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
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.
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The first author was partially supported by ANID (Agencia Nacional de Investigación y Desarrollo) Fondecyt Project no 3200422.
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Rey, C.A., Saintier, N. Non-local Equations and Optimal Sobolev Inequalities on Compact Manifolds. J Geom Anal 34, 17 (2024). https://doi.org/10.1007/s12220-023-01451-2
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DOI: https://doi.org/10.1007/s12220-023-01451-2