New type of solutions for the critical Lane-Emden system

Abstract

In this paper, we consider the critical Lane-Emden system$\{\begin{array}{cc}-\Delta u=K_1\left(y\right)v^p,y\in R^N,& \\ -\Delta v=K_2\left(y\right)u^q,y\in R^N,& \\ u,v>0,\end{array}$ where $N\ge 5$, $p,q\in (1,\infty )$ with $\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}$, $K_1\left(y\right)$ and $K_2\left(y\right)$ are positive radial potentials. Under suitable conditions on $K_1\left(y\right)$ and $K_2\left(y\right)$, we construct a new family of solutions to this system, which are centred at points lying on the top and the bottom circles of a cylinder.