A Liouville type theorem for Lane-Emden systems involving the fractional Laplacian

Abstract

We establish a Liouville type theorem for the fractional Lane-Emden system: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-Δ)^αu=v^q&{\rm in}\,\,\R^N,\\ (-Δ)^αv=u^p&{\rm in}\,\,\R^N, \end{array} \right. \end{eqnarray*} where $\alpha \in (0,1)$, $N>2\alpha$ and $p,q$ are positive real numbers and in an appropriate new range. To prove our result we will use the local realization of fractional Laplacian, which can be constructed as Dirichlet-to-Neumann operator of a degenerate elliptic equation in the spirit of Caffarelli and Silvestre \cite{CS}. Our proof is based on a monotonicity argument for suitable transformed functions and the method of moving planes in an infinity half cylinder based on some maximum principles which obtained by some barrier functions and a coupling argument using fractional Sobolev trace inequality.

References in corpus (3)

• A Liouville Theorem for the Fractional Laplacian
• On the fractional Lane-Emden equation
• On the uniqueness of solutions of an nonlocal elliptic system

Cited by in corpus (3)

• Liouville type results for systems of equations involving fractional Laplacian in exterior domains
• Existence results of positive solutions for nonlinear cooperative elliptic systems involving fractional Laplacian
• Radial symmetry results for fractional Laplacian systems