Singular Elliptic Equations with Variable Exponents

Abstract

This paper deals with study of the nonlinear singular elliptic equations in a bounded domain $\Omega\subset\mathbb{R}^N$, $(N\geq 2)$ with Lipschitz boundary $\partial\Omega$:
\[
Au=\frac{f}{u^{\gamma(\cdot)}}+\mu,
\]
Where $A:=-\mathrm{div}\left(\widehat{a}(\cdot,Du)\right)$ is a Leray-Lions type operator which maps continuously $W^{1,p(\cdot)}_0(\Omega)$ into its dual $W^{-1,p'(\cdot)}(\Omega)$ whose simplest model is the $p(\cdot)$-laplacian type operator (i.e. $\widehat{a}(\cdot,\xi)=|\xi|^{p(\cdot)-2}\xi$) such that $f$ is a nonnegative function belonging to the Lebesgue space with variable exponents $L^{m(\cdot)}(\Omega)$ with $m(\cdot)$ being small (or $L^{1}(\Omega)$) and $\mu$ is a nonnegative function belongs to $L^{1}(\Omega)$ as nonhomogeneous datum while $m:\overline{\Omega}\to (1,+\infty)$, $\gamma:\overline{\Omega}\to (0,1)$ are continuous functions satisfying certain conditions depend on $p(\cdot)$. We prove the existence, uniqueness, and regularity of nonnegative weak solutions for this class of problems with $p(\cdot)$-growth conditions. More precisely, we will discuss that the nonlinear singular term has some regularizing effects on the solutions of the problem which depends on the summability of $f$, $m(\cdot)$, and the value of $\gamma(\cdot)$. The functional framework involves Sobolev spaces with variable exponents as well as Lebesgue spaces with variable exponents. Our results can be seen as a generalization of some results given in the constant exponents case.