On a class of p(x)-Laplacian-like Dirichlet problem depending on three real parameters

This research establishes the existence of weak solution for a Dirichlet boundary value problem involving the p(x)-Laplacian-like operator depending on three real parameters, originated from a capillary phenomena, of the following form: -Δp(x)lu+δ|u|α(x)-2u=μg(x,u)+λf(x,u,∇u)inΩ,u=0on∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \displaystyle \left\{ \begin{array}{ll} \displaystyle -\Delta ^{l}_{p(x)}u+\delta \vert u\vert ^{\alpha (x)-2}u=\mu g(x, u)+\lambda f(x, u, \nabla u) &{} \mathrm {i}\mathrm {n}\ \Omega ,\\ \\ u=0 &{} \mathrm {o}\mathrm {n}\ \partial \Omega , \end{array}\right. \end{aligned}$$\end{document}where Δp(x)l\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta ^{l}_{p(x)}$$\end{document} is the p(x)-Laplacian-like operator, Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} is a smooth bounded domain in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{N}$$\end{document}, δ,μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta ,\mu $$\end{document}, and λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} are three real parameters, and p(·),α(·)∈C+(Ω¯)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\cdot ),\alpha (\cdot )\in C_{+}(\overline{\Omega })$$\end{document} and g, f are Carathéodory functions. Under suitable nonstandard growth conditions on g and f and using the topological degree for a class of demicontinuous operator of generalized (S+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(S_{+})$$\end{document} type and the theory of variable-exponent Sobolev spaces, we establish the existence of a weak solution for the above problem.

problem's energy is provided by |∇u| p(x) dx. This type of energy can also be found in elasticity problems [34,37]. Other applications relate to image processing [3,10], elasticity [35], the flow in porous media [7,14], and problems in the calculus of variations involving variational integrals with nonstandard growth [4,18,35]. Let be a smooth bounded domain in R N (N ≥ 2), with a Lipschitz boundary denoted by ∂ , and let δ, μ, and λ be three real parameters.
In this paper, we investigate the existence of weak solution for a Dirichlet boundary value problem involving the p(x)-Laplacian-like operator depending on three real parameters, arising from capillarity phenomena, of the following form: is the p(x)-Laplacian-like operator, p(·), α(·) ∈ C + ( ) with p(·) is log-Hölder continuous function, and g : × R → R and f : × R × R N → R are Carathéodory functions that satisfy the assumption of growth. The expression f (x, u, ∇u) is often referred to as a convection term. The motivation for this research originated from the application of similar problems in physics to model the behavior of elasticity [34,35] and electrorheological fluids (see [27,29]), which have the ability to modify their mechanical properties when exposed to an electric field (see [5,6,25,26,36]), specifically the phenomenon of capillarity, which depends on solid-liquid interfacial characteristics as surface tension, contact angle, and solid surface geometry.
Problems related to (1.1) have been studied by many scholars; for example, Ni et al. [19,20] studied the following equations: −div ∇u The operator −div ∇u 1 + |∇u| 2 is most often denoted by the specified mean curvature operator.
In [21], Obersnel et al. established the existence and multiplicity of positive solution of the problem where λ > 0 and f : × R → R is a Carathéodory function. Their discussion is based on variational and critical point theory.
In the case when μ = δ = 0, λ > 0, and f independent of ∇u, we know that the problem (1.1) has a nontrivial solution from [28].
It is well known that problems such as (1.1), (1.2), and (1.3) have a role in relativity theory and differential geometry.
In the present paper, we will generalize these works, by proving, under suitable growth conditions on g and f , the existence of a weak solution for the problem (1.1) using another approach based on the topological degree for a class of demicontinuous operator of generalized (S + ) type of [8] and the theory of the variable-exponent Sobolev spaces. To the best of our knowledge, this is the first research that discusses a Dirichlet boundary value problem involving p(x)-Laplacian-like operator depending on three real parameters with convection term via topological degree methods.
The remainder of the paper is organized as follows. In Sect. 2, we review some fundamental preliminaries about the functional framework where we will treat our problem. In Sect. 3, we introduce some classes of operator of generalized (S + ) type, as well as the Berkovits topological degrees. Finally, in Sect. 4, we give our basic assumptions and some technical lemmas, and we will state and prove the main result of the paper.

Preliminaries
In the analysis of problem (1.1), we will use the theory of the variable-exponent Lebesgue-Sobolev spaces L p(x) ( ) and W 1, p(x) ( ). For convenience, we only recall some basic facts with will be used later, we refer to [12,16,[22][23][24]33] for more details.
Let be a smooth bounded domain in R N (N ≥ 2), with a Lipschitz boundary denoted by ∂ . Set For each p ∈ C + ( ), we define For every p ∈ C + ( ), we define Proposition 2.1 [12] Let (u n ) and u ∈ L p(·) ( ). Then For any u ∈ L p(x) ( ) and v ∈ L p (x) ( ), we have the following Hölder-type inequality: for any x ∈ , then there exists the continuous embedding equipped with the norm ||u|| = |u| p(x) + |∇u| p(x) .
We also define W 1, p(·) 0 ( ) as the subspace of W 1, p(·) ( ), which is the closure of C ∞ 0 ( ) with respect to the norm || · ||. Proposition 2.6 [13,30] If the exponent p(·) satisfies the log-Hölder continuity condition, i.e., there is a constant α > 0, such that for every x, y ∈ , x = y with |x − y| ≤ 1 2 , one has then we have the Poincaré inequality, then there exists a constant C > 0 depending only on and the function p, such that In this paper, we will use the following equivalent norm on W 1, p(·) 0 ( ): which is equivalent to || · ||. Furthermore, we have the compact embedding W [16]).
where the infinimum is taken on all possible decompositions

A review on some classes of mappings and topological degree theory
Now, we give some results and properties from the theory of topological degree. The readers can find more information about the history of this theory in [1,2,8,11,15].
In what follows, let X be a real separable reflexive Banach space and X * be its dual space with dual pairing · , · and given a nonempty subset of X . Strong (weak) convergence is represented by the symbol → ( ).
compact, if it is continuous and the image of any bounded set is relatively compact.
(2) quasimonotone, if for any sequence (u n ) ⊂ with u n u, we have lim sup n→∞ Fu n , u n − u ≥ 0. In the sequel, we consider the following classes of operators: where T ∈ F 1 (E) is called an essential inner map to F.

Definition 3.5 Suppose that E is bounded open subset of a real reflexive Banach space
is called an admissible affine homotopy with the common continuous essential inner map T . Next, as in [15], we give the topological degree for the class F(X ).

Theorem 3.7 Let
Then, there exists a unique degree function d : M −→ Z that satisfies the following properties: (1) (Normalization) For any h ∈ E, we have

Assumptions and main results
In this section, we will discuss the existence of weak solution to (1.1).
We assume that ⊂ R N (N ≥ 2) is a bounded domain with a Lipschitz boundary ∂ , p ∈ C + ( ) satisfy the log-Hölder continuity condition (2.8) is well defined (see [28]).
the assumptions (A 2 ) and (A 4 ) and the given hypotheses about the exponents p, α, q and s, because: Then, by Remark 2.5, we can conclude that L p(x) → L r (x) , L p(x) → L β(x) and

This implies that the integral
Then, we shall use the definition of weak solution for problem (1.1) in the following sense:

Definition 4.2 We say that an element
Before giving our main result, we first give two lemmas that will be used later. Let us consider the following functional: From [28], it is obvious that J is a continuously Gâteaux differentiable and In addition, the following lemma summarizes the properties of the operator T (see [28,Proposition 3.1.]).

Lemma 4.3 The mapping
is a continuous, bounded, strictly monotone operator, and is of class (S + ).

Lemma 4.4 Assume that the assumptions (A 1 ) − (A 4 ) hold. Then, the operator
Proof To prove this lemma, we proceed in four steps.
In this step, we prove that the operator ϒ is bounded and continuous. First, let u ∈ W 1, p(x) 0 ( ), bearing (A 4 ) in mind and using Eqs. (2.5) and (2.6), we infer Then, we deduce from (2.9) and L p(x) → L κ(x) that Second, we show that the operator ϒ is continuous. To this purpose, let u n → u in W 1, p(x) 0 ( ). We need to show that ϒu n → ϒu in L p (x) ( ). We will apply the Lebesgue's theorem.
Note that if u n → u in W 1, p(x) 0 ( ), then u n → u in L p(x) ( ). Hence, there exist a subsequence (u k ) of (u n ) and φ in L p(x) ( ), such that for a.e. x ∈ and all k ∈ N. Hence, from (A 2 ) and (4.1), we have for a.e. x ∈ and for all k ∈ N.
On the other hand, thanks to (A 3 ) and (4.1), we get, as k −→ ∞ Seeing that then, from the Lebesgue's theorem and the equivalence (2.4), we have and consequently that is, ϒ is continuous.
Step 2 : We define the operator : We will prove that is bounded and continuous. It is clear that is continuous. Next, we show that is bounded. Let u ∈ W 1, p(x) 0 ( ), and using (2.5) and (2.6), we obtain Hence, we deduce from L p(x) → L β(x) and (2.9) that and consequently, is bounded on W 1, p(x) 0 ( ).
Step 3 : Let us define the operator : We will show that is bounded and continuous.
Step 4: Let I * : L p (x) ( ) → W −1, p (x) ( ) be the adjoint operator of the operator I : W We then define and On another side, taking into account that I is compact, then I * is compact. Thus, the compositions I * • ϒ, I * • , and I * • are compact, that means S = I * • ϒ + I * • + I * • is compact. With this last step, the proof of Lemma 4.4 is completed.
We are now in the position to get the existence result of weak solution for (1.1).
Consequently, the problem (1.1) is equivalent to the equation Taking into account that, by Lemma 4.3, the operator T is a continuous, bounded, strictly monotone and of class (S + ), then, by [32,Theorem 26 A], the inverse operator is also bounded, continuous, strictly monotone and of class (S + ). On another side, according to Lemma 4.4, we have that the operator S is bounded, continuous, and quasimonotone.
then, according to L p(x) → L α(x) , L p(x) → L s(x) and L p(x) → L q(x) , we get what implies that Lϕ : ϕ ∈ B is bounded.
On the other hand, we have that the operator is S is bounded, then SoLϕ is bounded. Thus, thanks to (4.5), we have that B is bounded in W −1, p (x) ( ).
On another side, taking into account that I , S, and L are bounded, then I + SoL is bounded. Hence, we infer that I + S • L ∈ F L,B (B τ (0)) and I = T • L ∈ F L,B (B τ (0)). Next, we define the homotopy Finally, we infer that u = Lϕ is a weak solution of (1.1). The proof is completed.

Conclusion
In this paper, we proved the existence of weak solution for a Dirichlet boundary value problem involving the p(x)-Laplacian-like operator using the theory of topological degree. In our next works, our goal is to study the parabolic case associated with problem (1.1) by adopting the approach used in [17] which combines the Fatou Lemma together with bounds conditions and regularity assumptions. Furthermore, we will attempt to obtain useful approximations of the solutions via Galerkin approximations as in [9].