Multiple interior and boundary peak solutions to singularly perturbed nonlinear Neumann problems under the Berestycki-Lions condition

Abstract

Let be a smooth bounded domain in , . We consider the following singularly perturbed nonlinear elliptic problem on Omega, epsilon(2) Delta v - v + f (v) = 0, v > 0 on Omega, partial derivative v/partial derivative v = 0 on partial derivative Omega, where is an exterior unit normal vector to and a nonlinearity f satisfies subcritical growth condition. It has been known that for any , , there exists a solution of the above problem which exhibits -boundary peaks and -interior peaks for small under rather strong conditions on f, such as the linearized non-degeneracy condition for a limiting problem. In this paper, we extend the previous result to more general class of f satisfying Berestycki-Lions conditions which we believe to be almost optimal.