Affiliation:
1. School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, P.R. China
Abstract
We prove the existence and asymptotic behavior of solutions to the following problem: − Δ u + V ( x ) u − g ( x ) u = ( I α ∗ F ( u ) ) f ( u ) , x ∈ R N ; u ∈ H 1 ( R N ) , where g ( x ) : = μ | x | is called the Coulomb potential, g ( x ) : = β | x | 2 is called the Hardy potential (the inverse-square potential). μ , β > 0 are parameters, I α : R N ⟶ R is the Riesz potential. Moreover, the nonlinearity f satisfies Berestycki–Lions type conditions which are introduced by Moroz and Van Schaftingen (Trans. Amer. Math. Soc. 367 (2015) 6557–6579). When μ ∈ ( 0 , α ( N − 2 ) / 2 ( α + 1 ) ) and β ∈ ( 0 , α ( N − 2 ) 2 / 4 ( 2 + α ) ), under some mild assumptions on V, we establish the existence and asymptotic behavior of solutions. Particularly, our results extend some relate ones in the literature.