Fractional p-Laplacian equations on Riemannian manifolds




Guo, Lifeng
Zhang, Binlin
Zhang, Yadong

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Texas State University, Department of Mathematics


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 (-∆g)spu(x) = ƒ(x, u) in Ω, u = 0 in M \ Ω, where N > ps with s ∈ (0, 1), p ∈ (1, ∞), (-∆g)sp 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.



Fractional p-Laplacian, Riemannian manifolds, Variational methods


Guo, L., Zhang, B., & Zhang, Y. (2018). Fractional p-Laplacian equations on Riemannian manifolds. <i>Electronic Journal of Differential Equations, 2018</i>(156), pp. 1-17.


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