SPIKE SOLUTIONS IN COUPLED NONLINEAR SCHRODINGER EQUATIONS WITH ATTRACTIVE INTERACTION

We consider the following elliptic system: {epsilon(2)Delta u - lambda(1)u + mu(1)u(3) + beta uv(2) = 0 in Omega, epsilon(2)Delta u - lambda(2)u + mu(2)v(3) + beta u(2)v = 0 in Omega, u, v > 0 in Omega, u = v = 0 on partial derivative Omega, where Omega subset of R-N(N <= 3) is a smooth and bounded domain, epsilon > 0 is a small parameter, lambda(1), lambda(2), mu(1), mu(2) > 0 are positive constants and beta not equal 0 is a coupling constant. We show that there exists an interval I = [a(0), b(0)] and a sequence of numbers 0 < beta(1) < beta(2) < ... < beta(n) < ... such that for any beta is an element of (0,+infinity)\ (I boolean OR {beta(1), ..., beta(n), ...}), the above problem has a solution such that both u and v develop a spike layer at the innermost part of the domain. Central to our analysis is the nondegeneracy of radial solutions in R-N.