SIGN-CHANGING SOLUTIONS OF COMPETITION-DIFFUSION ELLIPTIC SYSTEMS AND OPTIMAL PARTITION PROBLEMS

By Hugo Tavares, S. Terracini. 2011

91 citations recorded by the source corpus.

Abstract In this paper we prove the existence of infinitely many sign-changing solutions for the system of m Schrodinger equations with competition interactions − Δ u i + a i u i 3 + β u i ∑ j ≠ i u j 2 = λ i , β u i , u i ∈ H 0 1 ( Ω ) , i = 1 , … , m where Ω is a bounded domain, β > 0 and a i ⩾ 0 ∀i. Moreover, for a i = 0 , we show a relation between critical energies associated with this system and the optimal partition problem inf ω i ⊂ Ω open ω i ∩ ω j = ∅ ∀ i ≠ j ∑ i = 1 m λ k i ( ω i ) , where λ k i ( ω ) denotes the k i -th eigenvalue of −Δ in H 0 1 ( ω ) . In the case k i ⩽ 2 we show that the optimal partition problem appears as a limiting critical value, as the competition parameter β diverges to +∞.

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Citations in the corpus

Stable solitary waves with prescribed $L^2$-mass for the cubic Schr\"odinger system with trapping potentials (2014, background): “2) Up to our knowledge, only a few papers deal with the fixed masses approach: essentially [10, 25, 33, 29], all of which address the defocusing, competitive case.”
A natural constraint approach to normalized solutions of nonlinear Schr\"odinger equations and systems (2016, background): “Notice that, if μ1, μ2, β < 0 and Ω is bounded, the existence of a single normalized solution can be proved quite easily by minimization arguments, and indeed in [33,50] the authors are mainly interested in multiplicity results and occurrence of phaseseparation.”