Existence of solutions for p(x)- nonlinear elliptic problems with not uniformly coercive

Abstract

In this paper we study existence of renormalized solution to the fallowing problem:
\begin{equation}\label{appp}- \mbox{div} \> a(x,u,\nabla u)+ g(x,u)= f \quad \mbox{in} \ \ \Omega.
\end{equation} where $\Omega$ is a bounded open subset of $\mathbb{R}^{N}$, $N\geq 2$. $f \in L^{1}(\Omega)$ and the first term of (\ref{appp}) is not controlled on u, and which is not uniformly coercive.