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

We consider the positive solutions of the nonlinear eigenvalue problem $-{\mathrm{\Delta }}_{{\mathbb{H}}^n}u=\lambda u+u^p,$ with $p=\frac{n+2}{n-2}$ and $u \in H_0^1(\Omega),$where$\Omega$isageodesicballofradius$\theta_1$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 $n\mathrm{.}$ 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$L^$isthefirstpositivevalueof$L = -\ell(\ell+1)$forwhichasuitablydefinedassociatedLegendrefunction$P_{\ell}^{-\alpha}(\cosh\theta) >0$if$0 < \theta<\theta_1$and$P_{\ell}^{-\alpha}(\cosh\theta_1)=0,$with$\alpha = (2-n)/2.$