Existence of solution of the $p(x)$-Laplacian problem involving critical exponent and Radon measure

Abstract

In this paper we are proving the existence of a nontrivial solution of the $p(x)$- Laplacian equation with Dirichlet boundary condition. We will use the variational method and concentration compactness principle involving positive radon measure $\mu$. \begin{align*} \begin{split} -Δ_{p(x)}u & = |u|^{q(x)-2}u+f(x,u)+μ\,\,\mbox{in}\,\,Ω,\\ u & = 0\,\, \mbox{on}\,\, \partialΩ, \end{split} \end{align*} where $\Omega \subset {\mathbb{R}}^N$ is a smooth bounded domain, $\mu >0$ and $1<p^{-}:=\underset{x\in \Omega }{\inf }p(x)\le p^{+}:=\underset{x\in \Omega }{\sup }p(x)<q^{-}:=\underset{x\in \Omega }{\inf }q(x)\le q(x)\le p^{\ast }(x)<N$. The function $f$ satisfies certain conditions. Here, $q^{\prime }(x)=\frac{q(x)}{q(x)-1}$ is the conjugate of $q(x)$ and $p^{\ast }(x)=\frac{Np(x)}{N-p(x)}$ is the Sobolev conjugate of $p(x)$.