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.
On the critical points of semi-stable solutions on convex domains of Riemannian surfaces / Grossi, Massimo; Provenzano, Luigi. - In: MATHEMATISCHE ANNALEN. - ISSN 0025-5831. - (2023). [10.1007/s00208-023-02722-7]
On the critical points of semi-stable solutions on convex domains of Riemannian surfaces
Massimo Grossi;Luigi Provenzano
2023
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.File | Dimensione | Formato | |
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