A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian (Q389922)

Let \(s_1,s_2\in (0,1)\) and let \(F\in C^{1,1}_{\mathrm{loc}}(\mathbb{R}^n)\). The authors consider the following elliptic system NEWLINE\[NEWLINE\begin{cases} (-{\Delta})^{s_1}u=F_1(u,v),\\ (-{\Delta})^{s_2}v=F_2(u,v), \end{cases}NEWLINE\]NEWLINE \noindent where \(F_1,F_2\) are the derivatives of \(F\) with respect to the first and the second variable, and, for \(s\in (0,1)\), \(\displaystyle{(-\Delta)^s(u)(x)={P.V.} \int_{\mathbb{R}^2}\frac{u(x)-u(y)}{|x-y|^{2s+n}}dy}\) is the fractional Laplacian in \(\mathbb{R}^n\).NEWLINENEWLINE\noindent Using a Poisson kernel extension, the above system can be reduced to the following extension problem NEWLINE\[NEWLINE\begin{cases}\mathrm{div}(y^{1-2s_1}\nabla U)=0 \quad &\text{ on}\quad \mathbb{R}^{n}\times(0,+\infty),\\ \lim_{y\rightarrow 0^+}(-y^{1-2s_1}\partial_yU)=F_1(U,V) \quad &\text{on}\quad \mathbb{R}^{n}\times \{0\},\\\mathrm{div}(y^{1-2s_2}\nabla V)=0 \quad &\text{on}\quad \mathbb{R}^{n}\times(0,+\infty),\\ \lim_{y\rightarrow 0^+}(-y^{1-2s_2}\partial_yV)=F_2(U,V) \quad &\text{ on}\quad \mathbb{R}^{n}\times \{0\}.\end{cases}NEWLINE\]NEWLINE The authors establish some weighted Poincarè type inequalities for solutions \((U,V)\) of this problem that satisfy either a monotonicity or a stability condition. Then, these use this inequalities to prove, for \(n=2\), a symmetry result both for stable and for monotone solutions of the above system.