Existence and concentration properties of ground state solutions for elliptic systems

ABSTRACT

This article is concerned with following Hamiltonian elliptic system with gradient term: $\begin{array}{rl}& -{\epsilon }^2\mathrm{\Delta }u+\epsilon b\cdot \mathrm{\nabla }u+u+V\left(x\right)v=Q\left(x\right)F_v(u,v),\\ & -{\epsilon }^2\mathrm{\Delta }v-\epsilon b\cdot \mathrm{\nabla }v+v+V\left(x\right)u=Q\left(x\right)F_u(u,v),\end{array}$ where ϵ is a small positive parameter, b is a constant vector, $V,Q\in C({\mathbb{R}}^N,\mathbb{R})$, V is allowed to be sign-changing and $infQ>0$. Using a much more direct approach, we study above system and prove first the existence of ground state solutions for each $\epsilon \in (0,{\epsilon }_0]$, ${\epsilon }_0>0$ is a constant which depends on N,V,Q and F. We consider further the exponential decay and concentration phenomena of ground states for the system. Particularly, we show that these ground states concentrate around a global minimum point of the linear potential V or a global maximum point of the nonlinear potential Q as $\epsilon \to 0$.

COMMUNICATED BY:

• D. Repov$\breve{s}$

KEYWORDS:

• Hamiltonian elliptic system
• strongly indefinite functionals
• ground states
• exponential decay
• concentration properties

AMS SUBJECT CLASSIFICATIONS:

• 35J50
• 35E05
• 58E05

Acknowledgements

Part of this work was completed during a visit by D. Qin to the Department of Mathematical Sciences of UNLV and Chern Institute of Mathematics. He would like to thank the members of the DMS at UNLV and the CIM for their invitation and hospitality.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (NNSF) [grant numbers 11571370, 11801574, 11501190], Central South University Innovation-Driven Project for Young Scholars [grant number 2019CX022] and Chern Institute of Mathematics Visiting Scholar Project.