For an $N$-body system of linear Schrödinger 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 Schrödinger equation with space dependent nonlinearity
$\begin{align*}-\varphi''+V(x)h'(|\varphi|^2)\varphi = λ \varphi,\end{align*}$
where $V(x) = -χ_{[-1,1]} (x)$ is minus the characteristic function of the interval $[-1,1]$ and where $h'$ is any continuous strictly increasing function. In this article, for any negative value of $λ$ 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.
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