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} -\Delta_{p(x)}u & = |u|^{q(x)-2}u+f(x,u)+\mu\,\,\mbox{in}\,\,\Omega,\\ u & = 0\,\, \mbox{on}\,\, \partial\Omega, \end{split} \end{align*} where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $\mu > 0$ and $1 < p^{-}:=\underset{x\in \Omega}{\text{inf}}\;p(x) \leq p^{+}:= \underset{x\in \Omega}{\text{sup}}\;p(x) < q^{-}:=\underset{x\in \Omega}{\text{inf}}\;q(x)\leq q(x) \leq p^{\ast}(x) < N$. 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 $p(x)$.