Weak and renormalized solutions for anisotropic Neumann problems with degenerate coercivity

Abstract

In this work, we study the following quasilinear Neumann boundary-value problem $$\left\{\begin{array}{ll} \displaystyle -\sum^{N}_{i=1} D^{i}(a_{i}(x,u,\nabla u))+|u|^{p_{0}-2} u= f(x,u,\nabla u) & \mbox{in } \ \quad \Omega,\\ \displaystyle \sum^{N}_{i=1} a_{i}(x,u,\nabla u)\cdot n_{i} = g(x) & \mbox{on } \ \quad \partial\Omega, \end{array}\right.$$ where $\Omega$ is a bounded open domain in $\>I\!\!R^{N}$, $(N\geq 2)$. We prove the existence of a weak solution for $f \in L^{\infty}(\Omega)$ and $g\in L^{\infty}(\partial\Omega)$ and the existence of renormalized solutions for $L^{1}$-data $f$ and $g$. The functional setting involves anisotropic Sobolev spaces with constants exponents.