Entropy Solution for Some $p(x)$-Quasilinear Problem with Right-Hand Side Measure

Abstract

In this paper we study the existence of entropy solution for the following $p(x)$-quasilinear elliptic problem $$ \mbox{div}(a(x,u,\nabla u))+ g(x,u,\nabla u) = \mu$$ where the right-hand side $\mu$ is a measure, which admits a decomposition in $L^{1}(\Omega)+W^{-1,p'(x)}(\Omega)$ and $g(x,s,\xi)$ is a nonlinear term which has a growth condition with respect to $\xi$ and has no growth with respect to $s$ while satisfying a sign condition on $s$.