W 1,1(Ω) Solutions of Nonlinear Problems with Nonhomogeneous Neumann Boundary Conditions

Abstract

In this paper we study the existence of W 1,1(Ω) distributional solutions of the nonlinear problems with Neumann boundary condition. The simplest model is $$\left \{ \begin{array}{cc} -\Delta_{p}u + |u|^{s-1}u = 0, & {\rm in}\, \Omega;\\ |\nabla u|^{p-2}\nabla u . \eta = \psi, & {\rm on} \, \partial\Omega;\end{array}\right.$$ where Ω is a bounded domain in $${I\!R^{N}}$$ with smooth boundary $${\partial\Omega, 1 0, \eta}$$ is the unit outward normal on $${\partial\Omega {\rm and} \psi \in L^{m}(\partial\Omega), m > 1}$$ .