Ground state solutions for Choquard equations with Hardy potentials and critical nonlinearity

Abstract

We are concerned with discussing the ground state solutions of the Choquard equation with the Hardy potentials and critical Sobolev exponent: $\left\{\begin{array}{l}-\mathrm{\Delta }u+\left(a-\frac{\mu }{|x{|}^2}\right)u=(I_{\alpha }\ast F(u\left)\right)f\left(u\right)+|u{|}^{2^{\ast }-2}u,x\in {\mathbb{R}}^N\setminus \left\{0\right\},\\ u\in H^1\left({\mathbb{R}}^N\right),\end{array}\right.$ where $N\ge 3$, $\alpha \in (0,N)$, $0\le \mu <\overline{\mu }:=\frac{(N-2{)}^2}{4}$, $I_{\alpha }$ is the Riesz potential, $2^{\ast }:=2N/(N-2)$ is the critical Sobolev exponent, and $f\in \mathcal{C}(\mathbb{R},\mathbb{R})$ satisfies neither the usual Ambrosetti–Rabinowitz type condition nor any monotonicity condition. Using some new variational and analytic techniques, we obtain a ground state solution of Poho$\breve{z}$aev type for the given problem.

Keywords:

• Choquard equations
• Poho$\breve{z}$aev manifold
• Hardy potentials
• critical Sobolev exponent

AMS SUBJECT CLASSIFICATIONS:

• 35B33
• 47J30

Acknowledgments

We would like to thank the referees for their careful reading and helpful comments and suggestions which improve the presentation of the paper.

Disclosure statement

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

Data availability statement

The data that supports the findings of this study are available within the article [and its supplementary material].

Additional information

Funding

This work is partially supported by the National Natural Science Foundation of China [grant number 11971485].