Saddle solutions for the critical Choquard equation

Abstract

We study the saddle type nodal solutions for the Choquard equation

$$\begin{aligned} -\Delta u = (I_\alpha *|u|^{\frac{N+\alpha }{N-2}} )|u|^{\frac{N+\alpha }{N-2}-2}u \quad \text { in }\;{\mathbb {R}}^N, \end{aligned}$$

where \(N\ge 3\), \(I_\alpha \) is the Riesz potential of order \(\alpha \in (0, N)\) and \(\frac{N+\alpha }{N-2}\) refers to the upper critical exponent with respect to the Hardy–Littlewood–Sobolev inequality. By introducing the symmetric groups of Coxeter, we give a unified framework to construct saddle solutions with prescribed symmetric nodal structures. To overcome the difficulties arising from the lack of compactness, we give a fine analyzes on the profile decompositions of the symmetric Palais–Smale sequence. These results further feature the nonlocal nature of the Choquard equation, in contrast, the counterpart Yamabe equation \(-\Delta u=|u|^{\frac{4}{N-2}}u\) can not permit such type of nodal solutions.

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Acknowledgements

The authors are grateful to Professor Zhi-Qiang Wang for his useful and illuminating discussions which motivate the paper. This paper is partially supported by NSFC 11771324,11811540026, 11901456 and 11901582. J. Xia is also supported by the Natural Science Foundation of Shaanxi Province (Grant No. 2020JQ–120).

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Xia, J., Zhang, X. Saddle solutions for the critical Choquard equation. Calc. Var. 60, 53 (2021). https://doi.org/10.1007/s00526-021-01919-5

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Mathematics Subject Classification

  • 35A15
  • 35B06
  • 35B33
  • 35J20
  • 35J60