Construction of solutions for some localized nonlinear Schrödinger equations

For an \begin{document}$N$ \end{document} -body system of linear Schrodinger equation with space dependent interaction between particles, one would expect that the corresponding one body equation, arising as a mean field approximation, would have a space dependent nonlinearity. With such motivation we consider the following model of a nonlinear reduced Schrodinger equation with space dependent nonlinearity \begin{document}$\begin{align*}-\varphi''+V(x)h'(|\varphi|^2)\varphi = λ \varphi,\end{align*}$ \end{document} where \begin{document}$V(x) = -χ_{[-1,1]} (x)$ \end{document} is minus the characteristic function of the interval \begin{document}$[-1,1]$ \end{document} and where \begin{document}$h'$ \end{document} is any continuous strictly increasing function. In this article, for any negative value of \begin{document}$λ$ \end{document} we present the construction and analysis of the infinitely many solutions of this equation, which are localized in space and hence correspond to bound-states of the associated time-dependent version of the equation.