On Neumann Boundary Value Problems For Some Quasilinear Elliptic Equations

Abstract

. We study the role played by the indefinite weight function a(x) on the existence of positive solutions to the problem 8 ! : \\Gammadiv (jruj p\\Gamma2 ru) = a(x)juj p\\Gamma2 u + b(x)juj fl \\Gamma2 u; x 2\\Omega ; @u @n = 0; x 2 @\\Omega ; where\\Omega is a smooth bounded domain in R n , b changes sign, 1 ! p ! N , 1 ! fl ! Np=(N \\Gamma p) and fl 6= p. We prove that (i) if R\\Omega a(x) dx 6= 0 and b satisfies another integral condition, then there exists some such that R\\Omega a(x) dx ! 0 and, for strictly between 0 and , the problem has a positive solution and (ii) if R \\Omega a(x) dx = 0, then the problem has a positive solution for small provided that R \\Omega b(x) dx ! 0. 1. Introduction and results. In this paper we study the existence of positive solutions of the Neumann boundary value problem 8 ! : \\Gamma\\Delta p u + g(x; u) = 0; x 2\\Omega ; @u @n = 0; x 2 @\\Omega ; (1:1) on a bounded domain\\Omega ae R N with smooth boundary @ where \\D...