Existence and asymptotic behavior of solitary waves for a weakly coupled Schrodinger system

This paper deals with the following weakly coupled nonlinear Schrodinger system {-Delta u(1) + a(1)(x)u(1) = vertical bar u(1)vertical bar(2p-2)u(1) + b vertical bar u(1)vertical bar(p-2)vertical bar u(2)vertical bar(p)u(1), x is an element of R-N, -Delta u(2) + a(2)(x)u(2) = vertical bar u(2)vertical bar(2p-2)u(2) + b vertical bar u(1)vertical bar(p-2)vertical bar u(2)vertical bar(p)u(1), x is an element of R-N, where N >= 1, b is an element of R is a coupling constant, 2p is an element of (2, 2*), 2* = 2N / (N - 2) if N >= 3 and +infinity if N = 1, 2, a1(x) and a2(x) are two positive functions. Assuming that ai(x) (i = 1, 2) satisfies some suitable conditions, by constructing creatively two types of two-dimensional mountain-pass geometries, we obtain a positive synchronized solution for vertical bar b vertical bar > 0 small and a positive segregated solution for b < 0, respectively. We also show that when 1 < p < min{2, 2*/2}, the positive solutions are not unique if b > 0 is small. The asymptotic behavior of the solutions when b -> 0 and b ->-infinity is also studied.