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

We consider the positive solutions of the nonlinear eigenvalue problem $-Δ_{\mathbb{H}^n} u = λu + u^p, $ with $p=\frac{n+2}{n-2}$ and $u \in H_0^1(Ω),$ where $Ω$ is a geodesic ball of radius $θ_1$ 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 $λ\in \left(n(n-2)/4 +L^*\,,\, λ_1\right).$ Here $L^*$ is the first positive value of $L = -\ell(\ell+1)$ for which a suitably defined associated Legendre function $P_{\ell}^{-α}(\coshθ) >0$ if $0 < θ<θ_1$ and $P_{\ell}^{-α}(\coshθ_1)=0,$ with $α= (2-n)/2.$