Existence Of A Single Peak Solution For A Nonlinear Schrodinger System With Coupled Quadratic Nonlinearity

We study the following nonlinear Schrodinger system with coupled quadratic nonlinearity where ? is a small positive parameter,2?N<6,?>0 and ?>?.The system above arises from second-harmonic generation in quadratic optical media.We assume that the potential functions P(x)and Q(x)are positive functions and satisfy some suitable conditions.Applying the finite dimensional reduction method,we prove that if x0 ? ? is a non-degenerate critical point of ?(P + Q)on ?,then the system above has a single peak solution(V?:w?)concentrating at x0 for ?:small enough.Applying the ideas in reference[33],we extend the existence result of a single peak solution to the classical Schrodinger equation in[22]to our problem.