# critical system involving fractional laplacian

Research paper by Maoding Zhen, Jincun He, Haoyuan Xu

Indexed on: 22 Mar '17Published on: 22 Mar '17Published in: arXiv - Mathematics - Analysis of PDEs

#### Abstract

In this paper, we discuss the existence of solutions for the following non-local critical systems: \begin{equation*} \begin{cases} (-\Delta)^{s}u= \mu_{1}|u|^{2^{\ast}-2}u+\frac{\alpha\gamma}{2^{\ast}}|u|^{\alpha-2}u|v|^{\beta} \ \ \ in \ \ R^{N}, (-\Delta)^{s}v= \mu_{2}|v|^{2^{\ast}-2}v+\frac{\beta\gamma}{2^{\ast}}|u|^{\alpha}|v|^{\beta-2}v\ in \ \ R^{N}, u,v\in D_{s}(R^{N}). \end{cases} \end{equation*} By using the Nehari manifold,\ under proper conditions, we establish the existence and nonexistence of positive least energy solution.