Positive radial solutions for a class of elliptic systems concentrating on spheres with potential decay
Bull. Korean Math. Soc. 2013 Vol. 50, No. 3, 839-865
https://doi.org/10.4134/BKMS.2013.50.3.839
Printed May 31, 2013
Paulo Cesar Carri\~{a}o, Narciso Horta Lisboa, and Olimpio Hiroshi Miyagaki
Universidade Federal de Minas Gerais, Universidade Estadual de Montes Claros, Universidade Federal de Juiz de Fora
Abstract : 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
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