Ground states for a linearly coupled system of Schrödinger equations on R N

Abstract

We study the following class of linearly coupled Schrödinger elliptic systems $$\begin{array}{c}\left\{\begin{array}{cc}-\mathrm{\Delta }\mathit{u}+{\mathit{V}}_1\mathrm{(}\mathit{x}\mathrm{)}\mathit{u}=\mathit{\mu }|\mathit{u}{|}^{\mathit{p}-2}\mathit{u}+\mathit{\lambda }\mathrm{(}\mathit{x}\mathrm{)}\mathit{v}\mathrm{,}\hfill & \mathit{x}\in {\mathbb{R}}^{\mathit{N}}\mathrm{,}\hfill \\ -\mathrm{\Delta }\mathit{v}+{\mathit{V}}_2\mathrm{(}\mathit{x}\mathrm{)}\mathit{v}=|\mathit{v}{|}^{\mathit{q}-2}\mathit{v}+\mathit{\lambda }\mathrm{(}\mathit{x}\mathrm{)}\mathit{u}\mathrm{,}\hfill & \mathit{x}\in {\mathbb{R}}^{\mathit{N}}\mathrm{,}\hfill \end{array}\right.\end{array}$$ where $\mathit{N}⩾3$ , $2\mathrm{<}\mathit{p}⩽\mathit{q}⩽2^{\ast }=2\mathit{N}\mathrm{/}\mathrm{(}\mathit{N}-2\mathrm{)}$ and $\mathit{\mu }⩾0$ . We consider nonnegative potentials periodic or asymptotically periodic which are related with the coupling term $\mathit{\lambda }\mathrm{(}\mathit{x}\mathrm{)}$ by the assumption $|\mathit{\lambda }\mathrm{(}\mathit{x}\mathrm{)}|⩽\mathit{\delta }\sqrt{{\mathit{V}}_1\mathrm{(}\mathit{x}\mathrm{)}{\mathit{V}}_2\mathrm{(}\mathit{x}\mathrm{)}}$ , for some $0\mathrm{<}\mathit{\delta }\mathrm{<}1$ . We deal with three cases: Firstly, we study the subcritical case, $2\mathrm{<}\mathit{p}⩽\mathit{q}\mathrm{<}{2}^{\ast }$ , and we prove the existence of positive ground state for all parameter $\mathit{\mu }⩾0$ . Secondly, we consider the critical case, $2\mathrm{<}\mathit{p}\mathrm{<}\mathit{q}=2^{\ast }$ , and we prove that there exists ${\mathit{\mu }}_0\mathrm{>}0$ such that the coupled system possesses positive ground state solution for all $\mathit{\mu }⩾{\mathit{\mu }}_0$ . In these cases, we use a minimization method based on Nehari manifold. Finally, we consider the case $\mathit{p}=\mathit{q}=2^{\ast }$ , and we prove that the coupled system has no positive solutions. For that matter, we use a Pohozaev identity type.