Strongly interacting bumps for the Schrödinger–Newton equations 论文

摘要

We study concentrated bound states of the Schrödinger–Newton equations h2Δψ−E(x)ψ+Uψ=0, ψ>0, x∊R3; ΔU+12|ψ|2=0, x∊R3; 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 Δψ−ψ+Uψ=0, ψ>0, x∊R3; ΔU+12|ψ|2=0, x∊R3; ψ(x)→0, U(x)→0 as |x|→∞. We first prove that the linearized operator around the unique ground state radial solution (ψ0,U0) with ψ0(r)=(Ae−r/r)(1+o(1)) as r=|x|→∞, U0(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 Pi∊R3, i=1,2…,K, with Pi≠Pj 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 Pi. We also prove that given a local maximum point P0 of E(P) there exists a solution with K bumps which all concentrate at P0 and whose distances to P0 are at least O(h1/3).