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

Let ${\mathbb{H}}^n$ be the $n$-dimensional real hyperbolic space, $\mathrm{\Delta }$ 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 $${\mathrm{\partial }}_{t}u+{\mathrm{\Delta }}^{\sigma }u=e^{\beta t}\mathrm{\mid }u{\mathrm{\mid }}^{\gamma -1}u,$$ by proving that nontrivial positive global solutions exist if and only if $\gamma \ge 1+\beta \mathrm{/}{\lambda }_0^{\sigma }$. On the other hand, we prove the existence of non-negative, bounded and finite energy solutions of the semilinear fractional elliptic equation $${\mathrm{\Delta }}^{\sigma }v-{\lambda }^{\sigma }v-v^{\gamma }=0$$ for $0\le \lambda \le {\lambda }_0$ 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 Poincaré-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.