Localization of normalized solutions for saturable nonlinear Schrodinger equations

In this paper, we study the existence and concentration behavior of the semiclassical states with L-2-constraints for the following saturable nonlinear Schrodinger equation: -epsilon(2)Delta v + Gamma I(x) + v(2)/1 + I(x) + v(2) v = lambda v for x is an element of R-2. For a negatively large coupling constant Gamma, we show that there exists a family of normalized positive solutions (i.e., with the L-2-constraint) when epsilon is small, which concentrate around local maxima of the intensity function I(x) as epsilon -> 0. We also consider the case where I(x) may tend to -1 at infinity and the existence of multiple solutions. The proof of our results is variational and the novelty of the work lies in the development of a new truncation-type method for the construction of the desired solutions.