Weak and Entropy Solutions to Nonlinear Elliptic Problems with Variable Exponent

We study the boundary value problem $-div(a(x,\nabla u))=f(x,u)$ in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$ and $div(a(x,\nabla u))$ is a $p(x)$-Laplace type operator. We obtain the existence and uniqueness of an entropy solution for $L^{1}$-data $f$ independent of $u$, the existence of weak energy solution for general data $f$ dependent of $u$ where the variable exponent $p(.)$ is not necessarily continuous.