Symmetry and Monotonicity of Positive Solutions to Schr\"{o}dinger Systems with Fractional $p$-Laplacian

TL;DR

This paper extends the method of moving planes to fractional p-Laplacian systems, establishing symmetry, monotonicity, and nonexistence results for positive solutions under certain conditions.

Contribution

It introduces a narrow region principle and decay at infinity theorem to analyze qualitative properties of fractional p-Laplacian Schrödinger systems.

Findings

Positive solutions are radially symmetric in the unit ball and entire space.

Monotonicity of solutions in parabolic domains is established.

Nonexistence of positive solutions in half-space under certain conditions.

Abstract

In this paper, we first establish a narrow region principle and a decay at infinity theorem to extend the direct method of moving planes for general fractional $p$-Laplacian systems. By virtue of this method, we can investigate the qualitative properties of the following Schr\"{o}dinger system with fractional $p$-Laplacian \begin{equation*} \left\{\begin{array}{r@{\ \ }c@{\ \ }ll} \left(-\Delta\right)_{p}^{s}u+au^{p-1}& =&f(u,v), \\[0.05cm] \left(-\Delta\right)_{p}^{t}v+bv^{p-1}& =&g(u,v), \end{array}\right. \end{equation*} where $0<s,\,t<1$ and $2<p<\infty$. We obtain the radial symmetry in the unit ball or the whole space $\mathbb{R}^{N}(N\geq2)$, the monotonicity in the parabolic domain and the nonexistence on the half space for positive solutions to the above system under some suitable conditions on $f$ and $g$, respectively.