Blow-up exponents and a semilinear elliptic equation for the fractional Laplacian on hyperbolic spaces

Abstract

Let $\mathbb{H}^n$ be the $n$-dimensional real hyperbolic space, $\Delta$ its nonnegative Laplace--Beltrami operator whose bottom of the spectrum we denote by $\lambda_{0}$, and $\sigma \in (0,1)$. The aim of this paper is twofold. On the one hand, we determine the Fujita exponent for the fractional heat equation \[\partial_{t} u + \Delta^{\sigma}u = e^{\beta t}|u|^{\gamma-1}u,\] by proving that nontrivial positive global solutions exist if and only if $\gamma\geq 1 + \beta/ \lambda_{0}^{\sigma}$. On the other hand, we prove the existence of non-negative, bounded and finite energy solutions of the semilinear fractional elliptic equation \[ \Delta^{\sigma} v - \lambda^{\sigma} v - v^{\gamma}=0 \] for $0\leq \lambda \leq \lambda_{0}$ and $1<\gamma< \frac{n+2\sigma}{n-2\sigma}$. The two problems are known to be connected and the latter, aside from its independent interest, is actually instrumental to the former. \smallskip At the core of our results stands a novel fractional Poincar\'e-type inequality expressed in terms of a new scale of $L^{2}$ fractional Sobolev spaces, which sharpens those known so far, and which holds more generally on Riemannian symmetric spaces of non-compact type. We also establish an associated Rellich--Kondrachov-like compact embedding theorem for radial functions, along with other related properties.