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^2\Delta v-v+f(v)=0\hbox{in}\Omega,\quad v=0 \hbox{on}\partial \Omega,{equation*} where $\Omega $ is a smooth and bounded domain of $\R^N$, $\e$ 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 $\partial \Omega$.