Mountain pass solutions for singularly perturbed nonlinear Dirichlet problems

Abstract

We consider the following singularly perturbed nonlinear elliptic problem epsilon(2)Delta u - u + f(u) = 0, u > 0 in Omega, u = 0 on partial derivative Omega where the nonlinearity f is of subcritical growth and Omega is a bounded domain in R-n. Under certain conditions of f, there exists a mountain pass solution us for above problem and the solution u(epsilon) exhibits a spike layer as epsilon -> 0. To see the location of the spike layer for small epsilon > 0, some additional assumptions, certain nondegeneracy for a limiting problem or monotonicity of f(t)/t, have been assumed. In this paper, we characterize the location of the spike layer without such additional assumptions for n >= 3. (c) 2005 Elsevier Inc. All rights reserved.