A construction of peak solutions by a local mountain pass approach for a nonlinear Schrödinger system with three wave interaction

In this paper, we consider the following nonlinear Schrödinger system with three wave interaction: (Formula presented.) where N≤5, 1<p<2∗-1, 2∗=∞(N≤2), 2∗=2N/(N-2)(N≥3), ε>0, Vj(x)>0, μj>0(j=1,2,3) and α>0. We construct a peak solution that is concentrating at a local minimum point of a function c(V1(x),V2(x),V3(x)). Here c(λ1,λ2,λ3) is a mountain pass value of the following limit system (Formula presented.) When p∈(1,2), this limit system does not necessarily have a ground state. Hence a key of the construction is to use a local mountain pass approach.