Existence of normalized solutions to nonlinear Schrödinger equations with potential on lattice graphs

Abstract

We study the existence of ground state normalized solution of the following Schrödinger equation: \begin{equation*} \begin{cases} -Δu+V(x)u+λu=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 $\hat{I}\pm \in [0,\infty )$ such that there do not exist ground state normalized solutions for $a\in (0,\hat{I}\pm )$, and there exists a ground state normalized solution for $a\in (\hat{I}\pm ,\infty )$. Furthermore, we prove sufficient conditions for the positivity of $\hat{I}\pm$ that $\hat{I}\pm =0$ if $f(x,u)$ is mass-subcritical near 0, and $\hat{I}\pm >0$ if $f(x,u)$ is mass-critical or mass-supercritical near 0.