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

Abstract

In this article, we consider the following problem: \begin{align*} -Δ_{\mathbb{B}^N} u \, - \, λu = |u|^{p-1}u, \quad u \in H^{1}(\mathbb{B}^N), \end{align*} where $N \geq 3$, $λ> \frac{N(N-2)}{4}$, and $1 < p \leq 2^*-1$. 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 this work, we establish the existence and multiplicity of nonradial sign-changing solutions when $λ< \frac{(N-1)^2}{4}$, and $p = 2^*-1$. We also prove a partial non-existence result for a large class of symmetric solutions when $λ> \frac{(N-1)^2}{4}$.