Existence of normalized solutions to nonlinear Schr\"{o}dinger equations with potential on lattice graphs

Abstract

We study the existence of ground state normalized solution of the following Schr\"{o}dinger equation: \begin{equation*} \begin{cases} -\Delta u+V(x)u+\lambda u=f(x,u), & x\in\mathbb{Z}^d \\ \Vert u\Vert_2^2=a \end{cases} \end{equation*} where $V(x)$ is trapping potential or well potential, $f(x,u)$ satisfies Berestycki-Lions type condition and other suitable conditions. We show that there always exists a threshold $\alpha\in[0,\infty)$ such that there do not exist ground state normalized solutions for $a\in (0,\alpha)$, and there exists a ground state normalized solution for $a\in(\alpha,\infty)$. Furthermore, we prove sufficient conditions for the positivity of $\alpha$ that $\alpha=0$ if $f(x,u)$ is mass-subcritical near 0, and $\alpha>0$ if $f(x,u)$ is mass-critical or mass-supercritical near 0.