SPIKE VECTOR SOLUTIONS FOR SOME COUPLED NONLINEAR SCHRODINGER EQUATIONS

We consider spike vector solutions for the nonlinear Schrodinger system {-epsilon(2)Delta u + P(x)u = mu u(3) + beta v(2)u in R-3, -epsilon(2)Delta v + Q(x)v = nu v(3) + beta u(2)v in R-3, u, v > 0 in R-3, where epsilon > 0 is a small parameter, P(x) and Q(x) are positive potentials, mu > 0; nu > 0 are positive constants and beta not equal 0 is a coupling constant. We investigate the effect of potentials and the nonlinear coupling on the solution structure. For any positive integer k >= 2, we construct k interacting spikes concentrating near the local maximum point x 0 of P(x) and Q(x) when P(x(0)) = Q(x(0)) in the attractive case. In contrast, for any two positive integers k >= 2 and m >= 2, we construct k interacting spikes for u near the local maximum point x(0) of P(x) and m interacting spikes for v near the local maximum point (x) over bar (0) of Q(x) respectively when (x) over bar (0) not equal (x) over bar (0), moreover, spikes of u and v repel each other. Meanwhile, we prove the attractive phenomenon for beta < 0 and the repulsive phenomenon for beta > 0.