Singularly perturbed nonlinear Neumann problems with a general nonlinearity

Abstract

Let Omega be a bounded domain in R-n, n >= 3, 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 u - u + f(u) = 0, u > 0 on Omega, partial derivative u/partial derivative v = 0 on partial derivative Omega, where v is an exterior normal to partial derivative Omega and a nonlinearity f of subcritical growth. Under rather strong conditions on f, it has been known that for small epsilon > 0, there exists a solution u(epsilon) of the above problem which exhibits a spike layer near local maximum points of the mean curvature H on partial derivative Omega as epsilon -> 0. In this paper, we obtain the same result under some conditions on f (Berestycki-Lions conditions), which we believe to be almost optimal. (C) 2008 Elsevier Inc. All rights reserved.