Singularly perturbed nonlinear Neumann problems under the conditions of Berestycki and Lions

Abstract

Let Omega be a bounded domain in R-N with the boundary partial derivative Omega is an element of C-3. We consider the following singularly perturbed nonlinear elliptic problem on Omega, epsilon(2)Delta nu - v + f(v)=0, v > 0 on Omega, partial derivative v/partial derivative nu = 0 on partial derivative Omega, where nu is the exterior normal to partial derivative Omega and the nonlinearity f is of subcritical growth. It has been known that under Berestycki and Lions conditions for f is an element of C-1(R) and N >= 3, there exists a solution nu(epsilon) of the problem which develops a spike layer near a local maximum point of the mean curvature H on partial derivative Omega for small epsilon > 0. In this paper, we extend the previous result for f is an element of C-0(R) and N >= 2. (C) 2011 Published by Elsevier Inc.