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Non‐uniqueness of positive ground states of non‐linear Schrödinger equations
Author(s) -
Dávila Juan,
del Pino Manuel,
Guerra Ignacio
Publication year - 2013
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pds038
Subject(s) - uniqueness , mathematics , schrödinger equation , mathematical physics , mathematical analysis , combinatorics , pure mathematics
Existence of a positive, decaying radial solution to the problem Δ u − u + u p + λ u q = 0 in R N , when λ>0 and 1< q < p <( N +2)/( N −2) has been known for a long time. For λ=0, it is well known that this solution is unique. While uniqueness conditions for rather general non‐linearities have been found, the issue has remained elusive for this problem. We prove that uniqueness is in general not true. We find that if N =3, 1< q <3, λ is fixed sufficiently large, and p <5 is taken sufficiently close to 5, then there are at least three positive decaying radial solutions.