Normalized solutions to nonlocal Schrödinger systems with $L^2$-subcritical and supercritical nonlinearities

Abstract

We consider the following Schrödinger system with nonlocal Kirchhoff terms:

$$\begin{aligned} \left\{ \begin{array}{ll} - \left( {a + b\int _{{\mathbb {R}^N}} {|\nabla {u_1}{|^2}\textrm{d}x} } \right) \Delta {u_1} = {\lambda _1}{u_1} + {\mu _1}|{u_1}{|^{{p_1} - 2}}{u_1}+ \beta {r_1}|{u_1}{|^{{r_1} - 2}}{u_1}|{u_2}{|^{{r_2}}},\\ - \left( {a + b\int _{{\mathbb {R}^N}} {|\nabla {u_2}{|^2}\textrm{d}x} } \right) \Delta {u_2} = {\lambda _2}{u_2} + {\mu _2}|{u_2}{|^{{p_2} - 2}}{u_2} + \beta {r_2}|{u_1}{|^{{r_1}}}|{u_2}{|^{{r_2} - 2}}{u_2}, \end{array}\right. \end{aligned}$$

satisfying the normalization constraint

$$\begin{aligned} \int _{{\mathbb {R}^N}} {|{u_1}{|^2}\textrm{d}x} = {c_1},~\int _{{\mathbb {R}^N}} {|{u_2}{|^2}\textrm{d}x} = {c_2}. \end{aligned}$$

When $2 + \frac{8}{N}< {r_1} + {r_2} < {2^*}$ and $(p_1,p_2)$ belongs to a certain domain in ${\mathbb {R}^2}$, we prove the existence and multiplicity of positive radial vector solutions via variational method and constraint minimization argument, and our main results may be illustrated by the red areas and green areas shown in Fig. 1. Our work complements some related works and also extends some classical results [such as Bartsch and Jeanjean (Proc R Soc Edinb Sect A 148:225–242, 2018), Gou and Jeanjean (Nonlinearity 31:2319–2345, 2018)].