The solution gap of the Brezis-Nirenberg problem on the hyperbolic space

TL;DR

This paper investigates the existence and uniqueness of positive solutions to a nonlinear eigenvalue problem on hyperbolic space, revealing a precise parameter range where solutions exist for certain dimensions.

Contribution

It extends the Brezis-Nirenberg problem to hyperbolic space, identifying exact conditions for solution existence based on eigenvalues and Legendre functions for radial solutions.

Findings

Unique positive solutions exist for 2<n<4 within specific eigenvalue intervals.

The problem's solution structure is characterized by Legendre functions and their positivity.

Solution existence depends on the first positive zero of a Legendre function at the boundary.

Abstract

We consider the positive solutions of the nonlinear eigenvalue problem $-\Delta_{\mathbb{H}^n} u = \lambda u + u^p, $ with $p=\frac{n+2}{n-2}$ and $u \in H_0^1(\Omega),$ where $\Omega$ is a geodesic ball of radius $\theta_1$ on $\mathbb{H}^n.$ For radial solutions, this equation can be written as an ODE having $n$ as a parameter. In this setting, the problem can be extended to consider real values of $n.$ We show that if $2<n<4$ this problem has a unique positive solution if and only if $\lambda\in \left(n(n-2)/4 +L^*\,,\, \lambda_1\right).$ Here $L^*$ is the first positive value of $L = -\ell(\ell+1)$ for which a suitably defined associated Legendre function $P_{\ell}^{-\alpha}(\cosh\theta) >0$ if $0 < \theta<\theta_1$ and $P_{\ell}^{-\alpha}(\cosh\theta_1)=0,$ with $\alpha = (2-n)/2.$