Critical quasi-linear Schr\"{o}dinger system with $p$-Laplacian

Abstract

In this paper, we mainly consider positive solution to the $D^{1,p}(\R^{N})$-critical quasi-linear Schr\"{o}dinger system with $p$-Laplacian: \begin{equation*}\begin{cases} -\Delta_p u = u^{\alpha}v^{\beta} \, \ \ \ \ \ \text{in}\,\ \ \R^N, \\ -\Delta_p v = u^{\beta}v^{\alpha} \,\ \ \ \ \ \text{in}\,\ \ \R^N, \end{cases}\end{equation*} where $1<p<N$, $N\geq2$, $0\leq \alpha \leq \beta,$ and $u,v\in D^{1,p}(\R^N)$. We establish regularity and the sharp estimates on asymptotic behaviors for any positive solution $(u,v)$. Then, we prove that all positive solutions are radially symmetric and strictly decreasing about some point. Furthermore, we obtain the uniqueness and complete classification of positive solutions. Our results extend the uniqueness results in \cite{LM,QS} for $p=2$ to general cases $1<p<N$.