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 Ws,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 LKu + h|u| p−2u = f |u| q−2u, (1) where the operator LK u is an integro-differential operator a little more general than the fractional Laplacian, defined on Ws,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 Ws,2(M). This inequality gives us a sufficient existence condition for (1) with p = 2 and q = 2∗ = 2n n−2s the fractional critical Sobolev exponent.