A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system

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Multipulse Phases in k-Mixtures of Bose–Einstein Condensates (2008, background): “In the recent literature, systems of type (1) have been the object of an intensive research also in different ranges of the interaction parameters, for their possible applications to a number of other physical models, such as the study of incoherent solutionsinnon-linearoptics.Wereferthereadertother”
Infinitely many positive solutions for nonlinear equations with non-symmetric potentials (2012, background): “Phase separation has been proved in several cases with constant potentials such as in the work [5, 14, 22, 34, 43, 44] as the coupling constant β tends to negative infinity.”
Ground States and Bound States of a Nonlinear Schrödinger System (2010, background): “Schrödinger systems, nonlinear couplings, co-existing ground states, multiple bound states”
Segregated and Synchronized Vector Solutions for Nonlinear Schrödinger Systems (2013, methodology): “In [9,30] the minimax method is used to give infinitely many radial positive solutions for 2-systems (see also [26] for generalizations to the N -systems).”
A priori estimates and existence for quasi-linear elliptic equations (2008, background): “In a recent paper [ 2 ], the authors obtained a Liouville non-existence theorem for bounded non-negative solutions of a semi-linear elliptic two-system.”