The Emden-Fowler equation on a spherical cap of $\mathbb{S}^N$

TL;DR

This paper analyzes the bifurcation diagram of radial solutions to the Emden-Fowler equation on spherical caps in higher dimensions, revealing existence, uniqueness, and multiplicity results depending on parameters.

Contribution

It provides a detailed bifurcation analysis of the Emden-Fowler equation on spherical caps, including existence, uniqueness, and multiplicity of solutions, and studies asymptotic behaviors for different parameter regimes.

Findings

Existence of radial solutions for large enough geodesic radius when p > p_S.

Uniqueness of solutions near the sphere's boundary for certain p.

Presence of infinitely many solutions at specific radii for p between p_S and p_JL.

Abstract

Let $\mathbb{S}^N\subset\mathbb{R}^{N+1}$, $N\ge 3$, be the unit sphere, and let $S_{\Theta}\subset\mathbb{S}^N$ be a geodesic ball with geodesic radius $\Theta\in(0,\pi)$. We study the bifurcation diagram $\{(\Theta,\left\|U\right\|_{\infty})\}\subset\mathbb{R}^2$ of the radial solutions of the Emden-Fowler equation on $S_{\Theta}$ $\Delta_{\mathbb{S}^N}U+U^p=0$ in $S_{\Theta}$, $U=0$ on $\partial S_{\Theta}$, $U>0$ in $S_{\Theta}$, where $p>1$. Among other things, we prove the following: For each $p>p_{\rm S}:=(N-2)/(N+2)$, there exists $\underline{\Theta}\in(0,\pi)$ such that the problem has a radial solution for $\Theta\in(\underline{\Theta},\pi)$ and has no radial solution for $\Theta\in(0,\underline{\Theta})$. Moreover, this solution is unique in the space of radial functions if $\Theta$ is close to $\pi$. If $p_{\rm S}<p<p_{\rm JL}$, then there exists $\Theta^*\in(\underline{\Theta},\pi)$ such that the problem has infinitely many radial solutions for $\Theta=\Theta^*$, where $p_{\rm JL}= 1+\frac{4}{N-4-2\sqrt{N-1}}$ if $N\ge 11$, $p_{\rm JL}=\infty$ if $2\le N\le 10$. Asymptotic behaviors of the bifurcation diagram as $p\to\infty$ and $p\downarrow 1$ are also studied.