Variational approach for a Steklov problem involving nonstandard growth conditions

1.   Introduction

In this paper, we investigate the following $p(x)-$Kirchhoff type problem

where $\Omega \subset \mathbb{R} ^{N}$ $\left(N\geq 2\right)$ is a smooth bounded domain, $\lambda$ is a positive parameter, $p$ is continuous function on $\overline{\Omega }$ with $p^{-}: = \inf\limits_{x\in \overline{\Omega }}p\left(x\right)$, $\mathrm{div} \left(a\left(x, \nabla u\right) \right)$ is a $p\left(x\right) -$Laplace type operator and $a(x, \xi):\overline{\Omega }\times \mathbb{R} ^{N}\rightarrow \mathbb{R}$ is the continuous derivative with respect to $\xi$ of the mapping $A: \overline{\Omega }\times \mathbb{R} ^{N}\rightarrow \mathbb{R},$ $A = A(x, \xi)$, i.e. $a(x, \xi) = \nabla _{\xi }A(x, \xi)$. Furthermore, $f:\partial \Omega \times \mathbb{R} \rightarrow \mathbb{R}$ satisfies Carathéodory function and $M:\left(0, \infty \right) \rightarrow \left(0, \infty \right)$ is a continuous function.

Problem $(\bf{P})$ is generalization of a model, the so-called Kirchhoff equation, introduced by Kirchhoff ^[22]. Kirchhoff established a model given by the equation

where $0\leq x\leq L,$ $t\geq 0,$ $u$ is the lateral deflection, $\rho$ is the mass density, $h$ is the cross-sectional area, $L$ is the length, $E$ is the Young's modulus and $P_{0}$ is the initial axial tension. This equation is an extension of the classical D'Alambert's wave equation by considering the effects of the changes in the length of the strings during the vibrations.

Recently, equations with nonstandard growth condition have started to attract more attention due to their various physical applications. In fact, there are applications concerning image restoration ^[11], elastic mechanics ^[38], the image restoration or the motion of the so called electrorheological fluids ^[30], stationary thermo-rheological viscous flows of non-Newtonian fluids ^[3] and the mathematical description of the processes filtration of an idea barotropic gas through a porous medium ^[4].

As in the study of differential and partial differential equations, the investigate of Kirchhoff type equations under different boundary conditions has initially been extended to the case involving the $p-$growth conditions, and then the equations involving the $p(x)-$growth conditions. Especially, researchers have studied extensively the existence, multiplicity, uniqueness, nontrivial weak solution and regularity of solutions for various Kirchhoff type equations ^{[5,8,12,13,14,16,19,26,34,36]}. For example, Zhang and all in ^[36] studied the existence of nontrivial solutions and many solutions for a nonlocal $p\left(x\right) -$Kirchhoff problem with a $p^{+}-$superlinear subcritical Caratheodory reaction term, which does not satisfy the Ambrosetti–Rabinowitz condition using Mountain Pass theorem and Fountain theorem. Cao and all in ^[10] established the existence of nontrivial solutions for $p\left(x\right) -$Laplacian equations without any growth and Ambrosetti-Rabinowitz conditions. In ^[31], the author proved the existence of positive solutions using the Nehari manifold approach in $W_{0}^{1, p(x)}\left(\Omega \right)$.

The Steklov problems involving $p\left(x\right) -$Laplacian have been worked by some of the authors ^{[1,7,15,32,35]}. Especially, the authors have studied the problems of type $\bf{(P)}$ when $M(t) = 1.$ For instance, in ^[37], by applying the Ricceri's three critical points theorem, the authors investigated the existence of at least three solutions to the following elliptic problem:

where $\mu, \lambda \in \left[ 0, \infty \right),$ $\Omega \subset \mathbb{R} ^{N}$ $\left(N\geq 2\right)$ is a bounded domain of smooth boundary $\partial \Omega$, $\upsilon$ is the outward unit normal vector on $\partial \Omega,$ $p\left(x\right) \in C\left(\Omega \right)$ is the variable exponent and $f:\Omega \times \mathbb{R} \rightarrow \mathbb{R}$ and $g:\partial \Omega \times \mathbb{R} \rightarrow \mathbb{R}$ are two Carathéodory functions.

In ^[25], using Ricceri's variational principle and mountain pass theorem, they showed we prove in a different cases the existenceand multiplicity of a-harmonic solutions for the following elliptic problem:

where $\Omega \subset \mathbb{R} ^{N}$ $\left(N\geq 2\right)$ is a bounded domain of smooth boundary $\partial \Omega$ and $\upsilon$ is the outward unit normal vector on $\partial \Omega.$ $f:\partial \Omega \times \mathbb{R} \rightarrow \mathbb{R}$ and $a:\overline{\Omega }\times \mathbb{R} ^{N}\rightarrow \mathbb{R}$ satisfy appropriate conditions. In ^[24], the authors studied the problem (1.2) for case $f\left(x, u\right)$ $= \lambda |u|^{m(x)-2}u$, where the functions $m\left(x\right) \in L^{\infty }\left(\partial \Omega \right)$. If $a:\overline{\Omega }\times \mathbb{R} ^{N}\rightarrow \mathbb{R}$ satisfies the appropriate conditions, then the authors proved infinitely many positive eigenvalue sequences via the Ljusternik–Schnirelmann principle and a new variational technique. Now, we also mention some new paper that are related to our work. In ^[21], the authors concerned with a nontrivial weak solution under appropriate conditions a weighted Steklov problem involving the $p\left(x\right) -$Laplacian operator in Sobolev spaces with variable exponents by variational method and Ekeland's principle. Ourraoui in ^[28] proved some results on the existence and uniqueness of solutions concerned a class of elliptic problems involving $p\left(x\right) -$Laplacian with Steklov boundary condition. In ^[6], the author investigated the existence and multiplicity of solutions for Steklov problem with non-standard growth condition without using the Ambrosetti-Rabinowitz type condition. In ^[2], the authors obtained the existence and multiplicity of solutions for the nonlinear Steklov boundary value problem, using Mountain Pass, Fountain and Ricceri three critical points theorems for $M(t) = 1$ in problem $(\bf{P})$. In ^[20], the authors established the existence of infinitely many solutions for perturbed nonlocal problems with variable exponent and nonhomogeneous Neumann conditions using variational methods and critical point theory with $m_{1}\leq M\left(t\right) \leq m_{2},$ where for all $s > 0$ and $m_{1}$ and $m_{2}$ are positive constants and $A\left(x, \nabla u\right) = \int_{\Omega }\frac{1}{p\left(x\right) }\left(|\nabla u|^{p\left(x\right) }+a\left(x\right) |u|^{p\left(x\right) }\right) dx$ for problem $(\bf{P})$. Chammen and all in ^[9] studied the existence and the multiplicity of solutions is obtained by using variational methods, and mountain pass lemma combined with Ekeland variational principle for a class of Steklov Neumann boundary value problems involving $p(x)-$ Laplacian operator when $M(t) = 1$ for problem $(\bf{P}).$

Inspired by the papers above mentioned, we studied the Steklov problem involving the $p\left(x\right) -$Kirchhoff type operator. The present article is composed of three sections. In the second part, we introduce necessary notations, fundamental hypothesis and the variable exponent Lebesgue-Sobolev spaces on which we work. In the third part, after giving some basic results that will be useful for the proof of our principal theorems, we give the main theorems and their proofs.

2.   Preliminaries and main results

In order to discuss problem $\left(\bf{P}\right)$, we review some basic properties about the variable exponent Lebesgue- Sobolev spaces ($L^{p\left(x\right) }\left(\Omega \right), W^{1, p(x)}\left(\Omega \right)$ and $W_{0}^{1, p(x)}\left(\Omega \right))$, we refer to ^{[18,23,32,34]}.

Set

Denote $1 < p^{-}: = \inf\limits_{x\in \overline{\Omega }}p\left(x\right) \leq p\left(x\right) \leq p^{+}: = \sup\limits_{x\in \overline{\Omega }}p\left(x\right) < \infty$ for all $p\left(x\right) \in C_{+}\left(\overline{ \Omega }\right)$.

We define the variable exponent Lebesgue space by

with the norm

Moreover, we can define $C_{+}\left(\partial \Omega \right)$ and $p^{-}, p^{+}$ for any $p\left(x\right) \in C\left(\partial \Omega \right),$ and denote

endowed with the norm

where $d\sigma$ is the measure on the boundary. Moreover, if $p_{1}\left(x\right)$ and $p_{2}\left(x\right)$ are two functions in $C_{+}\left(\overline{\Omega }\right)$ such that $p_{1}\left(x\right) \leq p_{2}\left(x\right)$ almost everywhere $x\in \Omega$, then there exists a continuous embedding $L^{p_{2}\left(x\right) }\left(\Omega \right) \hookrightarrow L^{p_{1}\left(x\right) }\left(\Omega \right)$, and ıf $L^{p^{\prime }\left(x\right) }\left(\Omega \right)$ denotes the conjugate space of $L^{p\left(x\right) }\left(\Omega \right),$ where $\frac{1}{ p^{\prime }\left(x\right) }+\frac{1}{p\left(x\right) } = 1$, then we write H ölder-Type inequality

for any $u\in L^{p\left(x\right) }\left(\Omega \right)$ and $v\in$ $L^{p^{\prime }\left(x\right) }\left(\Omega \right)$.

The variable exponent Sobolev space $W^{1, p(x)}\left(\Omega \right)$ is defined by

with the norm

or

The space $W_{0}^{1, p(x)}\left(\Omega \right)$ is denoted by the closure of $C_{0}^{\infty }(\Omega)$ in $W^{1, p(x)}\left(\Omega \right)$ with respect to the norm $\left\Vert u\right\Vert _{1, p(x)}$. We can define an equivalent norm

If $p^{-} > 1$ and $p^{+} < \infty$, the $L^{p\left(x\right) }\left(\Omega \right)$, $L^{p\left(x\right) }\left(\partial \Omega \right)$, $W^{1, p(x)}\left(\Omega \right)$ and $W_{0}^{1, p(x)}\left(\Omega \right)$ are separable, reflexive and uniformly convex Banach spaces. An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the modular of the $L^{p\left(x\right) }\left(\Omega \right)$ space, which is the mapping $\psi \left(u\right) :L^{p\left(x\right) }\left(\Omega \right) \rightarrow \mathbb{R}$ defined by

Proposition 2.1 ^[18,23]. For all $u, u_{n}\in L^{p\left(x\right) }\left(\Omega \right) \left(\ n = 1, 2, ...\right)$ and $p^{+} < \infty,$ the following properties hold true:

(ⅰ) $|u|_{p\left(x\right) } > 1\left( = 1, < 1\right) \iff \psi \left(u\right) > 1\left( = 1, < 1\right),$

(ⅱ) $\min \left(|u|_{p\left(x\right) }^{p^{-}}, |u|_{p\left(x\right) }^{p^{+}}\right) \leq \psi \left(u\right) \leq \max \left(|u|_{p\left(x\right) }^{p^{-}}, |u|_{p\left(x\right) }^{p^{+}}\right),$

(ⅲ) $|u_{n}-u|_{p\left(x\right) }\rightarrow 0$ $\left(\rightarrow \infty \right) \Leftrightarrow \psi (u_{n}-u)\rightarrow 0$ $\left(\rightarrow \infty \right).$

Proposition 2.2 ^[15]. Let $\varphi \left(u\right) = \int_{\partial \Omega }\left\vert u\left(x\right) \right\vert ^{p\left(x\right) }d\sigma$. For all $u, u_{n}\in L^{p\left(x\right) }\left(\partial \Omega \right) \left(\ n = 1, 2, ...\right)$ , we have

(ⅰ) $|u|_{L^{p\left(x\right) }\left(\partial \Omega \right) } > 1\Rightarrow |u|_{L^{p\left(x\right) }\left(\partial \Omega \right) }^{p^{-}}\leq \varphi \left(u\right) \leq |u|_{L^{p\left(x\right) }\left(\partial \Omega \right) }^{p^{+}}, \ $

(ⅱ) $|u|_{L^{p\left(x\right) }\left(\partial \Omega \right) } < 1\Rightarrow |u|_{L^{p\left(x\right) }\left(\partial \Omega \right) }^{p^{+}}\leq \varphi \left(u\right) \leq |u|_{L^{p\left(x\right) }\left(\partial \Omega \right) }^{p^{-}},$

(ⅲ) $|u_{n}-u|_{p\left(x\right) }\rightarrow 0$ $\left(\rightarrow \infty \right) \Leftrightarrow \varphi (u_{n}-u)\rightarrow 0$ $\left(\rightarrow \infty \right).$

Proposition 2.3 ^[17]. Let $p(x)$ and$\; q(x)$ be measurable functions such that $1\leq p(x)q(x)\leq \infty$ and $p(x)\in L^{\infty }(\Omega)$ for a.e. $x\in \Omega$. Let $u\in L^{q(x)}(\Omega)$, $u\neq 0$ . Then

In particular, if $p(x) = p$ is constant, then we have

Proposition 2.4 ^[17,23,32].

(ⅰ) if $q\left(x\right) \in C_{+}\left(\overline{\Omega }\right)$ and $q\left(x\right) < p^{\ast }\left(x\right)$ for all $x\in \overline{\Omega }$, then the embedding $W_{0}^{1, p(x)}\left(\Omega \right) \hookrightarrow L^{q\left(x\right) }\left(\Omega \right)$ is compact and continuous, where

(ⅱ) if $q\left(x\right) \in C_{+}\left(\partial \Omega \right)$ and $q\left(x\right) < p^{\partial }\left(x\right)$ for all $x\in \partial \Omega$, then the trace embedding $W_{0}^{1, p(x)}\left(\Omega \right) \hookrightarrow L^{q\left(x\right) }\left(\partial \Omega \right)$ is compact and continuous, where

(ⅲ) Poincaré inequality; i.e.there is a positive constant $C > 0$ such that

Remark 2.5. If $N < p^{-}\leq p(x)$ for any $x\in \overline{\Omega },$ by Theorem 2.2 in ^[18] and Remark 1 in ^[29], we deduce that $W^{1, p(x)}\left(\Omega \right)$ is continuously embedded in $W^{1, p^{-}}\left(\Omega \right)$. Since $N < p^{-}$, it follows that $W^{1, p(x)}\left(\Omega \right)$ is compactly embedded in $C\left(\overline{\Omega }\right)$. Defining $\left\Vert u\right\Vert _{C\left(\overline{\Omega }\right) } = \sup\limits_{x\in \overline{\Omega }}\left\vert u\left(x\right) \right\vert,$ we find that there exists a positive constant $c_{\ast }$ $> 0$ such that

Theorem 2.6 (Mountain-Pass Geometry) ^[33]. Let $X$ be a Banach spaces and $J_{\lambda }\in C^{1}(X, \mathbb{R})$ satisfies Palais-Smale condition. Assume that $J_{\lambda }(0) = 0$, and there exist two positive real numbers $\eta$ and $r$ such that

(ⅰ) There exist two positive real numbers $\eta$ and $r$ such that $J_{\lambda }\left(u\right) \geq r > 0$ with $\left\Vert u\right\Vert = \eta$,

(ⅱ) There exists $u_{1}\in X$ such that $\left\Vert u_{1}\right\Vert > \eta$ and $J_{\lambda }\left(u_{1}\right) < 0.$

Put

Set $\beta = \inf \left\{ \max J_{\lambda }\left(\phi \left(\left[ 0, 1\right] \right) \right) :\phi \in G\right\}$. Then $\beta \geq r$ and $\beta$ is a critical value of $J_{\lambda }.$

Theorem 2.7 ^[8]. Let $X$ be a separable and reflexive real Banach space, $\Phi :X\rightarrow \mathbb{R}$ a continuous Gâteaux differentiable and sequentially weakly lower semi-continuous functional whose Gâteaux derivative admits a continuous inverse on $X^{\ast }$, $\Psi :X\rightarrow \mathbb{R}$ a continuous Gâteaux differentiable functional whose Gâteaux derivative is compact. Assume that,

(ⅰ) $\lim\limits_{\left\Vert u\right\Vert \rightarrow \infty }J_{\lambda }(u) = \lim\limits_{\left\Vert u\right\Vert \rightarrow \infty }\left(\Phi \left(u\right) +\lambda \Psi \left(u\right) \right) = \infty$ for all $\lambda > 0$,

(ⅱ) There are $r\in \mathbb{R}$ and $u_{0}$, $u_{1}\in X$ such that $\Phi \left(u_{0}\right) < r < \Phi \left(u_{1}\right)$,

(ⅲ) $\inf_{u\in \Phi ^{-1}\left(\left(-\infty, r\right] \right) }\Psi (u) > \frac{\left(\Phi \left(u_{1}\right) -r\right) \Psi (u_{0})+\left(r-\Phi \left(u_{0}\right) \right) \Psi \left(u_{1}\right) }{ \Phi \left(u_{1}\right) -\Phi \left(u_{0}\right) }$.

Then there exist an open interval $\Theta \subset \left(0, \infty \right)$ and a positive real number $\rho$ such that for each $\lambda \in \Theta$ the equation

has at least three solutions in $X$ whose norms are lees than $\rho$.

Throughout this paper, we consider the following assumptions:

$\left(\bf{M}_{1}\right)$ $M:\left(0, \infty \right) \rightarrow \left(0, \infty \right)$ is a continuous function such that

where $m_{1}, m_{2}$ and $\alpha$ are real numbers such that $0 < m_{1}\leq m_{2}$ and $\alpha > 1$.

$\left(\bf{A1}\right)$ There exists a constant $c_{0} > 0$ such that satisfies the following growth condition

$\left(\bf{A2}\right)$ The following inequalities hold

$\left(\bf{A3}\right)$ $A\left(x, 0\right) = 0,$ for all $x\in \overline{\Omega }.$

$\left(\bf{A4}\right)$ $A$ is $p(x)$-uniformly convex: There exists a constant $k_{0} > 0$ such that

Our main results in this paper are the proofs of the following theorems, which are based on the Mountain Pass Theorem ^[33] and the Ricceri Theorem ^[8]. Let $X$ denote the variable exponent Sobolev space $W_{0}^{1, p(x)}\left(\Omega \right)$.

Theorem 2.8.Suppose that $\left(\bf{M} _{1}\right), \left(\bf{A1}\right) -\left(\bf{A5}\right),$ $p^{+} < \alpha p^{-}$ and $f$ satisfies the following conditions hold

$\left(\bf{f}_{1}\right)$ There exits $c_{1}$ is a positive constant such that

where $\ m\left(x\right) \in C_{+}\left(\partial \Omega \right)$ such that $p^{+} < m^{-}: = \inf\limits_{x\in \partial \Omega }m\left(x\right) \leq m\left(x\right) \leq m^{+}: = \sup\limits_{x\in \partial \Omega }m\left(x\right) < p^{\partial }\left(x\right)$

$\left(\bf{f}_{2}\right)$ $f(x, t) = o\left(\left\vert t\right\vert ^{\alpha p^{+}-1}\right)$ as $t\rightarrow 0,$ for $x\in \partial \Omega$ and $\alpha p^{+} < \ m^{-}$

$\left(\bf{AR}\right)$ Ambrosetti-Rabinowitz's Condition holds, i.e., there exists $M > 0$ and $\theta > \frac{m_{2}\alpha p^{+}}{m_{1}}$ such that

Then there exists $\lambda ^{\ast } > 0$ such that for any $\lambda \in \left(0, \lambda ^{\ast }\right),$ problem $\left(\bf{P}\right)$ has a nontrivial weak solution in $X$.

Theorem 2.9. Assume that $(M_{1})$, $\left(\bf{A1}\right) -\left(\bf{A5}\right)$ and $f$ satisfies the following conditions hold

$\left(\bf{f}_{3}\right)$ $f:\partial \Omega \times \mathbb{R} \rightarrow \mathbb{R}$ satisfies Carathéodory condition and

where $\ k\geq 0$ is a constant, $n\left(x\right) \in L^{ \frac{\beta \left(x\right) }{\beta \left(x\right) -1}}\left(\partial \Omega \right)$, and $\beta (x)\in C_{+}\left(\partial \Omega \right)$ such that

$\left(\bf{f}_{4}\right)$ If $\left\vert t\right\vert \in \left(0, 1\right),$ then $F(x, t) < 0$ and $t\in \left(t_{0}, \infty \right)$ for $t_{0} > 1,$ then $F(x, t) > \varpi > 0$.

Then there exist an open interval $\Theta \subset \left(0, \infty \right)$ and a constant $\rho > 0$ such that for any $\lambda \in \Theta$, problem $(P)$ has at least three weak solutions in $X$ whose norms are less than $\rho$.

3.   Proofs of main results

Definition 3.1. We say that $u\in X$ is a weak solution of the boundary value problem $\left(\bf{P}\right)$ if and only if

for any $v\in X.$ We define the functionals; $\Phi, \Psi :X\rightarrow \mathbb{R}$

where $\widehat{M}\left(t\right), \Lambda \left(u\right)$ and $F\left(x, t\right)$ are denoted by

for all $t > 0$ and $\left(x, k\right) \in \partial \Omega \times \mathbb{R}.$

Proposition 3.2 ^[15]. Let $f:\partial \Omega \times \mathbb{R} \rightarrow \mathbb{R}$ is a Carathéodory function satisfying $\left(\bf{f} _{1}\right)$. For each $u\in X,$ set $\varkappa \left(u\right) = \int_{\partial \Omega }F\left(x, u\right) d\sigma.$ Then $\varkappa \left(u\right) \in C^{1}(X, \mathbb{R})$ and

for all $\upsilon \in X$. Moreover, the operator $\varkappa :X\rightarrow X^{\ast }$ is compact.

Lemma 3.3 ^[27].

(ⅰ) $A$ verifies the growth condition $\left\vert A\left(x, \xi \right) \right\vert \leq c_{0}(\left\vert \xi \right\vert +\left\vert \xi \right\vert ^{p\left(x\right) })$, for all $x\in \Omega$ and $\xi \in \mathbb{R} ^{N};$

(ⅱ) $A$ is $p(x)-$ homogeneous, $A\left(x, z\xi \right) \leq A\left(x, \xi \right) z^{p\left(x\right) }$, for all $z\geq 1,$ $\xi \in \mathbb{R} ^{N}$ and $x\in \Omega.$

Lemma 3.4 ^[27].

(ⅰ) The functional $\Lambda$ is well-defined on $X,$

(ⅱ) The functional $\Lambda$ is of class $C^{1}\left(X, \mathbb{R} \right)$ and

(ⅲ) The functional $\Lambda$ is weakly lower semi-continuous on $X,$

(ⅳ) For all $u, \upsilon \in X$

(ⅴ) For all $u, \upsilon \in X$

(ⅵ) $J_{\lambda }$ is weakly lower semi-continuous on $X.$

Then energy functional associated to the problem $\left(\bf{P}\right)$ is $J_{\lambda }\left(u\right) = \Phi \left(u\right) +\lambda \Psi \left(u\right)$. Furthermore, from Proposition 3.2, Lemma 3.3, Lemma 3.4, $\left(\bf{f}_{1}\right)$ and $(\bf{M}_{1}),$ it is easy to see that the functional $\Phi, \Psi \in C^{1}(X, \mathbb{R})$ and the derivate of $J_{\lambda }$ is the mapping $J_{\lambda }^{\prime }:X\rightarrow \mathbb{R}.$ Then, we have

for any $u, \upsilon \in X$ and we can infer that critical points of functional $J_{\lambda }$ are the weak solutions for problem $\left(\bf{ P}\right)$.

Lemma 3.5. Suppose $(\bf{M}_{1}), \left(\bf{f} _{1}\right), \left(\bf{A2}\right)$, $\left(\bf{AR} \right)$ and $\alpha p^{-} > p^{+}$ hold. Then, the functional $J_{\lambda }$ satisfies Palais-Smale $(PS)$ condition for any $\lambda \in \left(0, +\infty \right).$

Proof. Let assume that there exists a sequence $\left\{ u_{n}\right\} \subset X$ such that

Firstly, we prove that $\{u_{n}\}$ is bounded in $X$. Arguing by contradiction and passing to a subsequence, we have $\left\Vert u_{n}\right\Vert \rightarrow \infty$ as $n\rightarrow \infty$. From $(\bf{M}_{1})$, $\left(\bf{AR}\right), \left(3.1\right)$ and considering $\left\Vert u_{n}\right\Vert > 1$, for $n$ large enough, we get

From $\left(\bf{A3}\right)$ and Proposition 2.2 $(ⅱ)$, we have

If this last inequality is divided by $\left\Vert u_{n}\right\Vert ^{\alpha p^{-}}$ and pass to the limit as $n\rightarrow \infty,$ we obtain a contradiction with the condition $\left(\bf{AR}\right).$ So, $\alpha p^{-} > p^{+}$, $\{u_{n}\}$ is bounded in $X$. Thus, we may extract a subsequence $\left\{ u_{n}\right\} \subset X$ and $u\in X$ such that $u_{n}\rightharpoonup u$ in $X.$ Next, we will show that $u_{n}\rightarrow u$ in $X$. Taking into account relation $\left(3.1\right)$, we obtain that $\langle J_{\lambda }^{\prime }\left(u_{n}\right), u_{n}-u\rangle \rightarrow 0$. That is,

Moreover, using $\left(\bf{f}_{1}\right)$ and the inequality $\left(2.1\right)$, we deduce that

where $c_{2} > 0$ is a constant. On the other hand, from Proposition 2.3, if we consider the compact embedding $X\hookrightarrow L^{m\left(x\right) }\left(\partial \Omega \right),$ that is, $\left\vert u_{n}-u\right\vert _{L^{m\left(x\right) }\left(\partial \Omega \right) }\rightarrow 0$ as $n\rightarrow \infty$, we obtain

So, we use $\left(3.3\right)$ in the above inequality $\left(3.2\right)$, we have

Moreover, from $(\bf{M}_{1})$, we conclude that

that is, $\underset{n\rightarrow \infty }{\lim }\langle \Lambda ^{\prime }\left(u_{n}\right), u_{n}-u\rangle = 0.$ From Lemma 3.4 $($v$)$, we write

or

Thus, from Lemma 3.4 $(ⅲ)$ and the above inequality $\left(3.4\right)$, we have

Now, we assume by contradiction that $\left\{ u_{n}\right\}$ does not converge strongly to $u$ in $X.$ Then, there exists $\xi > 0$ and a subsequence $\left\{ u_{n_{k}}\right\}$ of $\left\{ u_{n}\right\}$ such that $\left\Vert u_{n_{k}}-u\right\Vert \geq \epsilon$. Moreover, by Lemma 3.4 $($iv$)$, we get

Letting $k\rightarrow \infty$ in the inequality $\left(3.6\right)$ and and using the inequalities $\left(3.5\right)$, we have

Moreover, we have $\{\dfrac{u_{n_{k}}+u}{2}\}$ converges weakly to $u$ in $X$. Using Lemma 3.4 $(ⅲ)$, we obtain

which is a contradiction. Therefore, it follows that $\left\{ u_{n}\right\}$ converges strongly to $u$ in $X.$ The proof of Lemma 3.5 is complete.

Lemma 3.6. Suppose $(\bf{M}_{1}), \left(\bf{f} _{1}\right), \left(\bf{f}_{2}\right), \left(\bf{A3}\right)$ and $\left(\bf{AR}\right)$ hold. Then the following statements hold;

(ⅰ) There exist two positive real numbers $\eta$ and $r$ such that $J_{\lambda }\left(u\right) \geq r > 0$, $u\in X$ with $\left\Vert u\right\Vert = \eta$.

(ⅱ) There exists $u\in X$ such that $\left\Vert u\right\Vert > \eta,$ $J_{\lambda }\left(u\right) < 0$.

Proof. (ⅰ) Let $\left\Vert u\right\Vert < 1$. From $\left(\bf{f}_{1}\right)$ and $\left(\bf{f}_{2}\right)$, we obtain

On the other hand, using the continuous embeddings $X\hookrightarrow L^{p\left(x\right) }\left(\Omega \right),$ $X\hookrightarrow L^{\alpha p^{+}}\left(\partial \Omega \right)$ and $X\hookrightarrow L^{m(x)}\left(\partial \Omega \right)$, there exists positive constants $c_{3}$, $c_{4}$ and $c_{5}$ such that

Therefore, by Proposition 2.1, Proposition 2.2, $\left(\bf{M} _{1}\right)$, the inequalities $\left(3.7\right)$ and $\left(3.8\right),$ we obtain

for $\left\Vert u\right\Vert$ small enought. Let $\varepsilon > 0$ be small enough such that $0 < \lambda \varepsilon c_{4}^{\alpha p^{+}}\leq \left(\frac{m_{1}}{2\alpha \left(p^{+}\right) ^{\alpha }}+\frac{c_{3}^{p^{+}}}{ p^{+}}\right),$ we have

As $\alpha p^{+} < m^{-}$, the functional $g:[0, 1]\rightarrow R$ defined by

is positive on neighborhood of the origin. It follows that there exist two positive real numbers $\eta$ and $r$ such that $J_{\lambda }\left(u\right) \geq r > 0$, $u\in X\ $with $\left\Vert u\right\Vert = \eta \in \left(0, 1\right)$.

(ⅱ) From $\left(\bf{AR}\right),$ we obtain that there exist positive constant $c_{6}$ such that

On the other hand, from $\left(\bf{M}_{1}\right)$ and $t > 1,$ we have

Moreover, we use the inequalities $\left(3.9\right)$ and $\left(3.10\right)$ together with Lemma 3.3, we obtain

From $\left(\bf{AR}\right),$ we show that $J_{\lambda }\left(t\phi \right) \rightarrow -\infty$ as $t\rightarrow +\infty$. Then, we can take $u_{1} = t\phi$ such that $\left\Vert u_{1}\right\Vert > \eta$ and $J_{\lambda }\left(u_{1}\right) < 0$. The proof of Lemma 3.6 is complete.

Proof of Theorem 2.8. By $\left(\bf{A3}\right)$, we obtain $J_{\lambda }\left(0\right) = 0$. If we also take into account Lemma 3.5 and Lemma 3.6, $J_{\lambda }$ satisfies the Mountain Pass theorem ^[33]. Hence, $J_{\lambda }$ has at least one nontrivial critical point, i.e., problem $\left(\bf{P}\right)$ has a nontrivial weak solution.

Proof of Theorem 2.9. To prove this theorem, it is sufficient to show that the conditions of Theorem 2.7 are satisfied. Since $X$ is continuous embedding $L^{m^{+}}\left(\partial \Omega \right)$ from Proposition 2.4, there exist positive constant $c_{7}$ such that

Also, consider the case $\left\Vert u\right\Vert > 1$ for $u\in X$. Then, using $(\bf{M}_{1})$ and Proposition 2.1, we get that

If we use the inequalities $\left(2.1\right)$ and $\left(3.11\right)$ together with the conditional $\left(\bf{f}_{3}\right)$, we can write

where $c_{8}$ and $c_{9}$ are positive constants. Combining $\left(3.12\right)$ and $\left(3.13\right)$, we obtain

From $p^{-} > \beta ^{+},$ it follows that

So, condition (ⅰ) of Theorem 2.7 is satisfied.

Now, we show the condition (ⅱ) of Theorem 2.7. By $f(x, t) = \frac{\partial F(x, t)}{\partial t}\ $ and from the condition $\left(\bf{f}_{3}\right)$, it is obtained that $F(x, t)$ is increasing for $t\in \left(t_{0}, \infty \right)$ and decreasing for $t\in \left(0, 1\right)$ uniformly for $x\in \partial \Omega$. Also, $F(x, 0) = 0$ is obvious and $F(x, t)$ $\rightarrow \infty$ when $t\rightarrow \infty$ because $F(x, t)\geq \varpi t$ uniformly on $x$. Then, there exists a real number $\gamma >$ $t_{0}$ such that

Let $\kappa, \delta$ be two real numbers such that $0 < \kappa < \min \left\{ 1, c\right\}$ where $c$ is a constant which satisfies

Since $N < p^{-},$we have from Remark 2.5 that compact embedding from $X$ to $C\left(\overline{\Omega }\right)$. If we choose $\delta > \gamma$ satisfying $\delta ^{p^{-}}\left\vert \Omega \right\vert > 1$ and using $\left(3.14\right),$ we obtain

Then, we have

On the other hand, we can write from $\delta > \gamma,$

and

So, combining $\left(3.16\right)$ and $\left(3.17\right)$, we show that

Let $u_{0}, u_{1}\in X$, $u_{0}\left(x\right) = 0$ and $u_{1}\left(x\right) = \delta$ for any $x\in \overline{\Omega }.$ If we define $\mu = \frac{1}{ \alpha \left(p^{+}\right) ^{\alpha }}.\left(\frac{\kappa }{c}\right) ^{p^{+}}$, then we have $\mu \in \left(0, 1\right), \Phi \left(u_{0}\right) = \Psi \left(u_{0}\right) = 0,$

and from $\left(3.18\right)$ we obtain

Thus we deduce that $\Phi \left(u_{0}\right) < r < \Phi \left(u_{1}\right),$ that is, condition (ⅱ) of Theorem 2.7 is satisfied.

Finally, we show the condition (ⅲ) of Theorem 2.7. we have

Let $u\in X$ such that $\Phi \left(u\right)$ $\leq \mu < 1$. Since $\frac{1 }{\alpha \left(p^{+}\right) ^{\alpha }}\psi \left(u\right) \leq \Phi \left(u\right) \leq \mu$ for $u\in X$, we obtain

So, by Proposition 2.1 for $\left\Vert u\right\Vert < 1$, we have

On the other hand, from $\left(3.15\right)$, $\left(3.20\right)$ and $\left(3.21\right)$ we show that

for all $u\in X$ and $x\in \overline{\Omega }$ with $\Phi \left(u\right)$ $\leq \mu.$

From $\left(3.22\right),$we obtain

Thus, by $\left(3.19\right)$ we have

or

So, condition (ⅲ) of Theorem 2.7 is obtained. Since all conditions of Theorem 2.7 are verified, there exists an open interval $\Theta \subset \left(0, \infty \right)$ and a positive real number $\rho$ such that for each $\lambda \in \Theta$ the equation

has at least three solutions in $X$ whose norms are lees than $\rho$. The proof of Theorem 2.9 is complete.

4.   Conclusions

In this paper, we studied a non-linear elliptic equation involving $p(x)$-growth conditions and satisfying Steklov boundary condition on a bounded domain $\Omega$. The study was carried out the existence of at least three solutions and a nontrivial weak solution of the problem under appropriate conditions. Our basic approach is to use Ricceri's three critical point theorem and Mountain Pass theorem together with the variational approach to investigate the existence of a multiplicity of solutions.

Conflict of interest

The authors declare no conflict of interest.