On the critical points of semi-stable solutions on convex domains of Riemannian surfaces

TL;DR

This paper proves that positive semi-stable solutions to certain semilinear equations on convex domains of Riemannian surfaces have exactly one maximum, using topological and analytical methods.

Contribution

It establishes a unique critical point property for semi-stable solutions on convex domains in two-dimensional Riemannian model spaces, extending previous results.

Findings

Semi-stable solutions have exactly one non-degenerate critical point.

The critical point is a maximum.

The proof uses topological degree and auxiliary functions.

Abstract

In this paper we consider semilinear equations $-\Delta u=f(u)$ with Dirichlet boundary conditions on certain convex domains of the two dimensional model spaces of constant curvature. We prove that a positive, semi-stable solution $u$ has exactly one non-degenerate critical point (a maximum). The proof consists in relating the critical points of the solution with the critical points of a suitable auxiliary function, jointly with a topological degree argument.