We study the following elliptic system concerning the fractional Laplacian operator (-Δ)siui=Hi(u1,…,um)inRn,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (- \Delta )^ {s_i} u_i = H_i ( u_1,\ldots ,u_m) \quad \text {in}\ \ \mathbb {R}^n, \end{aligned}$$\end{document}when 0<si<1\documentclass[12pt]{minimal}
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\begin{document}$$0<s_i<1$$\end{document}, ui:Rn→R\documentclass[12pt]{minimal}
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\begin{document}$$u_i: {\mathbb {R}}^n\rightarrow {\mathbb {R}}$$\end{document} and Hi\documentclass[12pt]{minimal}
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\begin{document}$$H_i$$\end{document} belongs to C1,γ(Rm)\documentclass[12pt]{minimal}
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\begin{document}$$C^{1,\gamma }(\mathbb {R}^m)$$\end{document} for γ>max(0,1-2minsi)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma > \max (0,1-2\min \left\{ s_i \right\} )$$\end{document} for 1≤i≤m\documentclass[12pt]{minimal}
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\begin{document}$$1\le i \le m$$\end{document}. The above system is called symmetric when the matrix H=(∂jHi(u1,…,um))i,j=1m\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {H}}=(\partial _j H_i(u_1,\ldots ,u_m))_{i,j=1}^m$$\end{document} is symmetric. The notion of symmetric systems seems crucial to study this system with a general nonlinearity H=(Hi)i=1m\documentclass[12pt]{minimal}
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\begin{document}$$H=(H_i)_{i=1}^m$$\end{document}. We establish De Giorgi type results for stable and H-monotone solutions of symmetric systems in lower dimensions that is either n=2\documentclass[12pt]{minimal}
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\begin{document}$$n=2$$\end{document} and 0<si<1\documentclass[12pt]{minimal}
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\begin{document}$$0<s_i<1$$\end{document} or n=3\documentclass[12pt]{minimal}
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\begin{document}$$n=3$$\end{document} and 1/2≤min{si}<1\documentclass[12pt]{minimal}
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\begin{document}$$1/2 \le \min \{s_i\}<1$$\end{document}. The case that n=3\documentclass[12pt]{minimal}
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\begin{document}$$n=3$$\end{document} and at least one of parameters si\documentclass[12pt]{minimal}
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\begin{document}$$s_i$$\end{document} belongs to (0, 1 / 2) remains open as well as the case n≥4\documentclass[12pt]{minimal}
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\begin{document}$$n \ge 4$$\end{document}. 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 ∇ui\documentclass[12pt]{minimal}
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\begin{document}$$\nabla u_i$$\end{document} and ∇uj\documentclass[12pt]{minimal}
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\begin{document}$$\nabla u_j$$\end{document} is exactly arccos|∂jHi(u)|/∂jHi(u)\documentclass[12pt]{minimal}
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\begin{document}$$\arccos \left( {|\partial _j H_i(u)|}/{\partial _j H_i(u)}\right) $$\end{document}. 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.
