Existence and concentration of ground states for saturable nonlinear Schrödinger equations with intensity functions in R2. (August 2018) - Wang, Xiaoming; Lin, Tai-Chia; Wang, Zhi-Qiang
Abstract: 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 R 2 : (0.1) − ε 2 Δ v + Γ I ( x ) + v 2 1 + I ( x ) + v 2 v = λ v, for x ∈ R 2 . 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 L 2 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.