TY  - JOUR
ID  - Zhu2022
LB  - Zhu2022
AU  - Zhu, Gaili
AU  - Duan, Chunping
AU  - Zhang, Jianjun
AU  - Zhang, Huixing
TI  - Ground states of coupled critical Choquard equations with weighted potentials
ST  - Ground states of coupled critical Choquard equations with weighted potentials
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2022
VL  - 42
IS  - 2
SP  - 337
EP  - 354
DO  - https://doi.org/10.7494/OpMath.2022.42.2.337
UR  - https://doi.org/10.7494/OpMath.2022.42.2.337
AB  - 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.
KW  - ground states
KW  - Choquard equations
KW  - Hardy-Littlewood-Sobolev inequality
KW  - lower critical exponent
SN  - 1232-9274
ER  -