Existence of solutions for Choquard equations with an L¹-integrable reciprocal potential and upper critical growth

This paper is dedicated to studying the Choquard-type equation $$ \{ {array}{l} - Δ u + V(x)u = ( {{Iα} * {{ | u |}ᵖ}} ){ | u |p - 2}u + λ { | u |q - 2}u, \\ u ∈ H¹ ({RN}), {array} . − Δ u + V ( x ) u = ( I α ∗ | u | p ) | u | p − 2 u + λ | u | q − 2 u , u ∈ H 1 ( R N ) , whereN≥ 3$ N ≥ 3 , $α ∈ ({0,N} )$ α ∈ ( 0 , N ) , $1< q<2*: = {{2N}}{{N - 2}}$ 1 < q < 2 ∗ : = 2 N N − 2 (the critical Sobolev exponent), $p=N+α /N-2$ p = N + α N − 2 is the upper critical Hardy-Littlewood-Sobolev exponent and $Iα$ I α is the Riesz potential. When the reciprocal of the potential V is integrable, we establish the existence of solutions to the above problem for both sublinear and superlinear cases by means of the variational methods.