NLS ground states on metric graphs with localized nonlinearities

We investigate the existence of ground states for the focusing subcritical NLS energy on metric graphs with localized nonlinearities. In particular, we find two thresholds on the measure of the region where the nonlinearity is localized that imply, respectively, existence or nonexistence of ground states. In order to obtain these results we adapt to the context of metric graphs some classical techniques from the Calculus of Variations.Comment: 18 page

De Giorgi's approach to hyperbolic Cauchy problems: the case of nonhomogeneous equations

In this paper we discuss an extension of some results obtained by E. Serra and P. Tilli, in [Serra&Tilli '12, Serra&Tilli '16], concerning an original conjecture by E. De Giorgi ([De Giorgi '96, De Giorgi '06]) on a purely minimization approach to the Cauchy problem for the defocusing nonlinear wave equation. Precisely, we show how to extend the techniques developed by Serra and Tilli for homogeneous hyperbolic nonlinear PDEs to the nonhomogeneous case, thus proving that the idea of De Giorgi yields in fact an effective approach to investigate general hyperbolic equations.Comment: 24 page

A general review on the NLS equation with point-concentrated nonlinearity

The paper presents a complete (to the best of the author's knowledge) overview on the existing literature concerning the NLS equation with point-concentrated nonlinearity. Precisely, it mainly covers the following topics: definition of the model, weak and strong local well-posedness, global well-posedness, classification and stability (orbital and asymptotic) of the standing waves, blow-up analysis and derivation from the standard NLS equation with shrinking potentials. Also some related problem is mentioned.Comment: 20 pages. Keywords: NLS equation, concentrated nonlinearity, delta potentials, well-posedness, standing waves, stability, blow-up, point-like limi

Nonlinear singular perturbations of the fractional Schr\"odinger equation in dimension one

The paper discusses nonlinear singular perturbations of delta type of the fractional Schr\"odinger equation $\imath\partial_t\psi=\left(-\triangle\right)^s\psi$, with $s\in(\frac{1}{2},1]$, in dimension one. Precisely, we investigate local and global well posedness (in a strong sense), conservations laws and existence of blow-up solutions and standing waves.Comment: 28 pages. Some minor revisions have been made with respect to the previous versio

Well-posedness of the Two-dimensional Nonlinear Schr\"odinger Equation with Concentrated Nonlinearity

We consider a two-dimensional nonlinear Schr\"odinger equation with concentrated nonlinearity. In both the focusing and defocusing case we prove local well-posedness, i.e., existence and uniqueness of the solution for short times, as well as energy and mass conservation. In addition, we prove that this implies global existence in the defocusing case, irrespective of the power of the nonlinearity, while in the focusing case blowing-up solutions may arise.Comment: 39 pages, pdfLaTex. Final version to appear in Ann. I. H. Poincar\'e - A

L2L^2-critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features

Carrying on the discussion initiated in (Dovetta-Tentarelli'18), we investigate the existence of ground states of prescribed mass for the $L^2$-critical NonLinear Schr\"odinger Equation (NLSE) on noncompact metric graphs with localized nonlinearity. Precisely, we show that the existence (or nonexistence) of ground states mainly depends on a parameter called reduced critical mass, and then we discuss how the topological and metric features of the graphs affect such a parameter, establishing some relevant differences with respect to the case of the extended nonlinearity studied by (Adami-Serra-Tilli'17). Our results rely on a thorough analysis of the optimal constant of a suitable variant of the $L^2$-critical Gagliardo-Nirenberg inequality.Comment: 22 pages, 7 figures. Keywords: metric graphs, NLS, ground states, localized nonlinearity, $L^2$-critical case. Some minor revisions have been made with respect to the previous version. Accepted for publication by Calc. Var. Partial Differential Equation

An existence result for dissipative nonhomogeneous hyperbolic equations via a minimization approach

We discuss a purely variational approach to the study of a wide class of second order nonhomogeneous dissipative hyperbolic PDEs. Precisely, we focus on the wave-like equations that present also a nonzero source term and a first-order-in-time linear term. The paper carries on the research program initiated in (Serra&Tilli'12), and developed in (Serra&Tilli'16), (Tentarelli&Tilli '18), on the De Giorgi approach to hyperbolic equations.Comment: 21 pages; keywords: minimization, nonhomogeneous hyperbolic equations, dissipation, De Giorgi conjectur

A note on the Dirac operator with Kirchoff-type vertex conditions on noncompact metric graphs

In this note we present some properties of the Dirac operator on noncompact metric graphs with Kirchoff-type vertex conditions. In particular, we discuss the specific features of the spectrum of the operator and, finally, we give some further details on the associated quadratic form (and on the form domain).Comment: 16 pages, 5 figures. Keywords: Dirac operator, metric graphs, spectral analysis, self-adjoint extensions, boundary triplets. arXiv admin note: text overlap with arXiv:1807.0693