Nehari‐type ground state solutions for a Choquard equation with lower critical exponent and local nonlinear perturbation

This paper deals with the following Choquard equation with a local nonlinear perturbation:− Δ u + u =I α ∗ | u |α N + 1| u |α N − 1 u + λ | u | p − 2 u , x ∈ R N ;u ∈ H 1 ( R N ) ,where N  ≥ 1, α ∈(0, N ), λ >0, 2< p <2 ∗ , andI α : R N → R is the Riesz potential. The exponentα N + 1 is critical with respect to the Hardy‐Littlewood‐Sobolev inequality. In the cases when 2 < p < 4 N + 2 , p = 4 N + 2 , and4 N + 2 < p < 2 ∗ , respectively, we prove the above equation admits a Nehari‐type ground state solution if λ > λ ∗ for some given number λ ∗ .