Ground states of nonlinear Schrödinger systems with mixed couplings: the critical case

Abstract

In this paper, we consider the following k-coupled nonlinear Schrödinger systems in the critical case: $\{\begin{array}{ll}{\displaystyle -\mathrm{\Delta }{u}_i+{\lambda }_i{u}_i={\mu }_i{u}_i^3+\sum _{j=1,j\ne i}^k{\beta }_{ij}{u}_i{u}_j^2}& \mathrm{in}\mathrm{\Omega },\\ u_i⩾0\mathrm{in}\mathrm{\Omega },u_i(x)=0& \mathrm{on}\mathrm{\partial }\mathrm{\Omega },i=1,2,\cdots ,k.\end{array}$ Here, $\mathrm{\Omega }\subset {\mathbb{R}}^4$ is a smooth bounded domain, $-{\lambda }_1(\mathrm{\Omega })<{\lambda }_i<0,{\mu }_i>0$ and ${\beta }_{ij}={\beta }_{ji}\ne 0$ for every $i\ne j$, where ${\lambda }_1(\mathrm{\Omega })$ is the first eigenvalue of $-\mathrm{\Delta }$ with the Dirichlet boundary condition. Note that the nonlinearity and the coupling terms are both critical in dimension 4 (i.e. $2N/(N-2)=4$ when $N=4$). We call the couplings ${\beta }_{ij}$ are attractive if ${\beta }_{ij}>0$, while repulsive stands for ${\beta }_{ij}<0$. Under the assumption that all the couplings ${\beta }_{ij}$ are purely attractive and large enough, we first show that this critical system has a fully nontrivial ground state solution, that is, a solution $\overrightarrow{u}=(u_1,\dots ,u_k)$ has all components nontrivial, under conditions providing additional $|{\lambda }_i-{\lambda }_{i^{\prime }}|<<1,|{\beta }_{ij}-{\beta }_{i^{\prime }{j}^{\prime }}|<<1$, while ground state solution may be semitrivial ($\overrightarrow{u}$ has null components) without the above additional conditions. When the systems admit mixed couplings, i.e. there exist $(i,j)$ and $(i^{\prime },j^{\prime })$ such that ${\beta }_{ij}>0$ and ${\beta }_{i^{\prime }{j}^{\prime }}<0$, we establish the existence of least energy positive solutions. The purpose of this paper is to solve some remaining open questions on Tavares and You (Calc. Var. Partial Differential Equations 59:26, 2020).

Keywords:

• Cubic Schrödinger systems
• ground states
• mixed coupling
• variational method

AMS Subject Classifications:

• 35B09
• 35J20
• 35Q55

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by NSFC-12171265.