Localization of normalized solutions for saturable nonlinear Schrödinger equations

In this paper, we study the existence and concentration behavior of the semiclassical states with L2-constraints for the following saturable nonlinear Schrödinger equation: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\varepsilon^{2}\Delta v+\Gamma{{I(x)+v^{2}}\over{1+I(x)+v^{2}}}v=\lambda v\;\;\;\;\;\text{for}\;x\in\mathbb{R}^{2}.$$\end{document} For a negatively large coupling constant Γ, we show that there exists a family of normalized positive solutions (i.e., with the L2-constraint) when ε is small, which concentrate around local maxima of the intensity function I(x) as ε → 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.