Variational construction of spike layer solutions for a singularly perturbed Neumann problem

Abstract

We consider the following singularly perturbed problem

\begin{equation*} $\varepsilon^2 \Delta u - u + f(u)=0$, $u>0 in \Omega$, $\frac{\partial u}{\partial \nu}=0 on \partial \Omega$. \end{equation*}

An existence of solutions with a spike layer near critical points of the mean curvature on the boundary $\partial \Omega$ is well known when a nondegeneracy for a limiting problem holds. In this dissertation, we develop a variational method for the construction of such solutions which does not depend on the nondengeneracy for the limiting problem. By the variational approach, we construct the solutions for an optimal class of nonlinearities f satisfying the Berestycki-Lions conditions.