On a class of elliptic equations with critical perturbations in the hyperbolic space

Abstract

We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space $$\begin{array}{c}-{\mathrm{\Delta }}_{{\mathbb{B}}^{\mathit{N}}}\mathit{u}-\mathit{\lambda }\mathit{u}=\mathit{a}\mathrm{(}\mathit{x}\mathrm{)}{\mathit{u}}^{\mathit{p}-1}+\mathit{\epsilon }{\mathit{u}}^{2^{\ast }-1}\text{in }{\mathbb{B}}^{\mathit{N}}\mathrm{,}\mathit{u}\in {\mathit{H}}^1\mathrm{\left(}{\mathbb{B}}^{\mathit{N}}\mathrm{\right)}\mathrm{,}\end{array}$$ where ${\mathbb{B}}^{\mathit{N}}$ denotes the hyperbolic space, $2\mathrm{<}\mathit{p}\mathrm{<}{2}^{\ast }:=\frac{2\mathit{N}}{\mathit{N}-2}$ , if $\mathit{N}⩾3;2\mathrm{<}\mathit{p}\mathrm{<}+\infty$ , if $\mathit{N}=2$ , $\mathit{\lambda }\mathrm{<}\frac{{\mathrm{(}\mathit{N}-1\mathrm{)}}^2}{4}$ , and $0\mathrm{<}\mathit{a}\in {\mathit{L}}^{\infty }\mathrm{(}{\mathbb{B}}^{\mathit{N}}\mathrm{)}$ . We first prove the existence of a positive radially symmetric ground-state solution for $\mathit{a}\mathrm{(}\mathit{x}\mathrm{)}\equiv 1$ . Next, we prove that for $\mathit{a}\mathrm{(}\mathit{x}\mathrm{)}⩾1$ , there exists a ground-state solution for ε 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 ${\mathbb{R}}^{\mathit{N}}$ . Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case $\mathit{a}\mathrm{(}\mathit{x}\mathrm{)}⩽1$ is considered where we first show that there is no ground-state solution, and prove the existence of a bound-state solution (high energy solution) for ε 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.