Normalized solutions for a Schrödinger system with critical Sobolev growth in R3$\mathbb {R}^3$

Mei‐Qi Liu and Wenming Zou

Mathematische Nachrichten, 2024, vol. 297, issue 5, 1694-1711

Abstract: We study the following critical Schrödinger system in R3$\mathbb {R}^3$: −Δu+λ1u=|u|4u+μ1|u|p−2u+αν|u|α−2u|v|β,−Δv+λ2v=|v|4v+μ2|v|p−2v+βν|u|α|v|β−2v,∫R3u2dx=a2and∫R3v2dx=b2,u,v∈H1(R3),$$\begin{equation*} {\begin{cases} -\Delta u+\lambda _1 u=|u|^{4}u+\mu _1|u|^{p-2}u+\alpha \nu|u|^{\alpha -2}u|v|^{\beta},\\ -\Delta v+\lambda _2 v=|v|^{4}v+\mu _2|v|^{p-2}v+\beta \nu|u|^{\alpha}|v|^{\beta -2}v,\\ \int _{\mathbb {R}^3} u^2dx=a^2\;\; \hbox{and}\;\; \int _{\mathbb {R}^3} v^2dx=b^2,\;\;u,v\in H^1(\mathbb {R}^3), \end{cases}} \end{equation*}$$where α,β>1,α+β=2∗=6$\alpha,\beta >1, \alpha +\beta =2^*=6$, p∈(2,6)$p\in (2,6)$, ν>0$\nu >0$, and μ1,μ2,λ1,λ2∈R$\mu _1, \mu _2, \lambda _1, \lambda _2\in \mathbb {R}$. Any (u,v)$(u,v)$ solving such system (for some λ1,λ2$\lambda _1,\lambda _2$) is called the normalized solution in the literature, where the normalization is settled in L2(R3)$L^2(\mathbb {R}^3)$. We show that this system has a positive ground state for p∈(2,103)$ p\in (2,\frac{10}{3})$ in the case of μ1,μ2>0$\mu _1,\mu _2>0$. For the case of 2

Date: 2024
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