exit

Mathematics   > Home   > Advances in Pure and Applied Mathematics   > Issue 3 (June 2022)   > Article

Existence Results for Singular p(x)-Laplacian Equation

Résultats d’existence pour l’equation du p(x)-laplacien singulier


R. Alsaedi
King Abdulaziz University
Saudi Arabia

K. Ben Ali
Gabès University
Tunisia

A. Ghanmi
Université de Tunis El Manar
Tunisie



Published on 1 June 2022   DOI : 10.21494/ISTE.OP.2022.0840

Abstract

Résumé

Keywords

Mots-clés

This paper is concerned with the existence of solutions for the following class of singular fourth order elliptic equations
$$$ \left\{ \begin{array}{ll} \Delta\Big(|x|^{p(x)}|\Delta u|^{p(x)-2}\Delta u\Big)=a(x)u^{-\gamma (x)}+\lambda f(x,u),\quad \mbox{in }\Omega, \\ u=\Delta u=0, \quad \mbox{on }\partial\Omega. \end{array} \right.$$$
where $$$\Omega$$$ is a smooth bounded domain in $$$\mathbb{R}^N, \gamma :\overline{\Omega}\rightarrow (0,1)$$$ be a continuous function, $$$f\in C^{1}( \overline{\Omega}\times \mathbb{R}), p:\; \overline{\Omega}\longrightarrow \;(1,\infty)$$$ and $$$a$$$ is a function that is almost everywhere positive in $$$\Omega$$$. Using variational techniques combined with the theory of the generalized Lebesgue-Sobolev spaces, we prove the existence at least one nontrivial weak solution.

This paper is concerned with the existence of solutions for the following class of singular fourth order elliptic equations
$$$ \left\{ \begin{array}{ll} \Delta\Big(|x|^{p(x)}|\Delta u|^{p(x)-2}\Delta u\Big)=a(x)u^{-\gamma (x)}+\lambda f(x,u),\quad \mbox{in }\Omega, \\ u=\Delta u=0, \quad \mbox{on }\partial\Omega. \end{array} \right.$$$
where $$$\Omega$$$ is a smooth bounded domain in $$$\mathbb{R}^N, \gamma :\overline{\Omega}\rightarrow (0,1)$$$ be a continuous function, $$$f\in C^{1}( \overline{\Omega}\times \mathbb{R}), p:\; \overline{\Omega}\longrightarrow \;(1,\infty)$$$ and $$$a$$$ is a function that is almost everywhere positive in $$$\Omega$$$. Using variational techniques combined with the theory of the generalized Lebesgue-Sobolev spaces, we prove the existence at least one nontrivial weak solution.

p(x)-biharmonic variable exponent Lebesgue space variable exponent Sobolev space

p(x)-biharmonic variable exponent Lebesgue space variable exponent Sobolev space