Symmetry of finite energy solutions to critical p-Laplacian systems in RN

Abstract

In this paper, we are concerned with the symmetry of finite energy solutions $(u,v)$ to the following critical p-Laplacian system:$\{\begin{array}{cc}-{\Delta }_{p}u=v^m{u}^r& \text{in}{R}^N,\\ -{\Delta }_{p}v=u^q{v}^s& \text{in}{R}^N,\\ u>0,v>0& \text{in}{R}^N,\end{array}$ where $1<p<N,N\ge 2$, $m,q>\max \{1,p-1\},r,s\ge 0$ satisfy $m-s\ge q-r>-p+1$ and $m+r+1=q+s+1=p^{⁎}:=\frac{pN}{N-p}$. Using decay estimates of the solutions at infinity obtained in [34, Theorem 1.3], we apply the moving planes method to prove that u and v are both radial and radially decreasing about some point $x_0\in R^N$.