Solutions of multi-component fractional symmetric systems

We study the following elliptic system concerning the fractional Laplacian operator $$(-\mathrm{\Delta }{)}^{s_i}{u}_i=H_i(u_1,\cdots ,u_m)\text{ in }{\mathbb{R}}^n,$$ 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$1\le i \le m$\mathrm{.}Theabovesystemiscalledsymmetricwhenthematrix$\mathcal H=(\partial_j H_i(u_1,\cdots,u_m))_{i,j=1}^m$ 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 $1\mathrm{/}2\le \min \{s_i\}<1$. The case that $n=3$ and at least one of parameters $s_i$ belongs to $(0,1\mathrm{/}2)$ 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 $\mathrm{\nabla }{u}_i$ and $\mathrm{\nabla }{u}_j$ is exactly $\arccos \left(\mathrm{\mid }{\mathrm{\partial }}_j{H}_i(u)\mathrm{\mid }\mathrm{/}{\mathrm{\partial }}_j{H}_i(u)\right)$. 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.