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

摘要

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 $$left { begin{array}{cc} -Delta_{p}u + |u|^{s-1}u = 0, & {rm in}, Omega; |nabla u|^{p-2}nabla u . eta = psi, & {rm on} , partialOmega;end{array}right.$$where Ω is a bounded domain in ({I!R^{N}}) with smooth boundary ({partialOmega, 1 < p < N, s > 0, eta}) is the unit outward normal on ({partialOmega {rm and} psi in L^{m}(partialOmega), m > 1}). Mathematics Subject Classification (2010) 35J60 Keywords Nonlinear problems Neumann boundary condition distributional solutions Lecture held in the Seminario Matematico e Fisico di Milano on February 22, 2013.