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 $\{(\Theta ,{\Vert U\Vert }_{\infty })\}\subset {\mathbb{R}}^2$ 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 $p>p_{\mathrm{S}}:=(N-2)/(N+2)$, 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 $p_{\mathrm{S}}<p<p_{\mathrm{JL}}$, then there exists ${\Theta }^{\ast }\in (\underline{\Theta },\pi )$ such that the problem has infinitely many radial solutions for $\Theta ={\Theta }^{\ast }$, where $p_{\mathrm{JL}}=1+\frac{4}{N-4-2\sqrt{N-1}}$ if $N\ge 11$, $p_{\mathrm{JL}}=\infty$ if $2\le N\le 10$. Asymptotic behaviors of the bifurcation diagram as $p\to \infty$ and $p\downarrow 1$ are also studied.

References in corpus (1)

• A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth