On a class of elliptic equations with Critical Perturbations in the hyperbolic space

We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space $$-\Delta_{\mathbb{B}^N} u-\lambda u=a(x)u^{p-1} \, + \, \varepsilon u^{2^*-1} \,\;\;\text{in}\;\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})},$$ where $\mathbb{B}^N$ denotes the hyperbolic space, $2<p<2^*:=\frac{2N}{N-2}$, if $N \geqslant 3; 2<p<+\infty$, if $N = 2,\;\lambda < \frac{(N-1)^2}{4}$, and $0< a\in L^\infty(\mathbb{B}^N).$ We first prove the existence of a positive radially symmetric ground-state solution for $a(x) \equiv 1.$ Next, we prove that for $a(x) \geq 1$, there exists a ground-state solution for $\varepsilon$ small. For proof, we employ ``conformal change of metric" which allows us to transform the original equation into a singular equation in a ball in $\mathbb R^N$. Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case $a(x) \leq 1$ is considered where we first show that there is no ground-state solution, and prove the existence of a \it bound-state solution \rm (high energy solution) for $\varepsilon$ small. We employ variational arguments in the spirit of Bahri-Li to prove the existence of high energy-bound-state solutions in the hyperbolic space.