Bulletin of the
Korean Mathematical Society BKMS

We deal with the existence of positive radial solutions concentrating on spheres for the following class of elliptic system \begin{equation} \left\{ \begin{array}{c} -\varepsilon ^{2}\Delta u+V_{1}(x)u=K(x)Q_{u}(u,v)\text{ in } \mathbb{R} ^{N}\text{,} \\ -\varepsilon ^{2}\Delta v+V_{2}(x)v=K(x)Q_{v}(u,v)\text{ in } \mathbb{R} ^{N}\text{,} \\ u,v\in W^{1,2}( \mathbb{R} ^{N})\text{, }u,v>0\text{ in } \mathbb{R} ^{N}\text{,} \end{array} \right. \tag{$S$} \end{equation} where $\varepsilon $ is a small positive parameter; $V_{1}$, $V_{2}\in C^{0}( \mathbb{R} ^{N},\left[ 0,\infty \right) )$ and $K\in C^{0}( \mathbb{R} ^{N},\left( 0,\infty \right) )$ are radially symmetric potentials; $Q$ is a $ (p+1)$-homogeneous function and $p$ is subcritical, that is, $1

Keywords: Schr\"odinger operator, radial solution, variational method, singular perturbation

MSC numbers: 35J50, 35B06, 35A15, 35B25