Strongly interacting bumps for the Schrödinger–Newton equations

We study concentrated bound states of the Schrödinger–Newton equations h 2 Δ ψ − E ( x ) ψ + U ψ = 0 , ψ > 0 , x ∊ R 3 ; Δ U + 1 2 | ψ | 2 = 0 , x ∊ R 3 ; ψ ( x ) → 0 , U ( x ) → 0 as | x | → ∞ . Moroz et al. [“An analytical approach to the Schrödinger-Newton equations,” Nonlinearity 12, 201 (1999)] proved the existence and uniqueness of ground states of Δ ψ − ψ + U ψ = 0 , ψ > 0 , x ∊ R 3 ; Δ U + 1 2 | ψ | 2 = 0 , x ∊ R 3 ; ψ ( x ) → 0 , U ( x ) → 0 as | x | → ∞ . We first prove that the linearized operator around the unique ground state radial solution ( ψ 0 , U 0 ) with ψ 0 ( r ) = ( A e − r / r ) ( 1 + o ( 1 ) ) as r = | x | → ∞ , U 0 ( r ) = ( B / r ) ( 1 + o ( 1 ) ) as r = | x | → ∞ for some A , B > 0 has a kernel whose dimension is exactly 3 (corresponding to the translational modes). Using this result we further show that if for some positive integer K the points P i ∊ R 3 , i = 1 , 2 … , K , with P i ≠ P j for i ≠ j are all local minimum or local maximum or nondegenerate critical points of E ( P ) , then for h small enough there exist solutions of the Schrödinger–Newton equations with K bumps which concentrate at P i . We also prove that given a local maximum point P 0 of E ( P ) there exists a solution with K bumps which all concentrate at P 0 and whose distances to P 0 are at least O ( h 1 / 3 ) .