Existence of high energy positive solutions for a class of elliptic equations in the hyperbolic space

TL;DR

This paper proves the existence of high energy positive solutions for a class of elliptic equations in hyperbolic space using a min-max method and new estimates involving hyperbolic bubbles.

Contribution

It introduces a min-max procedure adapted to hyperbolic space and develops new estimates with interacting hyperbolic bubbles for positive solutions.

Findings

Existence of positive solutions established for the scalar field problem.

Development of a min-max approach in hyperbolic space context.

New estimates involving interacting hyperbolic bubbles.

Abstract

We study the existence of positive solutions for the following class of scalar field problem on the hyperbolic space $$ -\Delta_{\mathbb{H}^N} u - \lambda u = a(x) |u|^{p-1} \, u\;\;\text{in}\;\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})}, $$ where $\mathbb{B}^N$ denotes the hyperbolic space, $1<p<2^*-1:=\frac{N+2}{N-2}$, if $N \geqslant 3; 1<p<+\infty$, if $N = 2,\;\lambda < \frac{(N-1)^2}{4}$, and $0< a\in L^\infty(\mathbb{B}^N).$ We prove the existence of a positive solution by introducing the min-max procedure in the spirit of Bahri-Li in the hyperbolic space and using a series of new estimates involving interacting hyperbolic bubbles.