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The Brezis–Nirenberg type double critical problem for the Choquard equation

Author

Listed:
  • Li Cai

    (Southeast University)

  • Fubao Zhang

    (Southeast University)

Abstract

In this paper, we study the following Choquard equation $$\begin{aligned} -\Delta u=\alpha |u|^{2^*-2}u+\beta \left( I_\mu *|u|^{2_\mu ^*}\right) |u|^{2_\mu ^* -2}u +\lambda u,\quad in\,\,\Omega , \end{aligned}$$ - Δ u = α | u | 2 ∗ - 2 u + β I μ ∗ | u | 2 μ ∗ | u | 2 μ ∗ - 2 u + λ u , i n Ω , where $$\Omega$$ Ω is a bounded domain of $${\mathbb {R}}^N$$ R N with Lipschitz boundary, $$N\ge 3,$$ N ≥ 3 , $$\alpha ,\beta ,\lambda$$ α , β , λ are real parameters satisfying suitable conditions, $$2^* =\frac{2N}{N-2}$$ 2 ∗ = 2 N N - 2 is the critical exponent for the embedding of $$H_0^1 (\Omega )$$ H 0 1 ( Ω ) to $$L^p (\Omega ),$$ L p ( Ω ) , $$2_\mu ^* =\frac{2N-\mu }{N-2}$$ 2 μ ∗ = 2 N - μ N - 2 is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. Using variational methods, we show the existence of nontrivial solutions for the Choquard equation with double critical exponents.

Suggested Citation

  • Li Cai & Fubao Zhang, 2020. "The Brezis–Nirenberg type double critical problem for the Choquard equation," Partial Differential Equations and Applications, Springer, vol. 1(5), pages 1-20, October.
  • Handle: RePEc:spr:pardea:v:1:y:2020:i:5:d:10.1007_s42985-020-00032-0
    DOI: 10.1007/s42985-020-00032-0
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