Fractional p-Laplacian equations on Riemannian manifolds

Abstract

<p>In this article we establish the theory of fractional Sobolev spaces on Riemannian manifolds. As a consequence we investigate some important properties, such as the reflexivity, separability, the embedding theorem and so on. As an application, we consider fractional p-Laplacian equations with homogeneous Dirichlet boundary conditions</p> <pre>(-∆_g)^s_pu(x) = ƒ(x, u) in Ω,<br> u = 0 in M \ Ω,</pre> <p>where N > ps with s ∈ (0, 1), p ∈ (1, ∞), (-∆_g)^s_p is the fractional p-Laplacian on Riemannian manifolds, (M, g) is a compact Riemannian N-manifold, Ω is an open bounded subset of M with smooth boundary ∂Ω, and ƒ is a Carathéodory function satisfying the Ambrosetti-Rabinowitz type condition. By using variational methods, we obtain the existence of nontrivial weak solutions when the nonlinearity ƒ satisfies sub-linear or super-linear growth conditions.</p>