Bound state positive solutions for a Hartree system with nonlinear couplings

Abstract

In this paper, we are interested in the following Hartree system with nonlinear couplings: { − ε 2 Δ u + V 1 ( x ) u = 1 ε N − μ [ ν 1 ( ∫ R N | u | p | x − y | μ d y ) | u | p − 2 u + β ( ∫ R N | v | q | x − y | μ d y ) | u | q − 2 u ] , − ε 2 Δv + V 2 ( x ) v = 1 ε N − μ [ ν 2 ( ∫ R N | v | p | x − y | μ d y ) | v | p − 2 v + β ( ∫ R N | u | q | x − y | μ d y ) | v | q − 2 v ] , u , v ∈ H 1 ( R N ) , u , v > 0 in R N , where N ≥ 3 , 0 < μ < N , 2 N − μ N ≤ p ≤ 2 N − μ N − 2 , 2 N − μ N ≤ q ≤ min { p , 2 } , ν 1 , ν 2 > 0 , ε is a small parameter and β < 0 is a coupling constant, and the potentials V 1 and V 2 have k 1 and k 2 isolated global minimum points, respectively. Using the Nehari manifold technique, the energy estimate method and the Lusternik–Schnirelmann theory, we find an interesting phenomenon that the problem possesses k 1 k 2 positive solutions when V 1 and V 2 do not have any common isolated global minimum points, and k 1 k 2 + m positive solutions when V 1 and V 2 have m common isolated global minimum points. Furthermore, the existence and nonexistence of the least energy positive solutions are also explored.