Existence and asymptotic behavior of normalized solutions to fractional Schrödinger equations with combined nonlinearities

Abstract

In the present paper, we consider the following fractional Schrödinger equations with combined nonlinearities

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^su+\lambda u=|u|^{q-2}u+|u|^{p-2}u\ \ \ \textrm{in}\ {\mathbb {R}}^N,\\ \int _{{\mathbb {R}}^N}u^2\textrm{d} x=a^2,\\ \end{array}\right. } \end{aligned}$$

where \(N\ge 2\), \(s\in (0,1)\), \(a>0\), \(2<q<p<2^{*}_{s}=\frac{2N}{N-2s}\), and \((-\Delta )^s\) is the fractional Laplace operator. Under various conditions on \(q<p\), \(a>0\), we investigate the existence of ground state normalized solutions by applying variational methods. Moreover, the asymptotic behavior of mountain pass type normalized solutions is also considered. We generalize the corresponding results in Qi and Zou (J Differ Equ 375:172–205, 2023), which concerns nonlinear Schrödinger equations with combined nonlinearities, to fractional nonlinear Schrödinger equations with combined nonlinearities.