We study existence and phase separation, and the relation between these two aspects, of positive bound states for the nonlinear elliptic system (Formula presented.).This system arises when searching for solitary waves for the Gross–Pitaevskii equations. We focus on the case of simultaneous cooperation and competition, that is, we assume that there exist two pairs (i1,j1) and (i2,j2) such that i1≠j1, i2≠j2, βi1j1>0 and βi2,j2<0. Our first main results establishes the existence of solutions with at least m positive components for every m≤d; any such solution is a minimizer of the energy functional J restricted on a Nehari-type manifoldN. At a later stage, by means of level estimates on the constrained second differential of J on N, we show that, under some additional assumptions, any minimizer of J on N has all nontrivial components. In order to prove this second result, we analyse the phase separation phenomena which involve solutions of the system in a not completely competitive framework.

On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition

SOAVE, NICOLA
2015

Abstract

We study existence and phase separation, and the relation between these two aspects, of positive bound states for the nonlinear elliptic system (Formula presented.).This system arises when searching for solitary waves for the Gross–Pitaevskii equations. We focus on the case of simultaneous cooperation and competition, that is, we assume that there exist two pairs (i1,j1) and (i2,j2) such that i1≠j1, i2≠j2, βi1j1>0 and βi2,j2<0. Our first main results establishes the existence of solutions with at least m positive components for every m≤d; any such solution is a minimizer of the energy functional J restricted on a Nehari-type manifoldN. At a later stage, by means of level estimates on the constrained second differential of J on N, we show that, under some additional assumptions, any minimizer of J on N has all nontrivial components. In order to prove this second result, we analyse the phase separation phenomena which involve solutions of the system in a not completely competitive framework.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
35B40; 35J50; 35Q55; Analysis; Applied Mathematics
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11311/1010711
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