Solutions of Fully Nonlinear Nonlocal Systems

TL;DR

This paper establishes symmetry, monotonicity, and non-existence results for solutions of fully nonlinear nonlocal systems using the method of moving planes, broadening understanding of such complex integro-differential equations.

Contribution

It introduces a narrow region principle and decay at infinity estimates, enabling the analysis of symmetry and non-existence for fully nonlinear nonlocal systems.

Findings

Positive solutions are radially symmetric and monotone in the whole space.

Non-existence of positive solutions on a half space is proved.

Method of moving planes is effectively applied to these systems.

Abstract

In this paper we consider the system involving fully nonlinear nonlocal operators: $$ \left\{ \begin{array}{ll} F_{\alpha}(u(x)) = C_{n,\alpha} PV \int_{{R}^n} \frac{G(u(x)-u(y))}{|x-y|^{n+\alpha}} dy=f(v(x)), F_{\beta}(v(x)) = C_{n,\beta} PV \int_{{R}^n} \frac{G(v(x)-v(y))}{|x-y|^{n+\beta}} dy=g(u(x)). \end{array} \right. $$ A \textit{narrow region principle} and a \textit{decay at infinity} for the system for carrying on the method of moving planes are established. Then we prove the radial symmetry and monotonicity for positive solutions to the nonlinear system in the whole space. Non-existence of positive solutions to the nonlinear system on a half space is proved.