}{l} u_{j} ⇀ u weakly in W^{1, h (x)} (Ω) . \end{array}$ Then there exists a countable set $I_{1}$ , positive numbers ${μ_{i}}_{i \in I_{1}}$ and ${ν_{i}}_{i \in I_{1}}$ and points ${x_{i}}_{i \in I_{1}} \subset {x \in \partial Ω; l (x) = h_{*} (x)}$ such that $\begin{array}{l} (3) & | u_{j} |^{l (x)} d S ⇀ ν = | u |^{l (x)} d S + \sum_{i \in I_{1}} ν_{i} δ_{x_{i}}, weakly-* in the sense of measures, \\ (4) & | \nabla u_{j} |^{h (x)} d x ⇀ μ ⩾ | \nabla u |^{h (x)} d x + \sum_{i \in I_{1}} μ_{i} δ_{x_{i}}, weakly-* in the sense of measures \end{array}$ and $\begin{array}{l} (5) & {\overline{T}}_{x_{i}} ν_{i}^{\frac{1}{h_{*} (x_{i})}} ⩽ μ_{i}^{\frac{1}{h (x_{i})}}, \forall i \in I_{1}, \end{array}$ where $\begin{array}{l} {\overline{T}}_{x_{i}} = sup_{ϵ > 0} T (h (\cdot), l (\cdot), Ω_{ϵ, i}, Λ_{ϵ, i}) \end{array}$ is the localized Sobolev trace constant, $\begin{array}{l} Ω_{ϵ, i} = Ω \cap B_{ϵ} (x_{i}) and Λ_{ϵ, i} = Ω \cap \partial B_{ϵ} (x_{i}) . \end{array}$

Proposition 10 (Bonder and Silva [4]).

Let $Ω \subset R^{N}$ be a bounded smooth domain, $h, m \in C_{+} (\overline{Ω}), h^{+} < N$ with $\begin{array}{l} m (x) ⩽ h^{*} (x), \forall x \in \overline{Ω}, \end{array}$ and $(u_{j})$ a sequence in $W^{1, h (x)} (Ω)$ such that $\begin{array}{l} u_{j} ⇀ u weakly in W^{1, h (x)} (Ω) . \end{array}$ Then there exists a countable set $I_{2}$ , positive numbers ${μ_{i}}_{i \in I_{2}}$ and ${ν_{i}}_{i \in I_{2}}$ and points ${x_{i}}_{i \in I_{2}} \subset {x \in \overline{Ω}; m (x) = h^{*} (x)}$ such that $\begin{array}{l} | u_{j} |^{m (x)} ⇀ ν = | u |^{m (x)} + \sum_{i \in I_{2}} ν_{i} δ_{x_{i}} weakly-* in the sense of measures, \\ | \nabla u_{j} |^{h (x)} ⇀ μ ⩾ | \nabla u |^{h (x)} + \sum_{i \in I_{2}} μ_{i} δ_{i} weakly-* in the sense of measures, \end{array}$ and $\begin{array}{l} σ_{m} ν_{i}^{1 / h^{*} (x_{i})} ⩽ μ_{i}^{1 / h (x_{i})}, \forall i \in I_{2}, \end{array}$ where $σ_{m}$ is the best constant in the Gagliardo–Nirenberg–Sobolev inequality for variable exponents, namely, $\begin{array}{l} σ_{m} = σ_{m} (Ω) = inf_{ϕ \in C_{0}^{\infty} (Ω)} \frac{‖ | \nabla ϕ | ‖_{L^{h (x)} (Ω)}}{‖ ϕ ‖_{L^{m (x)} (Ω)}} . \end{array}$

Definition 1.
Let X be a Banach space and a functional $J \in C^{1} (X, R)$ . Given a sequence $(u_{j})$ in X, if there exist $d \in R$ such that $\begin{array}{l} (6) & J (u_{j}) \to d and J^{'} (u_{j}) \to 0 in X^{'}, \end{array}$ we say that $(u_{j})$ is a Palais–Smale sequence with energy level d (or $(u_{j})$ is ${(P S)}_{d}$ for short). When any ${(P S)}_{d}$ sequence for J possesses some strongly convergent subsequence in X, we say that J satisfies the Palais–Smale condition at level d (or J is ${(P S)}_{d}$ for short).

3. Abstract framework

Let X be a real Banach space. Let us denote by Σ the class of all closed subsets $A \subset X ∖ {0}$ that are symmetric with respect to the origin, that is, $u \in A$ implies $- u \in A$ .

Definition 2.
Let $A \in Σ$ . The genus $γ (A)$ of A is defined as being the least positive integer k such that there is an odd mapping $ϕ \in C (A, R^{k})$ such that $ϕ (x) \neq 0$ for all $x \in A$ . If k does not exist we set $γ (A) = \infty$ . Furthermore, by definition $γ (\emptyset) = 0$ .

In the sequel we will quote only the properties of genus that will be used through this work. More information on this subject may be found in Ambrosetti and Malchiodi [2], Castro [5], G. Costa [12] and Krasnoselskii [24].
Theorem 2 (Rabinowitz [26]).

Let $X = R^{N}$ and $\partial Ω$ be the boundary of an open, symmetric and bounded subset $Ω \subset R^{N}$ with $0 \in Ω$ . Then $γ (\partial Ω) = N$ .

Proposition 11 (Rabinowitz [26]).

$γ (S^{N - 1}) = N$ .

As a consequence of this, if X is of infinite dimension and separable and S is the unit sphere in X, then $γ (S) = \infty$ .

Theorem 3 (Rabinowitz [26]).

Let $A, B \in Σ$ be, so:

If $γ (A) > 1$ , then A contains infinitely many distinct points;

If there is an odd application $h \in C (A, B)$ , then $γ (A) ⩽ γ (B)$ ;

If $A \subset B$ then $γ (A) ⩽ γ (B)$ ;

$γ (A \cup B) ⩽ γ (A) + γ (B)$ ;

If V is a subspace of E of codimension k and $γ (A) > k$ , then $A \cap V \neq \emptyset$ ;

If A is a compact, then $γ (A) < \infty$ and there is one $δ > 0$ such that $N_{δ} (A) \subset Σ$ and $γ (N_{δ} (A)) = γ (A)$ where $N_{δ} (A) = {u \in Σ; ‖ u - a ‖ ⩽ δ, \forall a \in A}$ ;

If $γ (B) < \infty$ , then $γ (\overline{A ∖ B}) ⩾ γ (A) - γ (B)$ ;

If W is a neighborhood of 0 in $R^{k}$ and there is an odd homeomorphism $h \in C (A, \partial W)$ then $γ (A) = k$ .

Theorem 4 (Clark [7]).

Let $J \in C^{1} (X, R)$ be a functional satisfying the Palais–Smale condition. Furthermore, let us suppose that:

J is bounded from below and even;

There is a compact set $K \in U$ such that $γ (K) = k$ and ${sup}_{x \in K} J (x) < J (0) = 0$ .

Then J possesses at least k pairs of distinct critical points and their corresponding critical values are less than $J (0)$ .

4. Proof of Theorem 1

The proof follows from several lemmas:

Lemma 1.

If $(u_{j}) \subset W^{1, p (x)} (Ω)$ is a Palais–Smale sequence, with energy level d, then $(u_{j})$ is bounded in $W^{1, p (x)} (Ω)$ .

Proof.

Since $(u_{j}) \subset W^{1, p (x)} (Ω)$ is a ${(P S)}_{d}$ for $J_{λ}$ , we have $J_{λ} (u_{j}) \to d$ e $J_{λ}^{'} (u_{j}) \to 0$ . Then, considering θ such that $\begin{array}{l} (7) & h^{+} < θ < min {\frac{(α + 1) {(r^{-})}^{α + 1}}{{(r^{+})}^{α}}, \frac{(β + 1) {(q^{-})}^{β + 1}}{{(q^{+})}^{β}}}, \end{array}$ and $C > 0$ such that $\begin{array}{l} (8) & C ⩾ J_{λ} (u_{j}), \forall j \end{array}$ we obtain $\begin{array}{l} C + ‖ u_{j} ‖ ⩾ & J_{λ} (u_{j}) - \frac{1}{θ} J_{λ}^{'} (u_{j}) u_{j} \\ ⩾ & (\frac{1}{p^{+}} - \frac{1}{θ}) ρ_{1, p (x)} (u_{j}) + (\frac{1}{h^{+}} - \frac{1}{θ}) \int_{Ω} | u_{j} |^{h (x)} d x + (\frac{λ}{θ} - \frac{λ}{v^{-}}) \int_{Ω} | u_{j} |^{v (x)} d x \\ + (\frac{1}{θ} \cdot \frac{1}{{(r^{+})}^{α}} - \frac{1}{α + 1} \cdot \frac{1}{{(r^{-})}^{α + 1}}) {[\int_{Ω} | u_{j} |^{r (x)} d x]}^{α + 1} \\ + (\frac{1}{θ} \cdot \frac{1}{{(q^{+})}^{β}} - \frac{1}{β + 1} \cdot \frac{1}{{(q^{-})}^{β + 1}}) {[\int_{\partial Ω} | u_{j} |^{q (x)} d Γ]}^{β + 1} . \end{array}$

Now, let us suppose that $(u_{j})$ is unbounded in $W^{1, p (x)} (Ω)$ . Passing to a subsequence if necessary and we get $‖ u_{j} ‖ > 1$ and we obtain $\begin{array}{l} C + ‖ u_{j} ‖ ⩾ (\frac{1}{p^{+}} - \frac{1}{θ}) ‖ u_{j} ‖^{p^{-}} + (\frac{λ}{θ} - \frac{λ}{v^{-}}) \int_{Ω} | u_{j} |^{v (x)} d x . \end{array}$

So, $\begin{array}{l} C + ‖ u_{j} ‖ ⩾ (\frac{1}{p^{+}} - \frac{1}{θ}) ‖ u_{j} ‖^{p^{-}} + (\frac{λ}{θ} - \frac{λ}{v^{-}}) | u_{j} |_{v (x)}^{v^{+}} . \end{array}$

Using embedding of Sobolev, we have $\begin{array}{l} C + ‖ u_{j} ‖ ⩾ (\frac{1}{p^{+}} - \frac{1}{θ}) ‖ u_{j} ‖^{p^{-}} + (\frac{λ}{θ} - \frac{λ}{v^{-}}) ‖ u_{j} ‖^{v^{\pm}}, \end{array}$ which is a contradiction because $p^{-} > v^{\pm} > 1$ . Hence $(u_{j})$ is bounded in $W^{1, p (x)} (Ω)$ . □

Lemma 2.

Let $(u_{j}) \subset W^{1, p (x)} (Ω)$ a Palais–Smale sequence with energy level d. If $\begin{array}{l} d < & min {L_{1} inf_{i \in I_{1}} {({\overline{c}}^{- \frac{1}{p (x_{i})}} {\overline{T}}_{x_{i}})}^{\frac{(β + 1) p_{*} (x_{i}) p (x_{i})}{p_{*} (x_{i}) - p (x_{i})}}, L_{2} inf_{i \in I_{2}} {({\tilde{c}}^{- \frac{1}{p (x_{i})}} S_{m})}^{\frac{(α + 1) p^{*} (x_{i}) p (x_{i})}{p^{*} (x_{i}) - p (x_{i})}}} \\ + K min {λ^{\frac{{(h / v)}^{-}}{{(h / v)}^{-} - 1}}, λ^{\frac{{(h / v)}^{+}}{{(h / v)}^{+} - 1}}} \end{array}$ where $\begin{array}{l} L_{1} = (\frac{1}{θ} \cdot \frac{1}{{(q_{A_{1}}^{+})}^{β}} - \frac{1}{β + 1} \cdot \frac{1}{{(q_{A_{1}}^{-})}^{β + 1}}) and L_{2} = (\frac{1}{θ} \cdot \frac{1}{{(r_{A_{2}}^{+})}^{α}} - \frac{1}{α + 1} \cdot \frac{1}{{(r_{A_{2}}^{-})}^{α + 1}}) . \end{array}$ Then, there exists $λ_{0} > 0$ such that, for all $0 < λ < λ_{0}$ the index sets $I_{1}$ and $I_{2}$ given in the Propositions 9 and 10 are empty and $\begin{array}{l} u_{j} \to u strongly in L^{q (x)} (\partial Ω) and in L^{r (x)} (Ω) . \end{array}$

Proof.

By Lemma 1, we have a subsequence, still denoted by $(u_{j})$ , such that $u_{j} ⇀ u$ weakly in $W^{1, p (x)} (Ω)$ , and so we can use the Propositions 9 and 10. Note that if $I_{1} \cup I_{2} \neq \emptyset$ , then $I_{1} \neq \emptyset$ or $I_{2} \neq \emptyset$ .

Firstly assume the case $I_{1} \neq \emptyset$ . Let $x_{i} \in A_{1}$ be a singular point of the measures μ and ν. We consider $ϕ \in C_{0}^{\infty} (R^{N})$ , such that $0 ⩽ ϕ (x) ⩽ 1$ , $ϕ (0) = 1$ and $s u p p ϕ \subset B_{1} (0)$ .

Consider for each $i \in I_{1}$ and $ε > 0$ the functions $\begin{array}{l} ϕ_{i, ε} (x) = ϕ (\frac{x - x_{i}}{ε}), \forall x \in R^{N} . \end{array}$

Since $J_{λ}^{'} (u_{j}) \to 0$ , we obtain $\begin{array}{l} J_{λ}^{'} (u_{j}) (ϕ_{i, ε} u_{j}) \to 0 . \end{array}$

That is, $\begin{array}{l} \int_{Ω} | \nabla u_{j} |^{p (x) - 2} \nabla u_{j} \nabla (ϕ_{i, ε} u_{j}) d x + \int_{Ω} | u_{j} |^{p (x) - 2} u_{j} ϕ_{i, ε} u_{j} d x + \int_{Ω} | u_{j} |^{h (x) - 2} u_{j} ϕ_{i, ε} u_{j} d x \\ - λ \int_{Ω} | u_{j} |^{v (x) - 2} u_{j} ϕ_{i, ε} u_{j} d x - {(\int_{Ω} \frac{1}{r (x)} | u_{j} |^{r (x)} d x)}^{α} \int_{Ω} | u_{j} |^{r (x) - 2} u_{j} ϕ_{i, ε} u_{j} d x \\ - {(\int_{\partial Ω} \frac{1}{q (x)} | u_{j} |^{q (x)} d Γ)}^{β} \int_{\partial Ω} | u_{j} |^{q (x) - 2} u_{j} ϕ_{i, ε} u_{j} d Γ \to 0, \end{array}$ which implies $\begin{array}{l} \int_{Ω} | \nabla u_{j} |^{p (x) - 2} \nabla u_{j} \nabla ϕ_{i, ε} u_{j} d x + \int_{Ω} | \nabla u_{j} |^{p (x)} ϕ_{i, ε} d x + \int_{Ω} | u_{j} |^{p (x)} ϕ_{i, ε} d x + \int_{Ω} | u_{j} |^{h (x)} ϕ_{i, ε} d x \\ - λ \int_{Ω} | u_{j} |^{v (x)} ϕ_{i, ε} d x - {(\int_{Ω} \frac{1}{r (x)} | u_{j} |^{r (x)} d x)}^{α} \int_{Ω} | u_{j} |^{r (x)} ϕ_{i, ε} d x \\ - {(\int_{\partial Ω} \frac{1}{q (x)} | u_{j} |^{q (x)} d Γ)}^{β} \int_{\partial Ω} | u_{j} |^{q (x)} ϕ_{i, ε} d Γ \to 0 . \end{array}$

When $j \to \infty$ we get $\begin{array}{l} 0 = & lim [\int_{Ω} | \nabla u_{j} |^{p (x) - 2} \nabla u_{j} \nabla ϕ_{i, ε} u_{j} d x + \int_{Ω} | u_{j} |^{p (x)} ϕ_{i, ε} d x + \int_{Ω} | u_{j} |^{h (x)} ϕ_{i, ε} d x \\ - λ \int_{Ω} | u_{j} |^{v (x)} ϕ_{i, ε} d x - {(\int_{Ω} \frac{1}{r (x)} | u_{j} |^{r (x)} d x)}^{α} \int_{Ω} | u_{j} |^{r (x)} ϕ_{i, ε} d x] \\ + \int_{Ω} ϕ_{i, ε} d μ - \int_{\partial Ω} ϕ_{i, ε} d \bar{ν}, \end{array}$ where $\begin{array}{l} (9) & lim_{j \to \infty} {(\int_{\partial Ω} \frac{1}{q (x)} | u_{j} |^{q (x)} d Γ)}^{β} | u_{j} |^{q (x)} ⇀ \bar{ν} . \end{array}$

We can show that ${lim}_{ε \to 0} \int_{Ω} u | \nabla u |^{p (x) - 2} \nabla u \nabla ϕ_{i, ε} d x = 0$ see Shang–Wang [30].

On the other hand, $\begin{array}{l} lim_{ε \to 0} \int_{Ω} ϕ_{i, ε} d μ = μ ϕ (0) and lim_{ε \to 0} \int_{\partial Ω} ϕ_{i, ε} d \bar{ν} = \bar{ν} ϕ (0) \end{array}$ and since $A_{1} \cap A_{2} = \emptyset$ , for $ε > 0$ sufficiently small $\begin{array}{l} {(\int_{Ω} \frac{1}{r (x)} | u_{j} |^{r (x)} d x)}^{α} \int_{Ω} | u_{j} |^{r (x)} ϕ_{i, ε} d x \to {(\int_{Ω} \frac{1}{r (x)} | u |^{r (x)} d x)}^{α} \int_{Ω} | u |^{r (x)} ϕ_{i, ε} d x, \\ \int_{Ω} | u_{j} |^{p (x)} ϕ_{i, ε} d x \to \int_{Ω} | u |^{p (x)} ϕ_{i, ε} d x, \int_{Ω} | u_{j} |^{h (x)} ϕ_{i, ε} d x \to \int_{Ω} | u |^{h (x)} ϕ_{i, ε} d x, and \\ λ \int_{Ω} | u_{j} |^{v (x)} ϕ_{i, ε} d x \to λ \int_{Ω} | u |^{v (x)} ϕ_{i, ε} d x . \end{array}$

Once that, when $ε \to 0$ , $\begin{array}{l} {(\int_{Ω} \frac{1}{r (x)} | u |^{r (x)} d x)}^{α} \int_{Ω} | u |^{r (x)} ϕ_{i, ε} d x \to 0, \int_{Ω} | u |^{p (x)} ϕ_{i, ε} d x \to 0, \\ \int_{Ω} | u |^{h (x)} ϕ_{i, ε} d x \to 0, and λ \int_{Ω} | u |^{v (x)} ϕ_{i, ε} d x \to 0 . \end{array}$ Thus, $\begin{array}{l} μ_{i} ϕ (0) = {\bar{ν}}_{i} ϕ (0), \end{array}$ and we have $\begin{array}{l} μ_{i} ⩽ \bar{c} ν_{i}, \end{array}$ where $\bar{c}$ comes from the convergence (9). By ${\overline{T}}_{x_{i}} ν_{i}^{\frac{1}{p_{*} (x_{i})}} ⩽ μ_{i}^{\frac{1}{p (x_{i})}}$ , we obtain ${({\overline{T}}_{x_{i}})}^{p (x_{i})} ν_{i}^{\frac{p (x_{i})}{p_{*} (x_{i})}} ⩽ μ_{i} ⩽ \bar{c} ν_{i}$ . Thus ${({\overline{T}}_{x_{i}})}^{p (x_{i})} ⩽ \bar{c} ν_{i}^{\frac{p_{*} (x_{i}) - p (x_{i})}{p_{*} (x_{i})}}$ , which implies ${\bar{c}}^{- \frac{1}{p (x_{i})}} {\overline{T}}_{x_{i}} ⩽ ν_{i}^{\frac{p_{*} (x_{i}) - p (x_{i})}{p (x_{i}) p_{*} (x_{i})}}$ . Therefore $\begin{array}{l} ν_{i} ⩾ {({\bar{c}}^{- \frac{1}{p (x_{i})}} {\overline{T}}_{x_{i}})}^{(p (x_{i}) p_{*} (x_{i}) / p_{*} (x_{i}) - p (x_{i}))} . \end{array}$

On the other hand, $\begin{array}{l} d = lim J_{λ} (u_{j}) = lim (J_{λ} (u_{j}) - \frac{1}{θ} J_{λ}^{'} (u_{j}) u_{j}), \end{array}$ thus, using θ satisfying (7), we obtain $\begin{array}{l} d ⩾ & lim {(\frac{1}{h^{+}} - \frac{1}{θ}) \int_{Ω} | u_{j} |^{h (x)} d x + (\frac{λ}{θ} - \frac{λ}{v^{-}}) \int_{Ω} | u_{j} |^{v (x)} d x \\ + (\frac{1}{θ} \cdot \frac{1}{{(q^{+})}^{β}} - \frac{1}{β + 1} \cdot \frac{1}{{(q^{-})}^{β + 1}}) {[\int_{\partial Ω} | u_{j} |^{q (x)} d Γ]}^{β + 1}} . \end{array}$

Now, setting $A_{δ} = ⋃_{x \in A_{1}} (B_{δ} (x) \cap Ω) = {x \in Ω : dist (x, A_{1}) < δ}$ , we obtain $\begin{array}{l} d ⩾ & (\frac{1}{h^{+}} - \frac{1}{θ}) \int_{Ω} | u |^{h (x)} d x + (\frac{λ}{θ} - \frac{λ}{v^{-}}) \int_{Ω} | u |^{v (x)} d x \\ + (\frac{1}{θ} \cdot \frac{1}{{(q_{A_{δ}}^{+})}^{β}} - \frac{1}{β + 1} \cdot \frac{1}{{(q_{A_{δ}}^{-})}^{β + 1}}) inf_{i \in I_{1}} {({\bar{c}}^{- \frac{1}{p (x_{i})}} {\overline{T}}_{x_{i}})}^{\frac{(β + 1) p (x_{i}) p_{*} (x_{i})}{p_{*} (x_{i}) - p (x_{i}))}} . \end{array}$

Since $δ > 0$ is arbitrary and q is continuous, we get $\begin{array}{l} d ⩾ & (\frac{1}{h^{+}} - \frac{1}{θ}) \int_{Ω} | u |^{h (x)} d x + (\frac{λ}{θ} - \frac{λ}{v^{-}}) \int_{Ω} | u |^{v (x)} d x \\ + (\frac{1}{θ} \cdot \frac{1}{{(q_{A_{1}}^{+})}^{β}} - \frac{1}{β + 1} \cdot \frac{1}{{(q_{A_{1}}^{-})}^{β + 1}}) inf_{i \in I_{1}} {({\bar{c}}^{- \frac{1}{p (x_{i})}} {\overline{T}}_{x_{i}})}^{\frac{(β + 1) p (x_{i}) p_{*} (x_{i})}{p_{*} (x_{i}) - p (x_{i}))}} . \end{array}$

Applying now Hölder inequality, we find $\begin{array}{l} d ⩾ & (\frac{1}{h^{+}} - \frac{1}{θ}) \int_{Ω} | u |^{h (x)} d x + (\frac{λ}{θ} - \frac{λ}{v^{-}}) \cdot (| | u |^{v (x)} |_{\frac{h (x)}{v (x)}} \cdot | Ω |^{\frac{h^{+} - v^{-}}{h^{-}}}) \\ + (\frac{1}{θ} \cdot \frac{1}{{(q_{A_{1}}^{+})}^{β}} - \frac{1}{β + 1} \cdot \frac{1}{{(q_{A_{1}}^{-})}^{β + 1}}) inf_{i \in I_{1}} {({\bar{c}}^{- \frac{1}{p (x_{i})}} {\overline{T}}_{x_{i}})}^{\frac{(β + 1) p (x_{i}) p_{*} (x_{i})}{p_{*} (x_{i}) - p (x_{i}))}} . \end{array}$

If $| | u |^{v (x)} |_{\frac{h (x)}{v (x)}} ⩾ 1$ , we have $\begin{array}{l} d ⩾ c_{1} | | u |^{v (x)} |_{\frac{h (x)}{v (x)}}^{{(h / v)}^{-}} - λ c_{2} | | u |^{v (x)} |_{\frac{h (x)}{v (x)}} + c_{3} \end{array}$ where $\begin{array}{l} c_{1} = (\frac{1}{h^{+}} - \frac{1}{θ}) > 0, c_{2} = (\frac{1}{v^{-}} - \frac{1}{θ}) \cdot | Ω |^{\frac{h^{+} - v^{-}}{h^{-}}} > 0 \end{array}$ and $\begin{array}{l} c_{3} = (\frac{1}{θ} \cdot \frac{1}{{(q_{A_{1}}^{+})}^{β}} - \frac{1}{β + 1} \cdot \frac{1}{{(q_{A_{1}}^{-})}^{β + 1}}) inf_{i \in I_{1}} {({\bar{c}}^{- \frac{1}{p (x_{i})}} {\overline{T}}_{x_{i}})}^{\frac{(β + 1) p (x_{i}) p_{*} (x_{i})}{p_{*} (x_{i}) - p (x_{i}))}} . \end{array}$

So, if $g_{1} (t) = c_{1} t^{{(h / v)}^{-}} - λ c_{2} t$ , this function attains its absolute minimum, for $t > 0$ , at the point $\begin{array}{l} \overline{t} = {(\frac{λ c_{2}}{c_{1} {(h / v)}^{-}})}^{\frac{1}{{(h / v)}^{-} - 1}} . \end{array}$

Note that, $\begin{array}{l} g_{1} (\overline{t}) = c_{1} {(\frac{λ c_{2}}{c_{1} {(h / v)}^{-}})}^{\frac{{(h / v)}^{-}}{{(h / v)}^{-} - 1}} - λ c_{2} {(\frac{λ c_{2}}{c_{1} {(h / v)}^{-}})}^{\frac{1}{{(h / v)}^{-} - 1}}, \end{array}$ which implies $\begin{array}{l} g_{1} (\overline{t}) = c_{1} {(\frac{λ c_{2}}{c_{1} {(h / v)}^{-}})}^{\frac{{(h / v)}^{-}}{{(h / v)}^{-} - 1}} - \frac{c_{1} {(h / v)}^{-}}{c_{1} {(h / v)}^{-}} \cdot λ c_{2} {(\frac{λ c_{2}}{c_{1} {(h / v)}^{-}})}^{\frac{1}{{(h / v)}^{-} - 1}} . \end{array}$

Thus, $\begin{array}{l} g_{1} (\overline{t}) = c_{1} {(\frac{λ c_{2}}{c_{1} {(h / v)}^{-}})}^{\frac{{(h / v)}^{-}}{{(h / v)}^{-} - 1}} - c_{1} {(h / v)}^{-} \cdot {(\frac{λ c_{2}}{c_{1} {(h / v)}^{-}})}^{\frac{1}{{(h / v)}^{-} - 1} + 1} . \end{array}$

So, $\begin{array}{l} g_{1} (\overline{t}) = c_{1} {(\frac{λ c_{2}}{c_{1} {(h / v)}^{-}})}^{\frac{{(h / v)}^{-}}{{(h / v)}^{-} - 1}} \cdot (1 - {(h / v)}^{-}) . \end{array}$

Using the fact that $v^{\pm} < h^{+}$ , we can write $\begin{array}{l} g_{1} (\overline{t}) = λ^{\frac{{(h / v)}^{-}}{{(h / v)}^{-} - 1}} \cdot k_{1}, \end{array}$ where $k_{1}$ is a negative constant depending only on h, v and Ω.

If $| | u |^{v (x)} |_{\frac{h (x)}{v (x)}} < 1$ , we have $\begin{array}{l} d ⩾ c_{1} | | u |^{v (x)} |_{\frac{h (x)}{v (x)}}^{{(h / v)}^{+}} - λ c_{2} | | u |^{v (x)} |_{\frac{h (x)}{v (x)}} + c_{3}, \end{array}$ where $\begin{array}{l} c_{1} = (\frac{1}{h^{+}} - \frac{1}{θ}) > 0, c_{2} = (\frac{1}{v^{-}} - \frac{1}{θ}) \cdot | Ω |^{\frac{h^{+} - v^{-}}{h^{-}}} > 0 \end{array}$ and $\begin{array}{l} c_{3} = (\frac{1}{θ} \cdot \frac{1}{{(q_{A_{1}}^{+})}^{β}} - \frac{1}{β + 1} \cdot \frac{1}{{(q_{A_{1}}^{-})}^{β + 1}}) inf_{i \in I_{1}} {({\bar{c}}^{- \frac{1}{p (x_{i})}} {\overline{T}}_{x_{i}})}^{\frac{(β + 1) p (x_{i}) p_{*} (x_{i})}{p_{*} (x_{i}) - p (x_{i}))}} . \end{array}$

So, if $g_{2} (t) = c_{1} t^{{(h / v)}^{+}} - λ c_{2} t$ , this function attains its absolute minimum, for $t > 0$ , at the point $\begin{array}{l} \underline{t} = {(\frac{λ c_{2}}{c_{1} {(h / v)}^{+}})}^{\frac{1}{{(h / v)}^{+} - 1}} . \end{array}$

Thus, we obtain $\begin{array}{l} g_{2} (\underline{t}) = λ^{\frac{{(h / v)}^{+}}{{(h / v)}^{+} - 1}} \cdot k_{2} \end{array}$ where $k_{2}$ is a negative constant depending only on h, v and Ω.

Then $\begin{array}{l} d ⩾ L_{1} inf_{i \in I_{1}} {({\bar{c}}^{- \frac{1}{p (x_{i})}} {\overline{T}}_{x_{i}})}^{\frac{(β + 1) p (x_{i}) p_{*} (x_{i})}{p_{*} (x_{i}) - p (x_{i}))}} + K min {λ^{\frac{{(h / v)}^{-}}{{(h / v)}^{-} - 1}}, λ^{\frac{{(h / v)}^{+}}{{(h / v)}^{+} - 1}}}, \end{array}$ where $L_{1} = (\frac{1}{θ} \cdot \frac{1}{{(q_{A_{1}}^{+})}^{β}} - \frac{1}{β + 1} \cdot \frac{1}{{(q_{A_{1}}^{-})}^{β + 1}})$ and $K = min {k_{1}, k_{2}}$ . Therefore $I_{1} = \emptyset$ .

Now, considering $I_{2} \neq \emptyset$ , following the same steps as for the case $I_{1}$ , we obtain $\begin{array}{l} d ⩾ L_{2} inf_{i \in I_{2}} {({\tilde{c}}^{- \frac{1}{p (x_{i})}} σ_{m})}^{\frac{(α + 1) p (x_{i}) p^{*} (x_{i})}{p^{*} (x_{i}) - p (x_{i})}} + K min {λ^{\frac{{(h / v)}^{-}}{{(h / v)}^{-} - 1}}, λ^{\frac{{(h / v)}^{+}}{{(h / v)}^{+} - 1}}}, \end{array}$ where $L_{2} = (\frac{1}{θ} \cdot \frac{1}{{(r_{A_{2}}^{+})}^{α}} - \frac{1}{α + 1} \cdot \frac{1}{{(r_{A_{2}}^{-})}^{α + 1}})$ and $K = min {k_{1}, k_{2}}$ . Therefore $I_{2} = \emptyset$ , and the proof is over. □

Lemma 3.

Let $(u_{j}) \subset W^{1, p (x)} (Ω)$ be a ${(P S)}_{d}$ sequence for $J_{λ}$ . If $\begin{array}{l} d < & min {L_{1} inf_{i \in I_{1}} {({\overline{c}}^{- \frac{1}{p (x_{i})}} {\overline{T}}_{x_{i}})}^{\frac{(β + 1) p_{*} (x_{i}) p (x_{i})}{p_{*} (x_{i}) - p (x_{i})}}, L_{2} inf_{i \in I_{2}} {({\tilde{c}}^{- \frac{1}{p (x_{i})}} σ_{m})}^{\frac{(α + 1) p^{*} (x_{i}) p (x_{i})}{p^{*} (x_{i}) - p (x_{i})}}} \\ + K min {λ^{\frac{{(h / v)}^{-}}{{(h / v)}^{-} - 1}}, λ^{\frac{{(h / v)}^{+}}{{(h / v)}^{+} - 1}}} \end{array}$ where $\begin{array}{l} L_{1} = (\frac{1}{θ} \cdot \frac{1}{{(q_{A_{1}}^{+})}^{β}} - \frac{1}{β + 1} \cdot \frac{1}{{(q_{A_{1}}^{-})}^{β + 1}}) and L_{2} = (\frac{1}{θ} \cdot \frac{1}{{(r_{A_{2}}^{+})}^{α}} - \frac{1}{α + 1} \cdot \frac{1}{{(r_{A_{2}}^{-})}^{α + 1}}), \end{array}$ there exists $u \in W^{1, p (x)} (Ω)$ such that $u_{j} \to u$ in $W^{1, p (x)} (Ω)$ .

Proof.

By Lemma 1, passing to a subsequence, there exists $u \in W^{1, p (x)} (Ω)$ such that $u_{j} ⇀ u$ in $W^{1, p (x)} (Ω)$ . Since $J_{λ}^{'} (u_{j}) \to 0$ , we obtain $J_{λ}^{'} (u_{j}) (u_{j} - u) \to 0$ , that is, $\begin{array}{l} \int_{Ω} | \nabla u_{j} |^{p (x) - 2} \nabla u_{j} \nabla (u_{j} - u) d x + \int_{Ω} | u_{j} |^{p (x) - 2} u_{j} (u_{j} - u) d x + \int_{Ω} | u_{j} |^{h (x) - 2} u_{j} (u_{j} - u) d x \\ λ \int_{Ω} | u_{j} |^{v (x) - 2} u_{j} (u_{j} - u) d x - {[\int_{Ω} \frac{1}{r (x)} | u_{j} |^{r (x)} d x]}^{α} \int_{Ω} | u_{j} |^{r (x) - 2} u_{j} (u_{j} - u) d x \\ - {[\int_{\partial Ω} \frac{1}{q (x)} | u_{j} |^{q (x)} d S]}^{β} \int_{\partial Ω} | u_{j} |^{q (x) - 2} u_{j} (u_{j} - u) d Γ \to 0 . \end{array}$

Using Hölder inequality, we have $\begin{array}{l} | \int_{Ω} | u_{j} |^{p (x) - 2} u_{j} (u_{j} - u) d x | ⩽ \int_{Ω} | u_{j} |^{p (x) - 1} | u_{j} - u | d x ⩽ C_{1} | | u_{j} |^{p (x) - 1} |_{\frac{p (x)}{p (x) - 1}} | u_{j} - u |_{p (x)}, \\ | \int_{Ω} | u_{j} |^{h (x) - 2} u_{j} (u_{j} - u) d x | ⩽ \int_{Ω} | u_{j} |^{h (x) - 1} | u_{j} - u | d x ⩽ C_{2} | | u_{j} |^{h (x) - 1} |_{\frac{h (x)}{h (x) - 1}} | u_{j} - u |_{h (x)}, \\ | \int_{Ω} | u_{j} |^{v (x) - 2} u_{j} (u_{j} - u) d x | ⩽ \int_{Ω} | u_{j} |^{v (x) - 1} | u_{j} - u | d x ⩽ C_{3} | | u_{j} |^{v (x) - 1} |_{\frac{v (x)}{v (x) - 1}} | u_{j} - u |_{v (x)}, \\ | \int_{Ω} | u_{j} |^{r (x) - 2} u_{j} (u_{j} - u) d x | ⩽ \int_{Ω} | u_{j} |^{r (x) - 1} | u_{j} - u | d x ⩽ C_{4} | | u_{j} |^{r (x) - 1} |_{\frac{r (x)}{r (x) - 1}} | u_{j} - u |_{r (x)}, \\ | \int_{\partial Ω} | u_{j} |^{q (x) - 2} u_{j} (u_{j} - u) d Γ | ⩽ \int_{\partial Ω} | u_{j} |^{q (x) - 1} | u_{j} - u | d Γ ⩽ C_{5} | | u_{j} |^{q (x) - 1} |_{\frac{q (x)}{q (x) - 1}} | u_{j} - u |_{q (x)}, \end{array}$ where $C_{1}$ , $C_{2}$ , $C_{3}$ , $C_{4}$ and $C_{5}$ are positive constants. Since the immersion $W^{1, p (x)} (Ω) ↪ L^{p (x)} (Ω)$ is compact, and by Lemma 2, $u_{j} \to u$ in $L^{r (x)} (Ω)$ and $u_{j} \to u$ in $L^{q (x)} (\partial Ω)$ , we derive $\begin{array}{l} | \int_{Ω} | u_{j} |^{p (x) - 2} u_{j} (u_{j} - u) d x | \to 0, | \int_{Ω} | u_{j} |^{h (x) - 2} u_{j} (u_{j} - u) d x | \to 0, \\ | \int_{Ω} | u_{j} |^{r (x) - 2} u_{j} (u_{j} - u) d x | \to 0, | \int_{Ω} | u_{j} |^{v (x) - 2} u_{j} (u_{j} - u) d x | \to 0 \end{array}$ and $\begin{array}{l} | \int_{\partial Ω} | u_{j} |^{q (x) - 2} u_{j} (u_{j} - u) d Γ | \to 0 . \end{array}$

So, considering $\begin{array}{l} L_{p (x)} (u_{j}) (u_{j} - u) = \int_{Ω} | \nabla u_{j} |^{p (x) - 2} \nabla u_{j} \nabla (u_{j} - u) d x, \end{array}$ we obtain $\begin{array}{l} L_{p (x)} (u_{j}) (u_{j} - u) \to 0 . \end{array}$ We also have $\begin{array}{l} L_{p (x)} (u) (u_{j} - u) \to 0 . \end{array}$

Therefore $\begin{array}{l} (L_{p (x)} (u_{j}) - L_{p (x)} (u), u_{j} - u) \to 0 \end{array}$ and from Proposition 8 it follows that $\begin{array}{l} u_{j} \to u in W^{1, p (x)} (Ω) . \end{array}$ □

Lemma 4.

The energy functional $J_{λ}$ associated to ( 1 ) is not bounded below.

Proof.

Take $0 < w \in W^{1, p (x)} (Ω)$ . For $t > 1$ , we obtain $\begin{array}{l} J_{λ} (t w) ⩽ & \frac{t^{p^{+}}}{p^{-}} ρ_{1, p (x)} (w) + \frac{t^{h^{+}}}{h^{-}} \int_{Ω} | w |^{h (x)} d x - λ \frac{t^{v^{+}}}{v^{+}} \int_{Ω} | w |^{v (x)} d x \\ - \frac{1}{α + 1} \frac{t^{r^{-} (α + 1)}}{{(r^{+})}^{α + 1}} {[\int_{Ω} | w |^{r (x)} d x]}^{α + 1} - \frac{1}{β + 1} \frac{t^{q^{-} (β + 1)}}{{(q^{+})}^{β + 1}} {[\int_{\partial Ω} | w |^{q (x)} d Γ]}^{β + 1} . \end{array}$

By inequality (7), we have $\begin{array}{l} lim_{t \to \infty} J_{λ} (t w) = - \infty, \end{array}$ and the proof is over. □

In what follows we will use a truncation, like in Garcia Azorero and Peral Alonso [21], and in Corrêa and Costa [8] to the space with variable exponent, on the functional $J_{λ}$ , to obtain a special bounded from below functional, as follows; $\begin{array}{l} J_{λ} (u) ⩾ & \frac{1}{p^{+}} ρ_{1, p (x)} (u) + \frac{1}{h^{+}} ρ_{h (x)} (u) - \frac{λ}{v^{-}} ρ_{v (x)} (u) \\ - \frac{1}{α + 1} \frac{1}{{(r^{-})}^{α + 1}} {(ρ_{r (x)} (u))}^{α + 1} - \frac{1}{β + 1} \frac{1}{{(q^{-})}^{β + 1}} {(ρ_{q (x)} (u))}^{β + 1} . \end{array}$

By Proposition 3, it follow that $\begin{array}{l} J_{λ} (u) ⩾ & \frac{1}{p^{+}} ρ_{1, p (x)} (u) + \frac{1}{h^{+}} ρ_{h (x)} (u) - \frac{λ}{v^{-}} max {| u |_{v (x)}^{v^{-}}, | u |_{v (x)}^{v^{+}}} \\ - \frac{1}{α + 1} \frac{1}{{(r^{-})}^{α + 1}} {(max {| u |_{r (x)}^{r^{-}}, | u |_{r (x)}^{r^{+}}})}^{α + 1} - \frac{1}{β + 1} \frac{1}{{(q^{-})}^{β + 1}} {(max {| u |_{q (x)}^{q^{-}}, | u |_{q (x)}^{q^{+}}})}^{β + 1} . \end{array}$ Since $\frac{1}{h^{+}} ρ_{h (x)} (u) ⩾ 0$ , we have $\begin{array}{l} J_{λ} (u) ⩾ & \frac{1}{p^{+}} ρ_{1, p (x)} (u) - \frac{λ}{v^{-}} max {| u |_{v (x)}^{v^{-}}, | u |_{v (x)}^{v^{+}}} - \frac{1}{α + 1} \frac{1}{{(r^{-})}^{α + 1}} {(max {| u |_{r (x)}^{r^{-}}, | u |_{r (x)}^{r^{+}}})}^{α + 1} \\ - \frac{1}{β + 1} \frac{1}{{(q^{-})}^{β + 1}} {(max {| u |_{q (x)}^{q^{-}}, | u |_{q (x)}^{q^{+}}})}^{β + 1} . \end{array}$ By Sobolev continuous embedding, $\begin{array}{l} J_{λ} (u) ⩾ & \frac{1}{p^{+}} ρ_{1, p (x)} (u) - \frac{λ}{v^{-}} (σ_{v}^{v^{-}} + σ_{v}^{v^{+}}) max {‖ u ‖^{v^{-}}, ‖ u ‖^{v^{+}}} \\ - \frac{1}{α + 1} \frac{{(σ_{r}^{r^{-}} + σ_{r}^{r^{+}})}^{α + 1}}{{(r^{-})}^{α + 1}} {(max {‖ u ‖^{r^{-}}, ‖ u ‖^{r^{+}}})}^{α + 1} \\ - \frac{1}{β + 1} \frac{{(σ_{q}^{q^{-}} + σ_{q}^{q^{+}})}^{β + 1}}{{(q^{-})}^{β + 1}} {(max {‖ u ‖^{q^{-}}, ‖ u ‖^{q^{+}}})}^{β + 1}, \end{array}$ or yet, $\begin{array}{l} J_{λ} (u) ⩾ & \frac{1}{p^{+}} ρ_{1, p (x)} (u) - λ K_{1} max {‖ u ‖^{v^{-}}, ‖ u ‖^{v^{+}}} \\ - \frac{1}{α + 1} \frac{K_{2}}{{(r^{-})}^{α + 1}} {(max {‖ u ‖^{r^{-}}, ‖ u ‖^{r^{+}}})}^{α + 1} \\ - \frac{1}{β + 1} \frac{K_{3}}{{(q^{-})}^{β + 1}} {(max {‖ u ‖^{q^{-}}, ‖ u ‖^{q^{+}}})}^{β + 1}, \end{array}$ where $K_{1} = \frac{σ_{v}^{v^{-}} + σ_{v}^{v^{+}}}{v^{-}}$ , $K_{2} = {(σ_{r}^{r^{-}} + σ_{r}^{r^{+}})}^{α + 1}$ and $K_{3} = {(σ_{q}^{q^{-}} + σ_{q}^{q^{+}})}^{β + 1}$ . By Proposition 5, we obtain $\begin{array}{l} J_{λ} (u) ⩾ & \frac{1}{p^{+}} ρ_{1, p (x)} (u) - λ K_{1} max {ρ_{1, p (x)} {(u)}^{\frac{v^{-}}{p^{+}}}, ρ_{1, p (x)} {(u)}^{\frac{v^{+}}{p^{-}}}} \\ - \frac{1}{α + 1} \frac{K_{2}}{{(r^{-})}^{α + 1}} {(max {ρ_{1, p (x)} {(u)}^{\frac{r^{-}}{p^{+}}}, ρ_{1, p (x)} {(u)}^{\frac{r^{+}}{p^{-}}}})}^{α + 1} \\ - \frac{1}{β + 1} \frac{K_{3}}{{(q^{-})}^{β + 1}} {(max {ρ_{1, p (x)} {(u)}^{\frac{q^{-}}{p^{+}}}, ρ_{1, p (x)} {(u)}^{\frac{q^{+}}{p^{-}}}})}^{β + 1} \\ = & ξ (ρ_{1, p (x)} (u)) . \end{array}$ Now we will show that there are $R > 0$ and $λ_{1} > 0$ such that $ξ (R) > 0$ , for all $λ \in (0, λ_{1})$ . In fact, consider the function $η : [0, \infty) \to R$ defined by $\begin{array}{l} η (x) = \frac{1}{p^{+}} x - λ K_{1} x^{\frac{v^{-}}{p^{+}}} - \frac{1}{α + 1} \frac{K_{2}}{{(r^{-})}^{α + 1}} x^{\frac{r^{-} (α + 1)}{p^{+}}} - \frac{1}{β + 1} \frac{K_{3}}{{(q^{-})}^{β + 1}} x^{\frac{q^{-} (β + 1)}{p^{+}}} . \end{array}$ Since $1 < \frac{r^{-} (α + 1)}{p^{+}}$ and $1 < \frac{q^{-} (β + 1)}{p^{+}}$ , there exists $R \in (0, 1)$ such that $\begin{array}{l} 0 ⩽ \frac{1}{2 p^{+}} R - \frac{1}{α + 1} \frac{K_{2}}{{(r^{-})}^{α + 1}} R^{\frac{r^{-} (α + 1)}{p^{+}}} - \frac{1}{β + 1} \frac{K_{3}}{{(q^{-})}^{β + 1}} R^{\frac{q^{-} (β + 1)}{p^{+}}}, \end{array}$ implying that $\begin{array}{l} η (R) ⩾ \frac{1}{2 p^{+}} R - λ K_{1} R^{\frac{v^{-}}{p^{+}}} . \end{array}$ Thus we can choose $λ_{1} > 0$ small enough, by checking $\begin{array}{l} \frac{1}{2 p^{+}} R - λ_{1} K_{1} R^{\frac{v^{-}}{p^{+}}} > 0 . \end{array}$ Taking $R_{0} = max {[0, R] \cap ξ^{- 1} ((- \infty, 0])}$ , so $0 < R_{0} < R$ and $ξ (R_{0}) = 0$ , for all $λ \in (0, λ_{1})$ .

Considering $τ : [0, \infty) \to [0, 1]$ of class $C^{\infty} ([0, \infty))$ such that $\begin{array}{l} τ (x) = \{\begin{matrix} 1 & if x ⩽ R_{0} \\ 0 & if x ⩾ R \end{matrix} \end{array}$ let us define for $x \in [0, \infty)$ $\begin{array}{l} \overline{ξ} (x) = & \frac{1}{p^{+}} x - λ K_{2} max {x^{\frac{v^{-}}{p^{+}}}, x^{\frac{v^{+}}{p^{-}}}} - \frac{1}{α + 1} \frac{K_{2}}{{(r^{-})}^{α + 1}} {(max {x^{\frac{r^{-}}{p^{+}}}, x^{\frac{r^{+}}{p^{-}}}})}^{α + 1} τ (x) \\ - \frac{1}{β + 1} \frac{K_{3}}{{(q^{-})}^{β + 1}} {(max {x^{\frac{q^{-}}{p^{+}}}, x^{\frac{q^{+}}{p^{-}}}})}^{β + 1} τ (x) . \end{array}$ Note that, $\overline{ξ}$ checks the following properties: $\begin{array}{l} \overline{ξ} (x) = ξ (x), if x ⩽ R_{0}; \\ \overline{ξ} (x) ⩽ ξ (x), for all x \in [0, \infty); \\ \overline{ξ} (x) = \frac{1}{p^{+}} x - λ K_{2} max {x^{\frac{v^{-}}{p^{+}}}, x^{\frac{v^{+}}{p^{-}}}}, if x ⩾ R; \\ \overline{ξ} (x) > 0, for x > R_{0} . \end{array}$

Now, we consider the truncated functional, with $0 < λ < λ_{1}$ , $\begin{array}{l} I_{λ} (u) = & \int_{Ω} \frac{1}{p (x)} (| \nabla u |^{p (x)} + | u |^{p (x)}) d x + \int_{Ω} \frac{1}{h (x)} | u |^{h (x)} d x - λ \int_{Ω} \frac{1}{v (x)} | u |^{v (x)} d x \\ - \frac{1}{α + 1} τ (ρ_{1, p (x)} (u)) {[\int_{Ω} \frac{1}{r (x)} | u |^{r (x)} d x]}^{α + 1} \\ - \frac{1}{β + 1} τ (ρ_{1, p (x)} (u)) {[\int_{\partial Ω} \frac{1}{q (x)} | u |^{q (x)} d Γ]}^{β + 1} . \end{array}$ where $u \in W^{1, p (x)} (Ω)$ . Moreover, $I_{λ} (u) ⩾ \overline{ξ} (ρ_{1, p (x)} (u))$ and $I_{λ} (u) = J_{λ} (u)$ , if $ρ_{1, p (x)} (u) ⩽ R_{0}$ .

We can see that $I_{λ}$ is coercive and, hence, $I_{λ}$ is bounded from below.

Lemma 5.

$I_{λ} \in C^{1} (W^{1, p (x)} (Ω), R)$ , if $I_{λ} (u) ⩽ 0$ then $ρ_{1, p (x)} (u) < R_{0}$ and $I_{λ} (v) = J_{λ} (v)$ for all v in a small enough neighborhood of u. Moreover, $I_{λ}$ satisfies a local Palais–Smale condition for $d ⩽ 0$ .

Proof.

It is immediate that $I_{λ} \in C^{1} (W^{1, p (x)} (Ω), R)$ . If $I_{λ} (u) ⩽ 0$ then $ρ_{1, p (x)} (u) < R_{0}$ by construction of truncated functional, it is enough to observe that $\overline{ξ} (ρ_{1, p (x)} (u)) ⩾ 0$ if $ρ_{1, p (x)} (u)) ⩾ R_{0}$ . Now, for all $u \in B_{R_{0}} (0)$ there exists $ϵ > 0$ such that $B_{ϵ} (u) \subset B_{R_{0}} (u)$ and $I_{λ} (v) = J_{λ} (v)$ for all $v \in B_{ϵ} (u)$ . To prove a local Palais–Smale condition, let $(u_{j})$ be a sequence ${(P S)}_{d}$ for $I_{λ}$ with $d ⩽ 0$ . Without loss of generality, we can admit that $I_{λ} (u_{j}) ⩽ 0$ , for all $j \in N$ . So $ρ_{1, p (x)} (u_{j}) < R_{0}$ , for all $j \in N$ . Hence we have $\begin{array}{l} I_{λ} (u_{j}) = J_{λ} (u_{j}) and I_{λ}^{'} (u_{j}) = J_{λ}^{'} (u_{j}) for all j \in N . \end{array}$ Therefore $(u_{j})$ is also a sequence ${(P S)}_{d}$ for $J_{λ}$ . Since $\begin{array}{l} d ⩽ 0 < & min {L_{1} inf_{i \in I_{1}} {({\bar{c}}^{- \frac{1}{p (x_{i})}} {\overline{T}}_{x_{i}})}^{\frac{(β + 1) p_{*} (x_{i}) p (x_{i})}{p_{*} (x_{i}) + p (x_{i})}}, L_{2} inf_{i \in I_{2}} {({\tilde{c}}^{- \frac{1}{p (x_{i})}} σ_{m})}^{\frac{(α + 1) p^{*} (x_{i}) p (x_{i})}{p^{*} (x_{i}) + p (x_{i})}})} \\ + K min {λ^{\frac{{(h / v)}^{-}}{{(h / v)}^{-} - 1}}, λ^{\frac{{(h / v)}^{+}}{{(h / v)}^{+} - 1}}}, \end{array}$ by Lemma 2, there exists $λ_{0}$ such that for $0 < λ < λ_{0}$ $\begin{array}{l} u_{j} \to u in L^{q (x)} (\partial Ω) and in L^{r (x)} (Ω) . \end{array}$ Hence, by Lemma 3, we conclude that $I_{λ}$ enjoys the local Palais–Smale condition. □

Lemma 6.

For every $n \in N$ there exists $ϵ > 0$ such that $γ (I_{λ}^{- ϵ}) ⩾ n$ , where $\begin{array}{l} I_{λ}^{- ϵ} = {u \in W^{1, p (x)} (Ω); I_{λ} (u) ⩽ - ϵ} \end{array}$ and γ is the Krasnoselskii’s genus.

Proof.

Let $E_{n} \subset W^{1, p (x)} (Ω)$ be an n-dimensional subspace. Hence we have for $u \in E_{n}$ such that $‖ u ‖ = 1$ and $0 < t < R_{0}$ , we get $\begin{array}{l} I_{λ} (t u) = & \int_{Ω} \frac{1}{p (x)} ({| \nabla (t u) |}^{p (x)} + | t u |^{p (x)}) d x + \int_{Ω} \frac{1}{h (x)} | t u |^{h (x)} d x - λ \int_{Ω} \frac{1}{v (x)} | t u |^{v (x)} d x \\ - \frac{1}{α + 1} {[\int_{Ω} \frac{1}{r (x)} | t u |^{r (x)} τ (ρ_{1, p (x)} (t u)) d x]}^{α + 1} \\ - \frac{1}{β + 1} {[\int_{\partial Ω} \frac{1}{q (x)} | t u |^{q (x)} τ (ρ_{1, p (x)} (t u)) d Γ]}^{β + 1} \\ ⩽ & \frac{t^{p^{-}}}{p^{-}} ρ_{1, p (x)} (u) + \frac{t^{h^{-}}}{h^{-}} \int_{Ω} | u |^{h (x)} d x - λ \frac{t^{v^{+}}}{v^{+}} \int_{Ω} | u |^{v (x)} d x \\ - \frac{t^{r^{+} (α + 1)}}{(α + 1) {(r^{+})}^{α + 1}} {(\int_{Ω} | u |^{r (x)} d x)}^{α + 1} - \frac{t^{q^{+} (β + 1)}}{(β + 1) {(q^{+})}^{β + 1}} {(\int_{\partial Ω} | u |^{q (x)} d Γ)}^{β + 1} \\ ⩽ & \frac{t^{p^{-}}}{p^{-}} + \frac{t^{h^{-}}}{h^{-}} \cdot a_{n} - λ \frac{t^{v^{+}}}{v^{+}} \cdot b_{n} - \frac{t^{r^{+} (α + 1)}}{(α + 1) {(r^{+})}^{α + 1}} \cdot c_{n} - \frac{t^{q^{+} (β + 1)}}{(β + 1) {(q^{+})}^{β + 1}} \cdot d_{n}, \end{array}$ where $\begin{array}{l} a_{n} = sup {(\int_{Ω} | u |^{h (x)} d x); u \in E_{n}, ‖ u ‖ = 1}, \\ b_{n} = inf {(\int_{Ω} | u |^{v (x)} d x); u \in E_{n}, ‖ u ‖ = 1}, \\ c_{n} = inf {{(\int_{Ω} | u |^{r (x)} d x)}^{α + 1}; u \in E_{n}, ‖ u ‖ = 1} \end{array}$ and $\begin{array}{l} d_{n} = inf {{(\int_{\partial Ω} | u |^{q (x)} d Γ)}^{β + 1}; u \in E_{n}, ‖ u ‖ = 1} . \end{array}$ Then, $\begin{array}{l} I_{λ} (t u) ⩽ \frac{t^{p^{-}}}{p^{-}} + \frac{t^{h^{-}}}{h^{-}} \cdot a_{n} - λ \frac{t^{v^{+}}}{v^{+}} \cdot b_{n} . \end{array}$

Note that $a_{n} > 0$ , $b_{n} > 0$ , $c_{n} > 0$ and $d_{n} > 0$ , because $E_{n}$ is finite dimensional and the norm $W^{1, p (x)} (Ω)$ in $L^{h (x)} (Ω)$ , $L^{v (x)} (Ω)$ , $L^{r (x)} (Ω)$ , $L^{q (x)} (\partial Ω)$ are equivalent on $E_{n}$ .

As $v^{+} < p^{-}$ and $0 < t < R_{0}$ we obtain that there exists positive constants ρ and ϵ such that $I_{λ} (ρ u) < - ϵ$ for $u \in E_{n}$ , $‖ u ‖ = 1$ .

Therefore, if we set $U_{ρ, n} = {u \in E_{n} : ‖ u ‖ = ρ}$ , we have that $U_{ρ, n} \subset I_{λ}^{- ϵ}$ . Hence, by monotonicity of genus, $γ (I_{λ}^{- ϵ}) ⩾ γ (U_{ρ, n}) = n$ . □

Lemma 7.

Let $\sum = {A \subset W^{1, p (x)} (Ω) - 0 : A is closed, A = - A}$ and $\sum_{k} = {A \subset \sum : γ (A) ⩾ k}$ where γ stands for the Krasnoselskii’s genus. Thus, $\begin{array}{l} d_{k} = inf_{A \in \sum_{k}} sup_{u \in A} J_{λ} (u) \end{array}$ is a negative critical value of $J_{λ}$ and moreover, if $d = d_{k} = \dots = d_{k + r}$ ,then $γ (K_{d}) ⩾ r + 1$ where $\begin{array}{l} K_{d} = {u \in W^{1, p (x)} (Ω) : J_{λ} (u) = d, J_{λ}^{'} (u) = 0} . \end{array}$

Proof.

Note that $- \infty < c < \infty$ . Indeed, by Lemma 6 for all $k \in N$ , there exists $ϵ > 0$ such that $\begin{array}{l} γ (I_{λ}^{- ϵ}) ⩾ k . \end{array}$ Since $I_{λ}$ is even and continuous, $I_{λ}^{- ϵ} \in \sum_{k}$ , so $\begin{array}{l} d_{k} ⩽ sup_{u \in I_{λ}^{- ϵ}} I_{λ} (u) ⩽ - ϵ < 0 \end{array}$ for all k. Moreover, $I_{λ}$ is bounded from below, hence $d_{k} > - \infty$ for all $k \in N$ . Since $d < 0$ , $I_{λ}$ verifies a local ${(P S)}_{d}$ condition. Then, $K_{d}$ is a compact and symmetric set, so $γ (K_{d})$ is well defined.

Suppose, by contradiction, that $d = d_{k} = \dots = d_{k + r}$ and $γ (K_{d}) ⩽ r + 1$ . By Theorem 3, we have a neighborhood K of the $K_{d}$ with $γ (K) = γ (K_{d}) < r + 1$ . By the deformation lemma, there exists an odd homeomorphism $η : W^{1, p (x)} (Ω) \to W^{1, p (x)} (Ω)$ such that $\begin{array}{l} η (I_{λ}^{d + δ} ∖ K) \subset I_{λ}^{d - δ}, \end{array}$ for some $δ > 0$ . Particularly, we can choose $0 < δ < - d$ , because $I_{λ}$ verifies the Palais–Smale condition on $I_{λ}^{0}$ . By definition, $\begin{array}{l} d = d_{k + r} = inf_{A \in \sum_{k + r}} sup_{u \in A} I_{λ} (u) . \end{array}$ Then, there exists $A \in \sum_{k + r}$ , such that ${sup}_{u \in A} I_{λ} (u) < d + δ$ , which implies $A \subset I_{λ}^{d + δ}$ , and $\begin{array}{l} (10) & η (A ∖ K) \subset η (I_{λ}^{d + δ} ∖ K) \subset I_{λ}^{d - δ} . \end{array}$ By Theorem 3, $\begin{array}{l} γ (η (\overline{A ∖ K})) ⩾ γ (\overline{A ∖ K}) ⩾ γ (A) - γ (K) ⩾ (k + r) - r = k . \end{array}$ So, $η (\overline{A ∖ K}) \in \sum_{k}$ . Then, ${sup}_{u \in η (\overline{A ∖ K})} I_{λ} (u) ⩾ d_{k} = d$ , and this contradicts (10). Therefore, if $d = d_{k} = \dots = d_{k + r}$ , then $γ (K_{d}) ⩾ r + 1$ .

Note that this ensures that $d_{k}$ is critical value because $γ (K_{d_{k}}) ⩾ 1$ , that is, $K_{d_{k}}$ is nonempty for all $k \in N$ . Moreover, if the value $d_{k}$ not all are distinct, we will have $γ (K_{d}) > 1$ and that means that $K_{d}$ is an infinite set. Thus, we reach an infinite number of critical points for $I_{λ}$ with negative energy, for $0 < λ < \overline{λ} = min {λ_{0}, λ_{1}}$ . By Lemma 5 these points are critical points of $J_{λ}$ . Thus, we conclude that there exist an infinite weak solutions to the problem (1). □

Footnotes

Acknowledgements

The authors would like to thank anonymous referees whose comments and suggestions contributed towards the improvement of this work.

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