Existence and non-existence of nonradial solutions to an elliptic equation on hyperbolic space with critical and subcritical nonlinearities

Abstract

In this article, we study the semi-linear elliptic problem: \begin{equation*} -Δ_{\mathbb{B}^N} u \, - \, λu = |u|^{p-1}u, \quad u \in H^{1}(\mathbb{B}^N), \end{equation*} where $N\ge 3$, $\lambda >\frac{N(N-2)}{4}$, and $1<p\le 2^{\ast }-1$. Here, ${\mathbb{B}}^N$ represents the Poincaré ball model of the hyperbolic space, and $H^1({\mathbb{B}}^N)$ denotes the Sobolev space on ${\mathbb{B}}^N$. In the critical case $p=2^{\ast }-1$, with $\lambda <\frac{(N-1{)}^2}{4}$ and $N\ge 4$, we establish the existence and multiplicity of nonradial sign-changing solutions. Moreover, our result answers an open question: ``the existence of sign-changing solutions in the lower dimensions $4\le N\le 6$". We also prove a partial non-existence result for a large class of symmetric solutions when $\lambda >\frac{(N-1{)}^2}{4}$, based on a quantitative orbit packing condition.