Multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent

This paper deals with the following system linearly coupled by nonlinear elliptic equations {-Delta u + lambda(1)u = vertical bar u vertical bar(2)*(-2)u + beta v, x is an element of Omega, -Delta u + lambda(2)u = vertical bar v vertical bar(2)*(-2)v + beta u, x is an element of Omega u = v = 0 on partial derivative Omega. Here Omega is a smooth bounded domain in R-N(N >= 3), lambda(1), lambda(2) > -lambda(1)(Omega) are constants, lambda(1)(Omega) is the first eigenvalue of (-Delta, H-0(1) (Omega)), 2* = 2N/N-2 is the Sobolev critical exponent and beta is an element of R is a coupling parameter. By variational method, we prove that this system has a positive ground state solution for some beta > 0. Via a perturbation argument, we show that this system also admits a positive higher energy solution when vertical bar beta vertical bar is small. Moreover, the asymptotic behaviors of the positive ground state and higher energy solutions as beta -> 0 are analyzed. (C) 2017 Elsevier Inc. All rights reserved.