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

[Forthcoming] Existence Results for Singular p(x)-Laplacian Equation


R. Alsaedi
King Abdulaziz University
Saudi Arabia

K. Ben Ali
Gabès University
Tunisia

A. Ghanmi
Université de Tunis El Manar
Tunisie



Publié le 19 octobre 2021   DOI :

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Abstract

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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