Competing nonlinearities in NLS equations as source of threshold phenomena on star graphs

Adami, Riccardo; Boni, Filippo; Dovetta, Simone

doi:10.1016/j.jfa.2022.109483

Cited by 16 publications

(15 citation statements)

References 71 publications

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“…Note that, by [1, theorem 1.1], if G is a star graph then μ p,q = μ p,q for every q = p 2 + 1. However, the analysis in [1] heavily relies on the fact that on star graphs an explicit characterization of the critical points of F p,q in H 1 μ (G) is available. Since this is clearly out of reach on general non-compact graphs, to understand whether, for every graph fulfilling the hypotheses of theorem 1.1, the two thresholds μ p,q , μ p,q coincide seems to be a challenging open question.…”

Section: Resultsmentioning

confidence: 99%

“…Let q < p 2 + 1. Then (a) There exists a graph G 1 satisfying assumption (H), with at least three half-lines and at least a vertex of degree 2, so that ground states of F p,q at mass μ exist for every μ > 0; (b) There exists a graph G 2 satisfying assumption (H), with at least three half-lines and at least a vertex of degree 2, and a value m > 0 so that ground states of F p,q at mass m on G 2 do not exist.…”

Section: Resultsmentioning

confidence: 99%

“…The first step towards the investigation of the interaction between a standard nonlinearity and a pointwise term can be found in [5,6], where a linear delta potential is located at the unique vertex of a star graph. Such a model amounts to take G as a star graph and q = 2 in (1). In this context, ground states are proved to exist for small masses only, as they bifurcate from the corresponding solution of the associated linear problem.…”

Section: Introductionmentioning

confidence: 99%

“…The ground states problem (2) has already been addressed both on the real line [26] and on star graphs [1]. On the real line, in the regime (3) where both the nonlinearities are L 2subcritical ground states exist for every value of the mass (see [26, theorem 1.3] and section 2 below).…”

Section: Introductionmentioning

confidence: 99%

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Doubly nonlinear Schrödinger ground states on metric graphs

Boni

Dovetta

2022

Nonlinearity

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We investigate the existence of ground states at prescribed mass on general metric graphs with half-lines for focussing doubly nonlinear Schrödinger equations involving both a standard power nonlinearity and delta nonlinearities located at the vertices. The problem is proved to be sensitive both to the topology and to the metric of the graph and to exhibit a phenomenology richer than in the case of the sole standard nonlinearity considered by Adami et al (2015 Calc. Var. 54 743–61; 2016 J. Funct. Anal. 271 201–23). On the one hand, we identify various topological features responsible for existence/non-existence of doubly nonlinear ground states in specific mass regimes. On the other hand, we describe the role of the metric in determining the interplay between these different topological properties.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Doubly nonlinear Schrödinger ground states on metric graphs

Boni

Dovetta

2022

Nonlinearity

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“…As the literature on the subject witnessed a massive growth, we refrain from overviewing it here, redirecting e.g. to [1,3,[9][10][11][12][13][14][15][16]21,26,29,32,33,[38][39][40] for some of the most recent developments and to the reviews [2,34] for more comprehensive discussions. Within the whole theory, prominent efforts have been devoted to the analysis of existence of positive standing wave solutions, with a particular focus on ground states.…”

Section: Introductionmentioning

confidence: 99%

On the notion of ground state for nonlinear Schrödinger equations on metric graphs

Coster¹,

Dovetta²,

Galant³

et al. 2023

Preprint

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We compare ground states for the nonlinear Schrödinger equation on metric graphs, defined as global minimizers of the action functional constrained on the Nehari manifold, and least action solutions, namely minimizers of the action among all solutions to the equation. In principle, four alternative cases may take place: ground states do exist (thus coinciding with least action solutions); ground states do not exist while least action solutions do; both ground states and least action solutions do not exist and the levels of the two minimizing problems coincide; both ground states and least action solutions do not exist and the levels of the two minimizing problems are different. We show that in the context of metric graphs all four alternatives do occur. This is accomplished by a careful analysis of doubly constrained variational problems. As a by-product, we obtain new multiplicity results for positive solutions on a wide class of noncompact metric graphs.

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NLS ground states on the half-line with point interactions

Boni

Carlone

2023

Nonlinear Differ. Equ. Appl.

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We investigate the existence and the uniqueness of NLS ground states of fixed mass on the half-line in the presence of a point interaction at the origin. The nonlinearity is of power type, and the regime is either $$L^{2}$$ L 2 -subcritical or $$L^{2}$$ L 2 -critical, while the point interaction is either attractive or repulsive. In the $$L^{2}$$ L 2 -subcritical case, we prove that ground states exist for every mass value if the interaction is attractive, while ground states exist only for sufficiently large masses if the interaction is repulsive. In the latter case, if the power is less or equal to four, ground states coincide with the only bound state. If instead, the power is greater than four, then there are values of the mass for which two bound states exist, and neither of the two is a ground state, and values of the mass for which two bound states exist, and one of them is a ground state. In the $$L^{2}$$ L 2 -critical case, we prove that ground states exist for masses strictly below a critical mass value in the attractive case, while ground states never exist in the repulsive case.

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Competing nonlinearities in NLS equations as source of threshold phenomena on star graphs

Cited by 16 publications

References 71 publications

Doubly nonlinear Schrödinger ground states on metric graphs

Doubly nonlinear Schrödinger ground states on metric graphs

On the notion of ground state for nonlinear Schrödinger equations on metric graphs

NLS ground states on the half-line with point interactions

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