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

Author:

Massimo Grossi, Luigi Provenzano

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG), Analysis of PDEs (math.AP), Spectral Theory (math.SP)

journal:

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date:

2023-06-17 16:00:00

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.