The role of the curvature of a surface in the shape of the solutions to elliptic equations

TL;DR

This work establishes that positive semi-stable solutions of the semilinear elliptic equation $-\Delta u=f(u)$ on domains in the sphere $\mathbb{S}^2$ or hyperbolic plane $\mathbb{H}^2$ possess a single, nondegenerate critical point under a natural geometric condition coupling boundary curvature with uniformly star-shapedness. The authors develop a vector-field method, constructing $V$ from two Killing fields $K_1,K_2$ and $\nabla u$ and applying the Poincaré–Hopf theorem to relate the zeros of $V$ to the critical points of $u$, with $\langle V,\nu\rangle>0$ on $\partial\Omega$ ensuring boundary well-posedness and index 1 for zeros. The results cover general nonlinearities under semi-stability and extend prior horoconvex/convex-domain theories to a broader class of geometries, including non-horoconvex hyperbolic domains and certain nonconvex spherical domains; optimality is demonstrated through explicit domain constructions yielding arbitrarily many maxima when the geometric condition fails. Collectively, the paper broadens the understanding of critical-point structure in curved geometries and provides tools and examples of broader applicability beyond the classical Euclidean setting.

Abstract

We prove uniqueness and non-degeneracy of the critical point of positive, semi-stable solutions of $-Δu=f(u)$ with Dirichlet boundary conditions for a class of star-shaped domains of the sphere and of the hyperbolic plane satisfying a geometric condition. In the spherical case, this condition is weaker than convexity, while in the hyperbolic case it is weaker than horoconvexity. Finally, we construct examples showing that this geometric condition is indeed optimal.

Figures (7)

• Figure 1: The strip $\mathcal{S}_b$ in the stereographic projection of $\mathbb S^2$ (left) and in the Poincaré disk model of $\mathbb H^2$ (right).
• Figure 2: In the stereographic projection (from the south pole) of the sphere, the origin corresponds to the north pole, the $x$ axis to a great circle passing through the north pole, the unit circle (dotted red circle) corresponds to the equator.
• Figure 3: The unperturbed domain $\mathcal{S}_b$ and the final domain $\Omega_{b}$ in the spherical case.
• Figure 4: The unperturbed domain $\mathcal{S}_b$ and the final domain $\Omega_b$ in the hyperbolic case.
• Figure 5: Some examples of non-convex domains with a unique critical point.

Theorems & Definitions (25)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1: Poincaré-Hopf
• Theorem 2.2
• Remark 3.1
• Remark 3.2
• Theorem 5.1
• Remark 5.1
• Lemma 5.2