Infinitely many solutions for a critical $p(x)$-Kirchhoff equation with Steklov boundary value

1.   Introduction

In this paper, we focus on a class of critical $\kappa(x)$-Kirchhoff-type problems formulated as follows:

where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, with the Lipschitz boundary denoted by $\partial\Omega$. The symbol $\frac{\partial}{\partial v}$ represents the outer unit normal derivative and $\Delta_{\kappa(x)}u = div\left(\left\vert\nabla u\right\vert^{\kappa(x)-2}\; {\nabla}u\right)$. $\mathfrak{S}(t)$ is a continuous function and the function $g: \Omega \times \mathbb{R} \rightarrow \mathbb{R}$ is a Carathéodory function that satisfies appropriate assumptions.

We assume that both $\kappa, r$ and $s$ are continuous functions in $\overline{\Omega}$, meaning they are defined and continuous on the closure of $\Omega$. Moreover, we consider the condition $1 < \kappa(x) < r(x) \leq \kappa^{*}(x)$ for all $x \in \Omega$, where $p^{*}(x)$ represents the critical Sobolev exponent. Additionally, we assume that the set $A = \{x\in \Omega : r(x) = \kappa^{*}(x)\}$ is non-empty.

Problems with critical growth, the concentration-compactness principle introduced by Lions (see ^[21]) has been widely recognized as a fundamental tool for establishing the existence of solutions. This principle is particularly crucial when considering equations involving Sobolev embeddings, which capture the critical growth behavior. For a more comprehensive understanding of this topic, we suggest referring to the references ^[3,4,18,24] and the additional sources mentioned therein.

The study of problems with variable exponents, critical growth, and problems involving fractional $p$-Laplacian has received significant attention in recent years. These problems have proven to be interesting and relevant in various applications, such as the modeling of electro-rheological fluids ^[20,23,25] and image processing ^[6]. Additionally, they give rise to challenging mathematical problems that require careful investigation.

In ^[5], the authors consider the critical variable exponents equation:

where $1 < p(x) < r(x)\leq p^{*}(x)$ for all $x\in \Omega$, where $p^{*}(x)$ denotes the critical Sobolev exponent associated with $p(x)$. The set $A = \{x\in \Omega : r(x) = p^{*}(x)\}$ is assumed to be non-empty, indicating the presence of critical growth behavior.

To establish the existence of solutions, the authors employ variational methods and make use of the mountain pass theorem. These techniques allow them to construct a suitable functional and apply critical point theory to find nontrivial solutions to the problem (1.2).

Due to their importance, problems involving variable exponents are attracting increasing interest from many researchers. Many authors studied the problem with Dirichlet, Neumann, or Steklov boundary conditions on a bounded domain. In particular, Chammem et al. ^[7] used the mountain pass theorem combined with Ekland's variational principle to study the following Steklov problem:

After that, Chammem and Sahbani ^[8] considered the following double-phase problem:

By means of variational methods, the mountain pass lemma and its $Z_{2}$ symmetric version, the existence and multiplicity of solutions for problem (1.4), was established.

Problems involving Kirchhoff-type with variable exponents are attracting attention and gaining prominence in several research groups for numerous theoretical and practical questions ^[9,10,12,13] and the references therein. On the other hand, it is also worth mentioning Kirchhoff's problems with fractional operators, which over the years has been increasing exponentially ^[1,17,29]. The $p(x)$-Laplacian possesses more complex nonlinearity which raises some of the essential difficulties. For example, in ^[30], Z. Yücedag consider the p(x)-Kirchhoff problem with Steklov boundary value conditions:

where $1 < p(x)\leq p^{*}(x)$ for all $x\in \Omega$.

Under suitable conditions on the functions $q$ and $f$, the authors employed variational methods and fountain theorem to establish the existence and multiplicity of solutions for problem (1.5).

Motivated by the results presented in references ^[5,7,8,30], our paper aims to contribute further by studying the critical case of the aforementioned problem. To this end, we utilize a recent concentration-compactness principle for spaces with variable exponents to investigate the weighted Kirchhoff problem (1.1). Our study provides a generalization, improvement, and extension of the aforementioned references under additional, appropriate conditions. Consequently, this research project holds significant importance and offers valuable insights.

In this paper, we consider problem (1.1), where $g(x, u) = a(x)h(u)$. Under specific hypotheses, we employ the variational method, the mountain pass theorem, and the symmetric mountain pass theorem to establish the existence and multiplicity of nontrivial weak solutions for problem (1.1). This rigorous approach ensures the robustness and reliability of our results.

In summary, our research significantly contributes to the existing literature by exploring the critical case of the Kirchhoff problem with Dirichlet boundary conditions. Through rigorous mathematical techniques and the utilization of recent concentration-compactness principles, we establish the existence and multiplicity of solutions for problem (1.1) under different scenarios, enhancing the overall understanding of this important topic.

In Section 2, we present some necessary preliminary. In Section 3 we give our main results, where we present and prove the existence and multiplicity of solutions for the weighted Kirchhoff problem.

2.   Preliminaries

In this section, we provide an overview of some important properties of variable exponent spaces. For more detailed information, we recommend referring to the works ^{[11,15,16,19,26]} and the references therein. We consider the set

For all $\kappa\in C_{+}({\Omega })$, consider

Additionally, we define

with the norm on $\mathscr{L}^{\kappa(\xi)}(\Omega)$ defined as

Also, we define

with the norm on $\mathscr{L}^{\kappa(\xi)}(\partial\Omega)$ defined as

The spaces $\left(\mathscr{L}^{\kappa(\xi)}(\Omega), |\cdot|_{\mathscr{L}^{\kappa(\xi)}(\Omega)}\right)$ and $\left(\mathscr{L}^{\kappa(\xi)}(\partial\Omega), |\cdot|_{\mathscr{L}^{\kappa(\xi)}(\partial\Omega)}\right)$ are a Banach spaces, which we refer to as variable exponent Lebesgue spaces.

The Sobolev space is defined as:

with the norm

Denote by $W_{0}^{1, \kappa(x)}(\Omega)$ the closure of $C_{0}^{\infty}(\Omega)$ in $W^{1, \kappa(x)}(\Omega)$.

The following proposition provides important properties of variable exponent spaces.

Proposition 1. ^[15] The spaces $\mathscr{L}^{\kappa(\xi)}(\Omega)$ and $W^{1, \kappa(x)}(\Omega)$ are separable and reflexive Banach spaces.

Proposition 2. ^[7,8,27]

(1) The Hölder inequality holds, that is, for any $u\in \mathscr{L}^{\kappa(\xi)}(\Omega)$ and $v\in \mathscr{L}^{\kappa'(\xi)}(\Omega),$ we have

(2) If $p_{1}, p_{2}\in C_{+}(\overline{\Omega})$ such that $p_{1}(\xi)\leq p_{2}(\xi)$ for all $\xi \in \overline{\Omega}$, then the embedding $\mathscr{L}^{p_{2}(\xi)}(\Omega)\hookrightarrow \mathscr{L}^{p_{1}(\xi)}(\Omega)$ is continuous.

Note that $\left\|u\right\|$ and $\left\|\nabla u \right\|_{\mathscr{L}^{\kappa(\xi)}(\Omega)}$ are equivalent in the space $W_{0}^{1, \kappa(x)}(\Omega)$, so let's use $\left\|u\right\| = \left\|\nabla u\right\|_{\mathscr{L}^{\kappa(\xi)}(\Omega)}$, for simplicity ^[11,28].

The following proposition highlights the properties of the variable exponent Sobolev spaces.

Proposition 3. ^[15]

(1) If $q\in C_{+}(\overline{\Omega})$ with $q(\xi) < \kappa^{\ast}(\xi)$ for all $\xi \in \overline{\Omega}$, then the embedding from $W_{0}^{1, \kappa(x)}(\Omega)$ into $\mathscr{L}^{q(\xi)}(\Omega)$ is compact and continuous. Here, $\kappa^{*}(\xi)$ is defined as follows:

(2) If $q\in C_{+}(\partial\Omega)$ with $q(\xi) < \kappa_{\ast}(\xi)$ for all $\xi \in \partial\Omega$, then the embedding from $W_{0}^{1, \kappa(x)}(\Omega)$ into $\mathscr{L}^{q(\xi)}(\partial\Omega)$ is compact and continuous. Here, $\kappa_{*}(\xi)$ is defined as follows:

For simplicity, let us denote

The following proposition provides important properties of the functional $\Gamma(u)$:

Proposition 4. ^[15]

(1) If $\Gamma(u)\geq 1$, then $\left\|u\right\|^{\kappa^{-}}\leq \Gamma(u)\leq \left\|u\right\|^{\kappa^{+}}$.

(2) If $\Gamma(u)\leq 1$, then $\left\|u\right\|^{\kappa^{+}}\leq \Gamma(u)\leq \left\|u\right\|^{\kappa^{-}}$.

(3) $\Gamma(u)\geq 1( = 1, \leq 1)\Leftrightarrow \left\|u\right\|\geq 1( = 1, \leq 1)$.

Let us define

The next proposition provides properties of the functional $\rho(u)$:

Proposition 5. ^[7,8] For all $u\in \mathscr{L}^{\kappa(\xi)}(\Omega)$, we have

(1) $|u|_{\mathscr{L}^{\kappa(\xi)}(\Omega)} < 1; (resp = 1, > 1)\Leftrightarrow \rho(u) < 1; (resp = 1, > 1)$.

(2) $|u|_{\mathscr{L}^{\kappa(\xi)}(\Omega)} > 1\Rightarrow |u|_{\mathscr{L}^{\kappa(\xi)}(\Omega)}^{\kappa^{-}}\leq \rho(u)\leq |u|_{\mathscr{L}^{\kappa(\xi)}(\Omega)}^{\kappa^{+}}.$

(3) $|u|_{\mathscr{L}^{\kappa(\xi)}(\Omega)} < 1\Rightarrow |u|_{\mathscr{L}^{\kappa(\xi)}(\Omega)}^{\kappa^{+}}\leq \rho(u)\leq |u|_{\mathscr{L}^{\kappa(\xi)}(\Omega)}^{\kappa^{-}}$.

The next proposition relates the norms of a function in variable exponent Lebesgue spaces with its pointwise behavior.

Proposition 6. ^[7,8] If $p$ and $q$ are measurable functions such that $p\in \mathscr{L}^{\infty}(\mathbb{R}^{N})$ and $1\leq \kappa(\xi)q(\xi)\leq\infty$ for all $\xi\in \mathbb{R}^{N}$, then for all $u\in \mathscr{L}^{q(\xi)}(\mathbb{R}^{N})$ with $u\neq 0$, we have

(1) $|u|_{\mathscr{L}^{\kappa(\xi)q(\xi)} (\Omega)}\leq1\Rightarrow |u|_{\mathscr{L}^{\kappa(\xi)q(\xi)} (\Omega)}^{q^{+}}\leq ||u|^{\kappa(\xi)}|_{\mathscr{L}^{q(\xi)}(\Omega) }\leq |u|_{\mathscr{L}^{\kappa(\xi)q(\xi)}(\Omega) }^{q^{-}}$.

(2) $|u|_{\mathscr{L}^{\kappa(\xi)q(\xi)}(\Omega) }\geq 1\Rightarrow |u|_{\mathscr{L}^{\kappa(\xi)q(\xi)}(\Omega) }^{q^{-}}\leq ||u|^{\kappa(\xi)}|_{\mathscr{L}^{q(\xi)}(\Omega) }\leq |u|_{\mathscr{L}^{\kappa(\xi)q(\xi)}(\Omega) }^{q^{+}}$.

Denote for $u\in \mathscr{L}^{p(\xi)}(\partial\Omega)$,

Proposition 7. ^[7,8] For all $u\in \mathscr{L}^{p(\xi)}(\partial\Omega)$, we have,

(1) $|u|_{\mathscr{L}^{p(\xi)}(\partial\Omega)} > 1\Rightarrow |u|_{\mathscr{L}^{p(\xi)}(\partial\Omega)}^{p^{-}}\leq \rho_{\partial}(u)\leq |u|_{\mathscr{L}^{p(\xi)}(\partial\Omega)}^{p^{+}}$,

(2) $|u|_{\mathscr{L}^{p(\xi)}(\partial\Omega)} < 1\Rightarrow |u|_{\mathscr{L}^{p(\xi)}(\partial\Omega)}^{p^{+}}\leq \rho_{\partial}(u)\leq |u|_{\mathscr{L}^{p(\xi)}(\partial\Omega)}^{p^{-}}$.

3.   Main results

In this section, we will present our main result of the paper. Firstly, we assume the following hypotheses:

${\bf(C_{1})}$ The function $g(x, u)$ can be expressed as $a(x)h(u)$, where $a$ and $h$ are measurable functions satisfying the following conditions: there exists $c_1 > 0$, $p, q\in C_{+}(\overline{\Omega })$ such that for all $(x, u)\in \Omega \times \mathbb{R}$, we have

and

($\bf C_{2}$) There exists $m_{0} > 0\; such \; that\; \mathfrak{S}(t)\geq m_{0}$.

($\bf C_{3}$) There exists $0 < \omega < 1$ such that, $1-\omega\geq\frac{1}{\kappa^{+}}$ and $\widehat{\mathfrak{S}}(t)\geq (1-\omega)\mathfrak{S}(t)t$, where $\widehat{\mathfrak{S}}(t) = \int_{0}^{t} \mathfrak{S}(s)ds$.

${\bf(C_{4})}$ There exist $M_{1} > 0$ and $\frac{\kappa^{+}}{1-\omega } < \theta < \min(r^{-}, s^{-})$, such that for all $x\in \Omega$, we have

${\bf(C_{5})}$ We have $\kappa^{+}\leq s(x) < \kappa_{\ast }(x)$.

${\bf(C_{6})}$ For all $x\in\overline{\Omega}$, we have $h(-u) = -h(u)$.

Next, we define a weak solution for the problem (1.1) as follows:

Definition 1. We say that $u\in X = W_{0}^{1, \kappa(x)}(\Omega)$ is a weak solution for Eq (1.1) if, for any $v\in X$, we have

Now, we are ready to state and prove the first main results:

Theorem 1. Under the hypotheses ${\bf(C_{1})}-{\bf(C_{5})}$, problem (1.1) has a nontrivial weak solution.

Theorem 2. Under the hypotheses ${\bf(C_{1})}-{\bf(C_{6})}$, problem (1.1) has infinitely many solutions.

Now, we introduce the functional $\mathfrak{J}(u)$ associated with problem (1.1), which characterizes the critical points and plays a key role in the existence of solutions.

where $\widehat {\mathfrak{S}}(t) = \int_{0}^{t}{\mathfrak{S}}(s)ds$, $L(u) = \widehat {\mathfrak{S}}\left(\int_{\Omega}\frac{1}{\kappa(x)} \left\vert\nabla u \right\vert ^{\kappa(x)} d x\right)$,

We recall from ^[7], that $L\in C^{1}(X, \mathbb{R}).$ Moreover, for all $u, v\in X$, we have

The functional $L'$ satisfies the following properties.

Proposition 8. ^[7]

Let ${\mathfrak{L}}:\; \; \; \; \mathcal{H}^{\varpi, \nu; \, \psi}_{\kappa(x)}(\Omega) \rightarrow (\mathcal{H}^{\varpi, \nu; \, \psi}_{\kappa(x)}(\Omega))^{*}$,

such that

(1) ${\mathfrak{L}}^{'}:X\rightarrow X^{\ast}$ is a continuous, bounded, and strictly monotone operator.

(2) ${\mathfrak{L}}^{'}$ is a mapping of $(S_{+})$ type, that is, if $u_{n}\rightharpoonup u$ in X and $\limsup\limits_{n\rightarrow \infty} < {\mathfrak{L}}^{'}(u_{n})-{\mathfrak{L}}^{'}(u), u_{n}-u > \leq 0$, then, $u_{n}\rightarrow u$ strongly in X.

Remark 1. It can be shown, using $(C_{1})$, Propositions 4, 6, and the Hölder inequality, that $J\in C^{1}(X, \mathbb{R})$. Furthermore, for all $u, v\in X$, we have

From Proposition 8 and Remark 1, it follows that $\mathfrak{J} \in C^{1}(X, \mathbb{R})$. Moreover, for all $u, v\in X$, we obtain

Hence, the weak solutions of problem (1.1) correspond to the critical points of the functional $\mathfrak{J}$.

Now, we establish a key result that provides a lower bound for the functional $\mathfrak{J}(u)$ associated with problem (1.1).

Lemma 1. Assume that ${\bf(C_{1})}$–${\bf(C_{5})}$ are satisfied. Then, there exist $m, \eta > 0$ such that, for $u\in X$,

Proof. Let $u\in X$, with $\left\Vert u\right\Vert < 1.$ Under the hypothesis ${\bf(C_{1})}$, we have for all $x\in \Omega$,

Since $1 < p(x) < \kappa^{*}(x),$ $1 < r(x)\leq \kappa^{*}(x)$, $1 < s(x) < \kappa_{*}(x)$ and according to Proposition 3, we obtain the existence of $c_{3},$ $c_{4}$, $c_{5} > 0,$ such that

On the other hand, under hypothesis ${\bf(C_{2})}$ and ${\bf(C_{3})}$ and by Proposition 4, we get,

Now, by (3.2)–(3.4) and using Propositions 2, 6 and 4, we obtain,

where

Since $q ^{-}$, $s ^{-}$ and $r ^{-}$ are both greater than $\kappa^{+}$, we can choose $\left\Vert u\right\Vert = \eta$ to be sufficiently small such that

Finally, we conclude that

□

Now, define the Palais$-$Smale $(PS)$ condition at a given level $c$:

Definition 2. Let $X$ be a Banach space and $\mathfrak{J} \in C^{1}(X, \mathbb{R})$, where $c\in \mathbb{R}$. We say that $\mathfrak{J}$ satisfies the Palais$-$Smale condition at level $c$ if any $\{u_{n}\}\subset X$, such that

contains a convergent subsequence.

In the following lemma, we establish a result regarding the boundedness of a Palais$-$Smale sequence in $X$.

Lemma 2. Suppose that conditions ${\bf(C_{2})}-{\bf(C_{5})}$ are satisfied. Let $\left\{ u_{n}\right\}$ be a Palais$-$Smale sequence of $\mathfrak{J}$ in $X.$ Then $\left\{ u_{n}\right\}$ is bounded in $X.$

Proof. Let $\left\{ u_{n}\right\}$ be a sequence in $X$ such that

where $c$ is a positive constant.

Since $\mathfrak{J} (u_{n})\rightarrow c$, there exists $T_{1} > 0,$ such that

On the other hand, the fact that $\mathfrak{J}'(u_{n})\rightarrow 0$ in $X^{\ast }$, implies that $< \mathfrak{J}'(u_{n}), u_{n} > \rightarrow 0.$ In particular, $< \mathfrak{J}'(u_{n}), u_{n} >$ is bounded. Thus, there exists $T_{2} > 0,$ such that

We claim that the sequence $\left\{u_{n}\right\}$ is bounded. If it is not true, by passing to a sub-sequence if necessary, we may assume that $\left\Vert u_{n}\right\Vert \rightarrow \infty$. Without loss of generality, we assume that $\left\Vert u_{n} \right\Vert \geq 1.$

From (3.5), (3.4) and using $\kappa^{+} < \theta < \min(r^{-}, s^{-})$, we obtain

On the other hand by (3.6) and assumption ${\bf(C_{2})}$, we have

By combining the above inequality and (3.7), we obtain

Hence, assumption ${\bf(C_{4})}$ implies

By ${\bf(C_{4})}$ and ${\bf(C_{2})}$, $\frac{\kappa^{+}}{1-\omega } < \theta$ and $m_{0} > 0$, then, we have $\Big(\frac{(1-\omega)\theta }{\kappa^{+}}-1\Big) m_{0} > 0$, so

By (3.8) this is absurder. Then, $\left\{ u_{n}\right\}$ is bounded in $X$. □

Now, we introduce the nonempty set $A$ define by $A = \{x\in \Omega : r(x) = \kappa^{*}(x)\}$. Also, define the set $A_{\delta} = \{x\in \Omega : dist((x, A) < \delta\}$ for $\delta > 0.$ We note $r_{\delta}^{-} = \inf_{\overline{A_{\delta} }} r(x)$, and $r_{A}^{-} = \inf_{\overline{A }} r(x).$

We will now introduce and recall several important theorem.

Theorem 3. (Concentration$-$compactness principle) (see ^[5]) Let $\kappa(x)$ and $r(x)$ be two continuous functions such that

Let $\{u_{j}\}_{j\in \mathbb{N}}$ be a weakly convergent sequence in $W_{0}^{1, \kappa(x)}(\Omega)$ with weak limit $u$ and such that:

● $|u_{j}|^{r(x)} \rightharpoonup \nu$ weakly in the sense of measures.

● $|\nabla u_{j}|^{\kappa(x)} \rightharpoonup \mu$ weakly in the sense of measures.

Also assume that $A = \{x\in \Omega : r(x) = \kappa^{*}(x)\}$ is nonempty. Then, for some countable index set $I$, we have:

Where $\{x_{i}\}_{i\in I}\subset A$ and $S$ is the best constant in the Gagliardo-Nirenberg-Sobolev inequality for variable exponents, namely

If $\left\{ u_{n}\right\}$ is a Palais–Smale sequence with energy level $c$, then according to Theorem 3, we have the following convergence results:

If $I = \emptyset,$ then $u_{n}\rightarrow u$ in $\mathscr{L}^{r(x)}(\Omega)$. It should be noted that $\left\{ x_{i}\right\}_{i\in A} \subset A$. We aim to demonstrate that if $c < (\frac{1}{\kappa^{+}}-\frac{1}{r_{A}^{-}})S^{n}$, then $I = \emptyset$, where $S$ is defined in Theorem 3.

The following lemma establishes an important result regarding the behavior of Palais$-$Smale sequences under certain conditions.

Lemma 3. If conditions ${\bf(C_{1})}-{\bf(C_{5})}$ are satisfied. Let $\left\{ u_{n}\right\}$ be a Palais$-$Smale sequence of $\mathfrak{J}$ in $X$ with energy level $c.$ If $c < (\frac{1}{\kappa^{+}}-\frac{1}{r_{A}^{-}})S^{n}$, then the index set $I$ is empty.

Proof. Suppose that $I\neq \emptyset$ and let $\varphi\in C_{0}^{\infty}(\mathbb{R}^{n})$ such that $\varphi(0)\neq 0$. Now, we consider the functions $\varphi_{i, \epsilon}(x) = \varphi(\frac{x-x_{i}}{\epsilon})$. We have $< \mathfrak{J}'(u_{n}), \varphi_{i, \epsilon}u_{n} > \rightarrow 0.$ Thus,

Passing to the limit as $n\rightarrow \infty$, we have

By Hölder's inequality and using hypothesis ${\bf(C_{2})}$ we can show that

On the other hand, we have

Then, $\Big(\mu_{i}-\nu_{i}\Big)\varphi(0) = 0,$ which implies $\mu_{i} = \nu_{i}.$ Consequently,

Thus, we conclude that $\nu_{i} = 0$ or $S^{n}\leq \nu_{i}.$

Now, since $r(x), s(x), \theta > \kappa^{+} > 1-\omega$, and by using hypothesis ${\bf(C_{4})}$, we have

On the other hand

Therefore, since $\delta$ is positive and arbitrary and $r$ is continuous, we have

Then if $c < (\frac{1}{\kappa^{+}}-\frac{1}{r_{A}^{-}})S^{n}$, the index set $I$ is empty. □

We now present the following lemma that establishes an important convergence result.

Lemma 4. If conditions ${\bf(C_{1})}-{\bf(C_{5})}$ are satisfied, let $\left\{ u_{n}\right\}$ be a Palais–Smale sequence of $\mathfrak{J}$ in $X$, with energy level $c.$ Then there exists a subsequence of $\left\{ u_{n}\right\}$ that converges strongly in $X.$

Proof. Let $\left\{ u_{n}\right\}$ be a $(PS)_{c}$ sequence in $X$. By Lemma 2, $\left\{ u_{n}\right\}$ is bounded in $X$. Then, there exists a subsequence of $\left\{ u_{n}\right\}$, such that $u_{n}\rightharpoonup u$.

Using Lemma 3, $p(x) < \kappa^{\ast }(x)$, $s(x) < \kappa_{* }(x)$ and Proposition 3, we have

Next, we will show that $u_{n}\rightarrow u$. We start by considering the inner product

By applying Hölder's inequality, Propositions 3 and 6, we can estimate the integral term as follows:

This leads to the conclusion

Similarly, we get

Now, by using ${(\bf C_{1})}$, propositions 3 and 6, and Hölder's inequality, we obtain

Hence, we have

By combining (3.13)–(3.15), and using the fact that $< \mathfrak{J}'(u_{n}), u_{n}-u > \rightarrow 0$, we conclude that

Hence by using hypothesis ${\bf(C_{2})}$, we have ${\mathfrak{S}}\Big(\int_{\Omega}\frac{1}{\kappa(x)} \left\vert\nabla u \right\vert ^{\kappa(x)}dx\Big) \neq 0$, then

Since $u_{n}\rightharpoonup u$, we get $< {\mathfrak{L}}^{'}(u), u_{n}-u > \rightarrow 0.$

Finally, we get

By Proposition 8, ${\mathfrak{L}}^{'}$ is of type $(S_{+})$, then $u_{n}\rightarrow u$. □

In order to further investigate the properties of the functional $\mathfrak{J}$ and its critical points, we establish the following lemma.

Lemma 5. If conditions ${(\bf C_{2})}-{(\bf C_{4})}$ hold. Then, there exists $u_{0}\in X$ such that

where $\eta$ is defined in Lemma 1.

Proof. By ${(\bf C_{4})}$, we get

By the conditions ${(\bf C_{2})}$ and ${(\bf C_{3})}$, the function $t\longmapsto \dfrac{\widehat{\mathfrak{S}}(t)}{t^{1/w-1}}$ is decreasing. So for all $t_{0} > 0$, when $t > t_0$, yields $\dfrac{\widehat{\mathfrak{S}}(t)}{t^{1/w-1}}\leq \dfrac{\widehat{\mathfrak{S}}(t_0)}{t_{0}^{1/w-1}}$, then,

Let $u\in X$ such that $\int_{\Omega}|u|^{\theta}\neq 0$ and let $t > 1$, be sufficiently large. Then, we have by (3.16) and (3.17)

Since $\theta > \frac{\kappa^{+}}{1-\omega}$, it follows that

Therefore, we can choose $t_{0} > 0$ and set $u_{0} = t_{0}e$, such that $||u_{0}|| > \eta$ and $\mathfrak{J}(u_{0}) < 0$. This completes the proof. □

Now, we establish the following lemma that provides a key result regarding the boundedness of a set under certain hypotheses.

Lemma 6. Under hypotheses ${\bf(C_{1})}-{\bf(C_{4})}$, if $F$ is a finite dimensional subspace of X, then the set

is bounded in $X.$

Proof. Let $u\in T.$ By (3.16) and (3.17), we have:

where $|.|_{\mathscr{L}^{\theta}}$ and $||.||$ are equivalent norms in $F$. Thus, there exists a positive constant $k$ such that

Therefore, we have

Hence, since $\frac{\kappa^{-}}{1-\omega } < \frac{\kappa^{+}}{1-\omega } < \theta$, we deduce that $T$ is bounded in X. □

In the context of our analysis and proofs, we will now introduce and recall several important theorems: The mountain pass theorem, its symmetric version for even functions. These theorems play a crucial role in establishing our results. Here are the statements of the theorems:

Theorem 4. (Mountain pass theorem) (see ^[2]) Let $X$ be a Banach space. Consider a functional $\mathfrak{J}\in C^1(X, \mathbb{R})$ satisfying the following conditions:

(1) $\mathfrak{J}(0) = 0$;

(2) $\mathfrak{J}$ satisfies the (PS) condition;

(3) There exist positive constants $\eta$ and $\rho$ such that if $||u|| = \eta$, then $\mathfrak{J} (u)\geq \rho$;

(4) There exists $e\in X$ with $||e|| > \eta$ such that $\mathfrak{J}(e)\leq 0$. Then, $\mathfrak{J}$ possesses a critical value $c\geq \rho$ which can be characterized as

where,

Theorem 5. (Symmetric mountain pass theorem) (see ^[2]) Let $X$ be an infinite dimensional real Banach space. Let $\mathfrak{J}\in C^{1}(X, \mathbb{R})$, satisfying the following conditions:

(1) $\mathfrak{J}$ is an even functional such that $\mathfrak{J}(0) = 0$.

(2) $\mathfrak{J}$ satisfies the (PS)-condition.

(3) There exist positive constants $\eta$ and $\rho$, such that if $||u|| = \eta$, then, $\mathfrak{J}(u)\geq \rho$.

(4) For each finite dimensional subspace $F\subset X$, the set $\{u\in F, \mathfrak{J}(u)\geq 0\}$ is bounded in X. Then $\mathfrak{J}$ has an unbounded sequence of critical values.

Proof of Theorem 1. Lemmas 1, 4, and 5 establish the fulfillment of all the conditions required by Theorem 4 (mountain pass theorem), ensuring the existence of a nontrivial solution to problem (1.1). With this, the proof of Theorem 1 is now concluded. □

Proof of Theorem 2. We observe that $\mathfrak{J}(0) = 0$, and due to ${(\bf C_{6})}$, the functional $\mathfrak{J}$ is even. Furthermore, Lemmas 1, 4, and 6 establish the fulfillment of all the conditions stated in Theorem 5 (symmetric mountain pass theorem). Consequently, we can conclude that problem (1.1) possesses an unbounded sequence of nontrivial solutions. With this, the proof of Theorem 2 is now completed. □

4.   Conclusions

In this paper, the existence of a solution and an infinite number of solutions for the Steklov problem have been proven under appropriate conditions on our parameters within variable exponent Sobolev spaces. An interesting perspective is to extend this work to Robin-type problems and to problems involving Leray-Lions type operators.

Author contributions

Khaled Kefi, Abdeljabbar Ghanmi, Abdelhakim Sahbani, and Mohammed M. Al-Shomrani contributed to the conception and design of the study; Abdelhakim Sahbani provided resources and support for the research; Abdeljabbar Ghanmi, Abdelhakim Sahbani, and Mohammed M. Al-Shomrani reviewed and edited the manuscript; Khaled Kefi acquired funding. All authors have read and approved the final version of the manuscript for publication.

Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at Northern Border University, Arar, KSA for funding this research work through the project number NBU-FPEJ-2024-1706-03. The Fourth author acknowledges the support of the Deanship of Science Research (DSR) at King Abdulaziz University, Jeddah, Saudi Arabia.

Conflicts of interest

The authors declare no conflict of interest.