Strongly interacting bumps for the Schroedinger-Newton equations
- Department of Mathematics, Chinese University of Hong Kong, Shatin (Hong Kong)
- Department of Mathematical Sciences, Brunel University, Uxbridge UB8 3PH (United Kingdom)
We study concentrated bound states of the Schroedinger-Newton equations h{sup 2}{delta}{psi}-E(x){psi}+U{psi}=0, {psi}>0, x(set-membership sign)R{sup 3}; {delta}U+(1/2)|{psi}|{sup 2}=0, x(set-membership sign)R{sup 3}; {psi}(x){yields}0, U(x){yields}0 as |x|{yields}{infinity}. Moroz et al. [''An analytical approach to the Schroedinger-Newton equations,'' Nonlinearity 12, 201 (1999)] proved the existence and uniqueness of ground states of {delta}{psi}-{psi}+U{psi}=0, {psi}>0, x(set-membership sign)R{sup 3}; {delta}U+(1/2)|{psi}|{sup 2}=0, x(set-membership sign)R{sup 3}; {psi}(x){yields}0, U(x){yields}0 as |x|{yields}{infinity}. We first prove that the linearized operator around the unique ground state radial solution ({psi}{sub 0},U{sub 0}) with {psi}{sub 0}(r)=(Ae{sup -r}/r)(1+o(1)) as r=|x|{yields}{infinity}, U{sub 0}(r)=(B/r)(1+o(1)) as r=|x|{yields}{infinity} 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{sub i}(set-membership sign)R{sup 3}, i=1,2...,K, with P{sub i}{ne}P{sub j} for i{ne}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 Schroedinger-Newton equations with K bumps which concentrate at P{sub i}. We also prove that given a local maximum point P{sub 0} of E(P) there exists a solution with K bumps which all concentrate at P{sub 0} and whose distances to P{sub 0} are at least O(h{sup 1/3})
- OSTI ID:
- 21175893
- Journal Information:
- Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 1 Vol. 50; ISSN JMAPAQ; ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
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