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,μ>0 and μ>γ.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 ∈ Γ is a non-degenerate critical point of Δ(P + Q)on Γ,then the system above has a single peak solution(Vε:wε)concentrating at x0 for ε: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.