Singularly perturbed quasilinear Choquard equations with nonlinearity satisfying Berestycki–Lions assumptions

In the present paper, we consider the following singularly perturbed problem: { − ε 2 Δ u + V ( x ) u − ε 2 Δ ( u 2 ) u = ε − α ( I α ∗ G ( u ) ) g ( u ) , x ∈ R N ; u ∈ H 1 ( R N ) , where ε > 0 is a parameter, N ≥ 3 , α ∈ ( 0 , N ) , G ( t ) = ∫ 0 t g ( s ) d s , I α : R N → R is the Riesz potential, and V ∈ C ( R N , R ) with 0 < min x ∈ R N V ( x ) < lim | y | → ∞ V ( y ) . Under the general Berestycki–Lions assumptions on g , we prove that there exists a constant ε 0 > 0 determined by V and g such that for ε ∈ ( 0 , ε 0 ] the above problem admits a semiclassical ground state solution u ˆ ε with exponential decay at infinity. We also study the asymptotic behavior of { u ˆ ε } as ε → 0 .