On the critical points of solutions of PDE in a non-convex settings: the case of concentrating solutions

TL;DR

This paper investigates the number and nature of critical points of solutions to nonlinear elliptic PDEs in non-convex planar domains, providing estimates and exact counts for multi-peak solutions of the Gel'fand problem.

Contribution

It offers new estimates and exact calculations for the critical points of solutions in non-convex domains, extending understanding of solution topology in these settings.

Findings

Derived bounds on the number of critical points

Calculated exact critical point counts for specific solutions

Analyzed the index and structure of solutions in non-convex domains

Abstract

In this paper we are concerned with the number of critical points of solutions of nonlinear elliptic equations. We will deal with the case of non-convex, contractile and non-contractile planar domains. We will prove results on the estimate of their number as well as their index. In some cases we will provide the exact calculation. The toy problem concerns the multi-peak solutions of the Gel'fand problem, namely $$\begin{cases} -\Delta u=\lambda e^{u}&\mbox{ in }\Omega u=0 & \mbox{ on }\partial\Omega, \end{cases} $$ where $\Omega\subset\mathbb{R}^2$ is a bounded smooth domain and $\lambda>0$ is a small parameter.