On nonhomogeneous Choquard equation with Hardy–Littlewood–Sobolev critical nonlinearity

In this paper, we investigate the existence of nontrivial solutions to the following critical nonhomogeneous Choquard equation: $$ \{ {aligned} -Δ u&=λ u +∫ Ω (\ {|u(y)|^{2*α}}{|x-y|α}dy )\ |u|^{2*α-2}u + f(x) in Ω ,\\ u ∈ &\, H¹₀(Ω ), {aligned} . $$ where $N≥ 4$, $λ ∈ R$, $0<α <N$ and $2*α=2N-α /N-2$ is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. By employing an abstract critical point theorem, we establish the existence of two distinct nontrivial solutions to the above problem when λ ≥ λ ₁. Our results extend results in the literature for 0<λ <λ ₁.