Existence of renormalized solutions for some quasilinear elliptic Neumann problems

Abstract

This paper is devoted to study some nonlinear elliptic Neumann equations of the type{Au+g(x,u,∇u)+|u|q(⋅)-2u=f(x,u,∇u)inΩ,∑i=1Nai(x,u,∇u)⋅ni=0on∂Ω,\left\{ {\matrix{ {Au + g(x,u,\nabla u) + |u{|^{q( \cdot ) - 2}}u = f(x,u,\nabla u)} \hfill & {{\rm{in}}} \hfill & {\Omega ,} \hfill \cr {\sum\limits_{i = 1}^N {{a_i}(x,u,\nabla u) \cdot {n_i} = 0} } \hfill & {{\rm{on}}} \hfill & {\partial \Omega ,} \hfill \cr } } \right. in the anisotropic variable exponent Sobolev spaces, where A is a Leray-Lions operator and g(x, u, ∇u), f (x, u, ∇u) are two Carathéodory functions that verify some growth conditions. We prove the existence of renormalized solutions for our strongly nonlinear elliptic Neumann problem.

Keywords

• renormalized solution
• strongly nonlinear elliptic equations
• anisotropic variable exponent sobolev spaces
• neumann problem
• 35j60
• 35d05