Bound state positive solutions for a Hartree system with nonlinear couplings

Abstract

In this paper, we are interested in the following Hartree system with nonlinear couplings: $\{\begin{array}{rl}& -{\epsilon }^2\mathrm{\Delta }u+V_1\left(x\right)u\\ & ={\displaystyle \frac{1}{{\epsilon }^{N-\mu }}}[{\nu }_1\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|u{|}^p}{|x-y{|}^{\mu }}}\mathrm{d}y}\right)|u{|}^{p-2}u\\ & +\beta \left({\displaystyle {\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|v{|}^q}{|x-y{|}^{\mu }}}\mathrm{d}y}\right)|u{|}^{q-2}u],\\ & -{\epsilon }^2\mathrm{\Delta v}+V_2\left(x\right)v\\ & ={\displaystyle \frac{1}{{\epsilon }^{N-\mu }}}[{\nu }_2\left({\displaystyle {\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|v{|}^p}{|x-y{|}^{\mu }}}\mathrm{d}y}\right)|v{|}^{p-2}v\\ & +\beta \left({\displaystyle {\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|u{|}^q}{|x-y{|}^{\mu }}}\mathrm{d}y}\right)|v{|}^{q-2}v],\\ & u,v\in H^1({\mathbb{R}}^N),u,v>0\mathrm{in}{\mathbb{R}}^N,\end{array}$where $N\ge 3$, $0<\mu <N$, $\frac{2N-\mu }{N}\le p\le \frac{2N-\mu }{N-2}$, $\frac{2N-\mu }{N}\le q\le min\{p,2\}$, ${\nu }_1,{\nu }_2>0$, ε is a small parameter and $\beta <0$ is a coupling constant, and the potentials $V_1$ and $V_2$ have $k_1$ and $k_2$ isolated global minimum points, respectively. Using the Nehari manifold technique, the energy estimate method and the Lusternik–Schnirelmann theory, we find an interesting phenomenon that the problem possesses $k_1{k}_2$ positive solutions when $V_1$ and $V_2$ do not have any common isolated global minimum points, and $k_1{k}_2+m$ positive solutions when $V_1$ and $V_2$ have m common isolated global minimum points. Furthermore, the existence and nonexistence of the least energy positive solutions are also explored.

Keywords:

• Hartree system
• positive solutions
• Nehari manifold
• Lusternik-Schnirelmann theory
• variational methods

2010 Mathematics Subject Classifications:

• Primary: 35J50
• 35A01
• Secondary:35J47

Disclosure statement

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

Additional information

Funding

G. Che was supported by the National Natural Science Foundation of China [grant number 12001114] and the Natural Science Foundation of Guangdong Province [grant number 2023A1515010755]. Y. Su was supported by the National Natural Science Foundation of China [grant number 12101006]. T. F. Wu was supported by the Ministry of Science and Technology, Taiwan [grant number 112-2115-M-390-001-MY3].