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

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

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