Solutions of multi-component fractional symmetric systems

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 Poincaré 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 Schrödinger system, that is a particular case of the above system, and we prove Liouville theorems and monotonicity formulae.

arXiv comment22 pages. Some Improvements for the version. Comments are welcome