Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions

Abstract

In this paper, we consider equations involving fully nonlinear nonlocal operators $$F_{\alpha }(u(x))\equiv C_{n,\alpha }PV{\int }_{{\mathbb{R}}^n}\frac{G(u(x)-u(z))}{|x-z{|}^{n+\alpha }}dz=f(x,u).$$ We prove a maximum principle and obtain key ingredients for carrying on the method of moving planes, such as narrow region principle and decay at infinity. Then we establish radial symmetry and monotonicity for positive solutions to Dirichlet problems associated to such fully nonlinear fractional order equations in a unit ball and in the whole space, as well as non-existence of solutions on a half space. We believe that the methods develop here can be applied to a variety of problems involving fully nonlinear nonlocal operators. We also investigate the limit of this operator as $\alpha \to 2$ and show that $$F_{\alpha }(u(x))\to a(-\Delta u(x))+b|▽u(x){|}^2.$$

27 pages. arXiv admin note: text overlap with arXiv:1411.1697

References in corpus (1)

• A Liouville Theorem for the Fractional Laplacian

Cited by in corpus (2)

• Symmetry and decay of traveling wave solutions to the Whitham equation
• Solutions of Fully Nonlinear Nonlocal Systems