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

Abstract

We study the existence of positive solutions for the following class of scalar field problem on the hyperbolic space $$-{\Delta }_{{\mathbb{H}}^N}u-\lambda u=a(x)|u{|}^{p-1}u\text{in}{\mathbb{B}}^N,u\in H^1({\mathbb{B}}^N),$$ where ${\mathbb{B}}^N$ denotes the hyperbolic space, $1<p<2^{\ast }-1:=\frac{N+2}{N-2}$, if $N⩾3;1<p<+\infty$, 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.

arXiv comment22 pages