Existence and uniqueness of weak and entropy solutions for homogeneous Neumann boundary-value problems involving variable exponents

In this article we study the nonlinear homogeneous Neumann boundary-value problem $$displaylines{ b(u)-hbox{div} a(x, abla u)=fquad hbox{in } Omegacr a(x, abla u).eta=0 quadhbox{on }partial Omega, }$$ where $Omega$ is a smooth bounded open domain in $mathbb{R}^{N}$, $N geq 3$ and $eta$ the outer unit normal vector on $partialOmega$. We prove the existence and uniqueness of a weak solution for $f in L^{infty}(Omega)$ and the existence and uniqueness of an entropy solution for $L^{1}$-data $f$. The functional setting involves Lebesgue and Sobolev spaces with variable exponents.