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 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 $\lambda \in \left(n(n-2)/4+L^{\ast },{\lambda }_1\right).$ Here $L^{\ast }$ 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 $\alpha =(2-n)/2.$

The final publication is available at Springer via http://dx.doi.org/10.1007/s00605-015-0861-1