Solutions of multi-component fractional symmetric systems

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.