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

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 $N\ge 3$, $\lambda >\frac{N(N-2)}{4}$, 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 this work, we establish the existence and multiplicity of nonradial sign-changing solutions when $\lambda <\frac{(N-1{)}^2}{4}$, and $p=2^{\ast }-1$. We also prove a partial non-existence result for a large class of symmetric solutions when $\lambda >\frac{(N-1{)}^2}{4}$.