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

TL;DR

This paper proves the existence of multiple sign-changing spike solutions with alternating signs along a boundary geodesic in a semilinear Dirichlet problem, under symmetry and convexity conditions.

Contribution

It introduces new methods to establish the existence of multiple interior spike solutions with alternating signs aligned along a boundary geodesic.

Findings

Existence of solutions with multiple alternating sign peaks near a boundary geodesic.

Solutions are constructed for large even numbers of peaks.

The domain's symmetry and convexity are crucial for the results.

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 $\partial \Omega$.