Electron. J. Differential Equations, Vol. 2018 (2018), No. 156, pp. 1-17.

Fractional p-Laplacian equations on Riemannian manifolds

Lifeng Guo, Binlin Zhang, Yadong Zhang

Abstract:
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
$$\displaylines{ (-\Delta_g)^s_p u(x)= f(x,u) \quad \text{in } \Omega,\cr u=0 \quad \text{in } M\setminus\Omega, }$$
where $N> ps$ with $s\in(0,1)$, $p\in (1, \infty)$, $(-\Delta_g)^s_p$ is the fractional p-Laplacian on Riemannian manifolds, (M,g) is a compact Riemannian $N-$manifold, $\Omega$ is an open bounded subset of M with smooth boundary $\partial\Omega$, and f is a Caratheodory function satisfying the Ambrosetti-Rabinowitz type condition. By using variational methods, we obtain the existence of nontrivial weak solutions when the nonlinearity f satisfies sub-linear or super-linear growth conditions.

Submitted August 18, 2017. Published August 22, 2018.
Math Subject Classifications: 35R01, 35R11, 35A15.
Key Words: Fractional p-Laplacian; Riemannian manifolds; variational methods.

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Lifeng Guo
School of Mathematics and Statistics
Northeast Petroleum University
Daqing 163318, China
email: lfguo1981@126.com
Binlin Zhang
Department of Mathematics
Heilongjiang Institute of Technology
Harbin 150050, China
email: zhangbinlin2012@outlook.com
Yadong Zhang
School of Mathematics and Statistics
Northeast Petroleum University
Daqing 163318, China
email: chichidong@foxmail.com

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