Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian

Abstract

In this paper, we consider the following system ofpseudo-differential nonlinear equations in $R^n$\begin{equation}\left\{\begin{array}{ll}(-\Delta)^{\alpha/2} u_i (x)= f_i( u_1(x), \cdots u_m(x)), & i=1, \cdots, m, \\u_i \geq 0 , & i=1, \cdots, m,  &nbsp  &nbsp  &nbsp  &nbsp （1）\end{array}\right.\label{b1}\end{equation}where $\alpha$ is any real number between $0$ and $2$. We obtain radial symmetry in the critical case and non-existencein the subcritical case for positive solutions. To this end, we first establish the equivalence between (1)and the corresponding integral system$$ \left\{\begin{array}{ll}u_i(x) = \int_{R^n} \frac{c_n}{|x-y|^{n-\alpha}} f_i( u_1(y), \cdots, u_m(y)), & i=1, \cdots, m, \\u_i(x) \geq 0, & i=1, \cdots, m.\end{array}\right.$$A new idea is introduced in the proof, which may hopefully beapplied to many other problems. Combining this equivalence with theexisting results on the integral system, we obtained much moregeneral results on the qualitative properties of the solutions for(1).