EXISTENCE OF NONTRIVIAL SOLUTIONS TO SCHRODINGER SYSTEMS WITH LINEAR AND NONLINEAR COUPLINGS VIA MORSE THEORY

In this paper, we use Morse theory to study existence of non-trivial solutions to the following Schrodinger system with linear and non-linear couplings which arises from Bose-Einstein condensates: {Delta u + lambda(1)u + kappa v = mu(1)u(3) + beta uv(2) in Omega, -Delta v + lambda(1)v + kappa u = mu(1)v(3) + beta vu(2) in Omega, u = v = 0 on partial derivative Omega, where Omega is a bounded smooth domain in R-N (N = 2, 3), lambda(1), lambda(2), mu(1), mu(2) is an element of R \ {0}, beta, kappa is an element of R. In two cases of kappa = 0 and kappa not equal 0, by transferring an eigenvalue problem into an algebraic problem, we compute the Morse index and critical groups of the trivial solution. Furthermore, even when the trivial solution is degenerate, we show a local linking structure of energy functional at zero within a suitable parameter range and then get critical groups of the trivial solution. As an application, we use Morse theory to get an existence theorem on existence of nontrivial solutions under some conditions.