Solutions of multi-component fractional symmetric systems

TL;DR

This paper investigates solutions to multi-component fractional elliptic systems with symmetric nonlinearities, establishing geometric and analytical properties such as gradient parallelism, Liouville theorems, and monotonicity formulas in lower dimensions.

Contribution

It introduces De Giorgi type results for stable and H-monotone solutions, and derives geometric inequalities and identities for symmetric fractional systems, extending understanding of their structure.

Findings

Gradients of solution components are parallel in lower dimensions.

Established Liouville theorems and monotonicity formulas for these systems.

Derived explicit angle relations between gradients based on system nonlinearities.

Abstract

We study the following elliptic system concerning the fractional Laplacian operator $$(- \Delta)^ {s_i} u_i = H_i ( u_1,\cdots,u_m) \ \ \text{in}\ \ \mathbb{R}^n,$$ when $0<s_i<1$, $u_i: \mathbb R^n\to R$ and $H_i$ belongs to $C^{1,\gamma}(\mathbb{R}^m)$ for $\gamma > \max(0,1-2\min \left \{s_i \right \})$ for $1\le i \le m$. The above system is called symmetric when the matrix $\mathcal H=(\partial_j H_i(u_1,\cdots,u_m))_{i,j=1}^m$ is symmetric. The notion of symmetric systems seems crucial to study this system with a general nonlinearity $H=(H_i)_{i=1}^m$. We establish De Giorgi type results for stable and $H$-monotone solutions of symmetric systems in lower dimensions that is either $n=2$ and $0<s_i<1$ or $n=3$ and $1/2 \le \min\{s_i\}<1$. The case that $n=3$ and at least one of parameters $s_i$ belongs to $(0,1/2)$ remains open as well as the case $n \ge 4$. Applying a geometric Poincar\'{e} inequality, we conclude that gradients of components of solutions are parallel in lower dimensions when the system is coupled. More precisely, we show that the angle between vectors $\nabla u_i$ and $\nabla u_j$ is exactly $\arccos\left({|\partial_j H_i(u)|}/{\partial_j H_i(u)}\right)$. In addition, we provide Hamiltonian identities, monotonicity formulae and Liouville theorems. Lastly, we apply some of our main results to a two-component nonlinear Schr\"{o}dinger system, that is a particular case of the above system, and we prove Liouville theorems and monotonicity formulae.