Positive solutions for a class of elliptic systems with singular potentials

Abstract

We deal with the existence of positive solutions for the following class of elliptic system $$\left\{\begin{array}{lll}-\varepsilon^{2}\Delta u+V_{1}(x)u = K(x)Q_{u}(u, v) & {\rm in} \, \mathbb{R}^{N},\\ -\varepsilon^{2}\Delta v+V_{2}(x)v = K(x)Q_{v}(u, v) & {\rm in}\,\mathbb{R}^{N},\\ u,v \in W^{1,2}(\mathbb{R}^{N}),\quad u,v > 0 & {\rm in}\,\mathbb{R}^{N},\\ {\rm \lim}_{|x|\rightarrow \infty}u(x) = {\rm \lim}_{|x|\rightarrow \infty}v(x)=0,\end{array}\right.\quad\quad\quad{(S)}$$ where $${\varepsilon}$$ is a small positive parameter, V 1, V 2, K are nonnegative potentials, Q is a (p + 1)-homogeneous function and p is subcritical; that is, 1 < p < 2* − 1, where 2* = 2N/(N − 2) is the critical Sobolev exponent for $${N\geq 3}$$ .