Multiple positive normalized solutions for nonlinear Schrödinger systems

Abstract

We consider the existence of multiple positive solutions to the nonlinear Schrödinger systems set onHere a 1 , a 2 > 0 are prescribed, µ 1 , µ 2 , β > 0, and the frequencies λ 1 , λ 2 are unknown and will appear as Lagrange multipliers. Two cases are studied, the first when N ≥ 1, 2 < p 1 , p 2 < 2 + 4 N , r 1 , r 2 > 1, 2 + 4 N < r 1 + r 2 < 2 * , the second when N ≥ 1, 2 + 4 N < p 1 , p 2 < 2 * , r 1 , r 2 > 1, r 1 + r 2 < 2 + 4 N . In both cases, assuming that β > 0 is sufficiently small, we prove the existence of two positive solutions. The first one is a local minimizer for which we establish the compactness of the minimizing sequences and also discuss the orbital stability of the associated standing waves. The second solution is obtained through a constrained mountain pass and a constrained linking respectively.

Stationary Waves with Prescribed $L^2$-Norm for the Planar Schrödinger--Poisson System

“…Also the second solution u − corresponds to a critical point of mountain-pass type for F on S(c). The existence of two critical points on S(c), one being a local minimizer and the second one of mountain-pass type is reminiscent of recent works [7,16,19,26] where a similar structure have been observed for prescribed norm problems.…”

Stable standing waves for cubic nonlinear Schrödinger systems with partial confinement

“…Bellazzini et al in ([9], Lemma 3.1) as well as Gou and Jeanjean in [20] proved a similar inequality as (22). Following [9,20], one may replace (22)…”

“…We would like to emphasize that the main difficulty is the lack of compactness due to the translation invariance in x 3 -direction. Indeed, on one hand, if one removes x 2 1 + x 2 2 in system (1), then one can restrict the functional I to a subspace of Σ × Σ, which consists of radially symmetric functions, see for example [5,6,20,31]; On the other hand, if one replaces x 2 1 + x 2 2 in system (1) by the harmonic potential…”

“…However, for systems with two or more components, it is difficult to check (21) for the constrained minimization problem (18). In addition, in [20], the local minimum energy level of the constrained energy functional corresponding to problem (12) is negative, this property plays a key role in rulling out dichotomy. Unfortunately, m χ c1,c2 > 0 in our paper (see Lemma 2.9).…”

Normalized solutions for nonlinear Schrödinger systems on bounded domains

“…Apart when global minimization can be applied, see [35], as far as we know the first result in the literature is due to Jeanjean [24], for the superlinear, Sobolev-subcritical NLS single equation on R N with a non-homogeneous nonlinearity. In recent years, other papers appeared, dealing with the NLS equation or system, always in the Sobolev subcritical regime, either on R N [5,21,7,9,4,22,6] or on a bounded domain [31,32,33,15,34,10]. These two settings are rather different in nature: each one requires a specific approach, and the results are in general not comparable.…”