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

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.