Normalized solutions for nonlinear Schrödinger systems with critical exponents

Abstract

In this paper, we consider the following nonlocal Schrödinger system−a+b∫R3|∇u1|2dxΔu1=λ1u1+μ1|u1|p1−2u1+βr1|u1|r1−2u1|u2|r2,−a+b∫R3|∇u2|2dxΔu2=λ2u2+μ2|u2|p2−2u2+βr2|u1|r1|u2|r2−2u2,∫R3|u1|2dx=c1,∫R3|u2|2dx=c2. $$\begin{cases}-\left(a+b{\int }_{{\mathbb{R}}^{3}}\vert \nabla {u}_{1}{\vert }^{2}\mathrm{d}x\right){\Delta}{u}_{1}={\lambda }_{1}{u}_{1}+{\mu }_{1}\vert {u}_{1}{\vert }^{{p}_{1}-2}{u}_{1}+\beta {r}_{1}\vert {u}_{1}{\vert }^{{r}_{1}-2}{u}_{1}\vert {u}_{2}{\vert }^{{r}_{2}},\quad \hfill \\ -\left(a+b{\int }_{{\mathbb{R}}^{3}}\vert \nabla {u}_{2}{\vert }^{2}\mathrm{d}x\right){\Delta}{u}_{2}={\lambda }_{2}{u}_{2}+{\mu }_{2}\vert {u}_{2}{\vert }^{{p}_{2}-2}{u}_{2}+\beta {r}_{2}\vert {u}_{1}{\vert }^{{r}_{1}}\vert {u}_{2}{\vert }^{{r}_{2}-2}{u}_{2},\quad \hfill \\ {\int }_{{\mathbb{R}}^{3}}\vert {u}_{1}{\vert }^{2}\mathrm{d}x={c}_{1}, {\int }_{{\mathbb{R}}^{3}}\vert {u}_{2}{\vert }^{2}\mathrm{d}x={c}_{2}.\quad \hfill \end{cases}$$ In the case of b = 0, 2 0, p 1 = p 2 = 2*, 2 0 respectively. Here we focus on these unknown case. We investigate the existence of positive normalized solutions under different assumptions on β > 0. Figure 1, Tables 1 and 2 will illustrate our main results and the relationship between our work and some related works in the literature.

Keywords

• schrödinger system
• normalized solution
• critical exponent
• monotonic technique
• 35j60
• 35a15
• 35b38