Non-local Equations and Optimal Sobolev Inequalities on Compact Manifolds

Abstract

This paper deals with the theory of fractional Sobolev spaces on a compact Riemannian manifold (M, g). Our first main result shows that the fractional Sobolev spaces $W^{s,p}(M)$ introduced by Guo et al. (Electron J Differ Equ 2018(156): 1–17, 2018) coincide with the classical Triebel–Lizorkin spaces (which in turn coincide with the Besov spaces). As an application, we study a non-local elliptic equation of the form

$$\begin{aligned} {\mathcal {L}}_{{\mathcal {K}}}u + h|u |^{p-2}u = f|u |^{q-2}u, \end{aligned}$$

(1)

where the operator ${\mathcal {L}}_{K} u$ is an integro-differential operator a little more general than the fractional Laplacian, defined on $W^{s,p}(M)$. We use the Mountain Pass Theorem to show an existence result under a coercivity condition when we have a sub-critical non-linearity on the right-hand side of the Eq. (1). Our second main result is a Sobolev inequality in the critical range with an optimal constant for the fractional Sobolev spaces $W^{s,2}(M)$. This inequality gives us a sufficient existence condition for (1) with $p=2$ and $q=2^*=\frac{2n}{n-2s}$ the fractional critical Sobolev exponent.