A radial symmetry and Liouville theorem for systems involving fractional Laplacian

Abstract

We investigate the nonnegative solutions of the system involving the fractional Laplacian:

\{\begin{array}{l}{(-\Delta )}^{\alpha }{{\displaystyle u}}_i(x)={{\displaystyle f}}_i(u),\hspace{1em}x\in {ℝ}^n,i=\mathrm{1},\mathrm{2},\mathrm{...},m,\\ u(x)=(u_{\mathrm{1}}(x),u_{\mathrm{2}}(x),\mathrm{...},u_m(x)),\end{array}

Where \mathrm{0}⟨\alpha ⟨\mathrm{1},n⟩\mathrm{2},{{\displaystyle f}}_i(u),1≤i≤m , are real-valued nonnegative functions of homogeneous degree p_i≥0 and nondecreasing with respect to the independent variables u₁, u₂, . . . , u_m. By the method of moving planes, we show that under the above conditions, all the positive solutions are radially symmetric and monotone decreasing about some point x₀ if {{\displaystyle p}}_i=(n+\mathrm{2}\alpha )/(n-\mathrm{2}\alpha ) for each 1≤i≤m; and the only nonnegative solution of this system is u ≡ 0 if \mathrm{1}⟨{{\displaystyle p}}_i⟨(n+\mathrm{2}\alpha )/(n-\mathrm{2}\alpha ) for all 1≤i≤m.

Key words： Fractional Laplacian; method of moving planes; Kelvin transform; Liouville theorem

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