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

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 ordinary differential equation 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 $\alpha = (2-n)/2$.