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

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.