Non-uniqueness of positive ground states of non-linear Schrödinger equations

Juan Dávila, Manuel Del Pino, Ignacio Guerra

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)

Abstract

Existence of a positive, decaying radial solution to the problem 훥푢−푢+푢푝+휆푢푞=0    in ℝ푁, 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.
Original languageEnglish
Pages (from-to)318-344
JournalProceedings of the London Mathematical Society
Volume106
Issue number2
DOIs
Publication statusPublished - 1 Feb 2013

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