A weak solution for a (p(x),q(x))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p(x),q(x))$\end{document}-Laplacian elliptic problem with a singular term

Here, we consider the following elliptic problem with variable components: −a(x)Δp(x)u−b(x)Δq(x)u+u|u|s−2|x|s=λf(x,u),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ -a(x)\Delta _{p(x)}u - b(x) \Delta _{q(x)}u+ \frac{u \vert u \vert ^{s-2}}{|x|^{s}}= \lambda f(x,u), $$\end{document} with Dirichlet boundary condition in a 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} with a smooth boundary. By applying the variational method, we prove the existence of at least one nontrivial weak solution to the problem.


Introduction
The quasilinear operator (p, q)-Laplacian has been used to model steady-state solutions of reaction-diffusion problems arising in biophysics, plasma physics, and in the study of chemical reactions. These problems appear, for example, in a general reaction-diffusion system: where D(u) = |∇u| p-2 + |∇u| q-2 is the diffusion coefficient, function u describes the concentration, and the reaction term h(x, u) has a polynomial form with respect to the concentration u. The differential operator p + q is known as the (p, q)-Laplacian operator, if p = q, where j , j > 1 denotes the j-Laplacian defined by j u := div(|∇u| j-2 ∇u). It is not homogeneous, thus some technical difficulties arise in applying the usual methods of the theory of elliptic equations (for further details, see [1,2,5,7,8,10,[12][13][14][15][16][19][20][21][22][23] and references therein).
Our main interest in this work is to prove the existence of a weak solution of the weighted (p(x), q(x))-Laplacian problem where ⊂ R N is a bounded domain with a smooth boundary, a, b ∈ L ∞ ( ) are positive functions with a(x) ≥ 1 a.e. on , λ > 0 is a real parameter, r(x) u = div(|∇u| r(x)-2 ∇u) denotes r(x)-Laplacian operator, for r ∈ {p, q}, where p, q ∈ C + (¯ ), 1 < s < q(x) < p(x) < ∞ a.e. on and f : × R → R is a Carathéodory function satisfying the following growth condition: a.e. in and In [11] the existence and multiplicity of solutions for the following problem have been In [4] the authors proved the existence of two weak solutions for the problem where 2 ≤ q < p < N . Motivated by their works, we want to verify the existence of at least one solution for the weighted problem (1.1). To this end, we introduce our notations and also bring some definitions and results. Throughout this note, ⊂ R N is a bounded domain with a smooth boundary. We set endowed with the following norm: For any u ∈ L p(·) (B) and v ∈ L p (·) (B), where L p (·) (B) is the conjugate space of L p(·) (B), the Hölder type inequality and the norm in W 1,p(·) ( ) is taken to be with the norm |∇u| p .
for any x ∈¯ , then the embedding a.e. on , then, by Proposition 1.1, one has Here, we recall the classical Hardy inequality (see [3,17]).  Assume that there exists r > 0 and x ∈ X, with 0 < (x) < r, such that (i) In the sequel we set X := W 1,p(x) 0 ( ) endowed with the norm u = |∇u| p .

Existence of a solution
In this section we prove the existence of at least one nontrivial weak solution of the problem (1.1).
Let : X → R be a functional defined by Remark 2.1 Under the above assumptions, we get Proof First, let u > 1. So, we have Now, let u ≤ 1. Then we have Thus the proof is completed.
It is known that is a continuously Gâteaux differentiable functional; moreover, for u, v ∈ X, see [18]. Let f : × R → R be a Carathéodory function and define Then the functional : for every u ∈ X is continuously Gâteaux differentiable, with the following compact derivative: for every u, v in X, see [18]. Moreover, define for every u, v ∈ X, and then the critical points of I are the weak solutions of problem (1.1). Set Obviously, there exists x 0 ∈ such that For γ > 0 and h ∈ C + ( ) with 1 < h -, we define and similarly, Further, , where is the Euler function.
The following is the main result of this paper. Let X, and be as above and fix λ ∈]0, λ * [. By (2.3), there exists We define u ∈ X such that where | · | is Euclidean norm on R N . By Remark 2.1, we have Clearly, 0 < (u) < 1. Moreover, thanks to (2.4), one has and so