Normalized Ground States and Multiple Solutions for Nonautonomous Fractional Schrödinger Equations

Abstract

In this paper, we consider the following fractional Schrödinger equations with prescribed $L^2$-norm constraint:

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^s u=\lambda u+h(\varepsilon x)f(u)&{}\text{ in }\ \mathbb R^N, \\ \int _{\mathbb R^N}|u|^2dx=a^2,\\ \end{array} \right. \end{aligned}$$

where $0<s<1$, $N\ge 3$, $a, \varepsilon >0$, $h\in C(\mathbb {R}^{N},\mathbb {R^+})$ and $f\in C(\mathbb {R},\mathbb {R})$. In the mass subcritical case but under general assumptions on f, we prove the multiplicity of normalized solutions to this problem. Specifically, we show that the number of normalized solutions is at least the number of global maximum points of h when $\varepsilon$ is small enough. Before that, without any restrictions on $\varepsilon$ and the number of global maximum points, the existence of normalized ground states can be determined. In this sense, by studying the relationship between $h_0:=\inf _{x\in \mathbb {R}^{N}}h(x)$ and $h_{\infty }:=\lim _{|x|\rightarrow \infty }h(x)$, we establish new results on the existence of normalized ground states for nonautonomous elliptic equations.