On the uniqueness of solutions of a nonlocal elliptic system
Tóm tắt
We consider the following elliptic system with fractional Laplacian
$$\begin{aligned} -(-\Delta )^su=uv^2,\, \, -(-\Delta )^sv=vu^2,\, \, u,v>0 \ \mathrm{on}\, {\mathbb R}^n, \end{aligned}$$
where
$$s\in (0,1)$$
and
$$(-\Delta )^s$$
is the s-Lapalcian. We first prove that all positive solutions must have polynomial bound. Then we use the Almgren monotonicity formula to perform a blown-down analysis. Finally we use the method of moving planes to prove the uniqueness of the one dimensional profile, up to translation and scaling.
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