Normalized Solutions of Nonlinear Schrödinger Equations with Potentials and Non-autonomous Nonlinearities

Abstract

We study the existence and multiplicity of normalized solutions to the following Schrödinger equations with potentials and non-autonomous nonlinearities:

$$\begin{aligned} \left\{ \begin{aligned}&-\Delta u+V(x)u+\lambda u=f(x,u) \quad \text {in}~{\mathbb {R}}^N, \\&\int _{{\mathbb {R}}^N} |u(x)|^2\mathrm {d}x=a, \quad u\in H^1({\mathbb {R}}^N) , \end{aligned}\right. \end{aligned}$$

where $V(x)\le \lim _{|x|\rightarrow \infty } V(x){:=}V_{\infty }\in (-\infty ,+\infty ]$ and f(x, s) satisfies Berestycki–Lions type conditions with mass sub-critical growth. In the case $V_{\infty }=+\infty$, we prove that for all $a>0$, the equation has a ground state solution, and if additionally f is odd, the equation has infinitely many normalized solutions with increasing energy. While in the case $V_{\infty }<+\infty$, we prove that there exists $a_0\ge 0$ such that the ground state energy can be attained when $a>a_0$, but not when $0<a<a_0$. To this end, We develop robust arguments to show the conditional strict subadditivity of the ground state energy with respect to a. We also investigate the multiplicity of normalized radial solutions by index theory in this case. These results can be extended to other types of Schrödinger equations.