Existence and concentration of ground states for saturable nonlinear Schrödinger equations with intensity functions in R2

In this paper, we study the existence and concentration behavior of semiclassical ground states for a class of saturable nonlinear Schrödinger equations with intensity functions in R2: (0.1)−ε2Δv+ΓI(x)+v21+I(x)+v2v=λv,forx∈R2.We show that for sufficiently negatively large coupling constant Γ and sufficiently small ε there exists a family of normalized ground states (i.e., with the L2 constraint) of the problem. We prove that the family of solutions concentrate around the maxima of the intensity function as ε→0. Our method is variational and depends upon a convexity technique together with the concentration-compactness method.