Solutions with multiple alternate sign peaks along a boundary geodesic to a semilinear Dirichlet problem

Abstract

We study the existence of sign-changing multiple interior spike solutions for the following Dirichlet problem {equation*}\e^2Δv-v+f(v)=0\hbox{in}Ω,\quad v=0 \hbox{on}\partial Ω,{equation*} where $\Omega$ is a smooth and bounded domain of ${\mathbb{R}}^N$, $\e$ is a small positive parameter, $f$ is a superlinear, subcritical and odd nonlinearity. In particular we prove that if $\Omega$ has a plane of symmetry and its intersection with the plane is a two-dimensional strictly convex domain, then, provided that $k$ is even and sufficiently large, a $k$-peak solution exists with alternate sign peaks aligned along a closed curve near a geodesic of $\partial \Omega$.