Opuscula Math. 42, no. 2 (2022), 337-354
https://doi.org/10.7494/OpMath.2022.42.2.337

 
Opuscula Mathematica

Ground states of coupled critical Choquard equations with weighted potentials

Gaili Zhu
Chunping Duan
Jianjun Zhang
Huixing Zhang

Abstract. In this paper, we are concerned with the following coupled Choquard type system with weighted potentials \[\begin{cases} -\Delta u+V_{1}(x)u=\mu_{1}(I_{\alpha}\!\ast\![Q(x)|u|^{\frac{N+\alpha}{N}}])Q(x)|u|^{\frac{\alpha}{N}-1}u+\beta(I_{\alpha}\!\ast\![Q(x)|v|^{\frac{N+\alpha}{N}}])Q(x)|u|^{\frac{\alpha}{N}-1}u,\\ -\Delta v+V_{2}(x)v=\mu_{2}(I_{\alpha}\!\ast\![Q(x)|v|^{\frac{N+\alpha}{N}}])Q(x)|v|^{\frac{\alpha}{N}-1}v+\beta(I_{\alpha}\!\ast\![Q(x)|u|^{\frac{N+\alpha}{N}}])Q(x)|v|^{\frac{\alpha}{N}-1}v,\\ u,v\in H^{1}(\mathbb{R}^{N}),\end{cases}\] where \(N\geq3\), \(\mu_{1},\mu_{2},\beta\gt 0\) and \(V_{1}(x)\), \(V_{2}(x)\) are nonnegative functions. Via the variational approach, one positive ground state solution of this system is obtained under some certain assumptions on \(V_{1}(x)\), \(V_{2}(x)\) and \(Q(x)\). Moreover, by using Hardy's inequality and one Pohozǎev identity, a non-existence result of non-trivial solutions is also considered.

Keywords: ground states, Choquard equations, Hardy-Littlewood-Sobolev inequality, lower critical exponent.

Mathematics Subject Classification: 35B25, 35B33, 35J61.

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  • Gaili Zhu
  • Chongqing Jiaotong University, College of Mathematica and Statistics, Chongqing 400074, China
  • Chunping Duan
  • Chongqing Jiaotong University, College of Mathematica and Statistics, Chongqing 400074, China
  • Jianjun Zhang (corresponding author)
  • Chongqing Jiaotong University, College of Mathematica and Statistics, Chongqing 400074, China
  • Huixing Zhang
  • China University of Mining and Technology, School of Mathematics, Xuzhou 221116, China
  • Communicated by Binlin Zhang.
  • Received: 2021-11-15.
  • Revised: 2021-12-21.
  • Accepted: 2021-12-29.
  • Published online: 2022-02-25.
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Cite this article as:
Gaili Zhu, Chunping Duan, Jianjun Zhang, Huixing Zhang, Ground states of coupled critical Choquard equations with weighted potentials, Opuscula Math. 42, no. 2 (2022), 337-354, https://doi.org/10.7494/OpMath.2022.42.2.337

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