arXiv subjects

Filippo Boni

Publications and source records attributed to Filippo Boni.

17 recordsLinked to original sources

arXiv2026

Point interactions and singular solutions to semilinear elliptic equations

We investigate the connection between semilinear elliptic PDEs with isolated singularities and stationary nonlinear Schr\"odinger equations with point interactions. In dimensions $d=2,3$, we establish a rigorous correspondence between their solutions, revealing two regimes depending on whether a boundary condition at the singularity can be imposed. This connection enables us to exploit operator-theoretic and variational methods that have not previously been applied to the study of isolated singularities. In the source regime, we prove the existence of infinitely many radial singular solutions, by applying the symmetric mountain pass theorem of Ambrosetti and Rabinowitz to the action functional associated with the point interaction. When $d=2$, a suitable uniqueness result allows us to characterize singular ground states (positive solutions) as action minimizers and to prove the existence of infinitely many nodal singular solutions.

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arXiv2025

Doubly nonlinear Schr\"odinger normalized ground states on 2D grids: existence results and singular limits

We investigate the existence and the singular limit of normalized ground states for focusing doubly nonlinear Schr\"odinger equations with both standard and concentrated nonlinearities on two-dimensional square grids. First, we provide existence and non-existence results for such ground states depending on the values of the nonlinearity powers and on the structure of the set of vertices where the concentrated nonlinearities are located. Second, we prove that suitable piecewise-affine extensions of such states converge strongly in $H^1(\R^2)$ to ground states of corresponding doubly nonlinear models defined on the whole plane as the length of the edges in the grid tends to zero. This convergence is proved both for limit models with standard nonlinearities only and for models combining standard and singular nonlinearities concentrated on a line or on a strip.

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arXiv2025

Trapping by repulsion: the NLS with a delta-prime

We establish the existence and provide explicit expressions for the stationary states of the one-dimensional Schr\"odinger equation with a repulsive delta-prime potential and a focusing nonlinearity of power type. Furthermore, we prove that, if the nonlinearity is subcritical, then ground states exist for any strength of the delta-prime interaction and for every positive value of the mass. This result supplies an example of ground states arising from a repulsive potential, a counterintuitive phenomenon explained by the emergence of an additional dimension in the energy space, induced by the delta-prime interaction. This new dimension contains states of lower energy and is thus responsible for the existence of nonlinear ground states that do not originate from linear eigenfunctions. The explicit form of the ground states is derived by addressing the ancillary problem of minimizing the action functional on the Nehari manifold. We solve such problem in the subcritical, critical, and supercritical regimes.

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arXiv2025

Normalized solutions of one-dimensional defocusing NLS equations with nonlinear point interactions

We investigate normalized solutions for doubly nonlinear Schr\"odinger equations on the real line with a defocusing standard nonlinearity and a focusing nonlinear point interaction of $\delta$-type at the origin. We provide a complete characterization of existence and uniqueness for normalized solutions and for energy ground states for every value of the nonlinearity powers. We show that the interplay between a defocusing standard and a focusing point nonlinearity gives rise to new phenomena with respect to those observed with single nonlinearities, standard combined nonlinearities, and combined focusing standard and pointwise nonlinearities.

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arXiv2025

An explicitly solvable NLS model with discontinuous standing waves

We study the NLS Equation on the line with a point interaction given by the superposition of an attractive delta potential with a dipole interaction, in the cases of $L^2$-subcritical and $L^2$-critical nonlinearity. For a subcritical nonlinearity we prove the existence and the uniqueness of Ground States at any mass. If the mass exceeds an explicit threshold, then there exists a positive excited state too. For the critical nonlinearity we prove that Ground States exist only in a specific interval of masses, while in a different interval excited states exist. We provide the value of the optimal constant in the Gagliardo-Nirenberg estimate and describe in the dipole case the branches of the stationary states as the strength of the interaction varies. Since all stationary states are explicitly computed, ours is a solvable model involving a non-standard interplay of a nonlinearity with a point interaction.

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arXiv2024

Normalized NLS ground states on a double plane hybrid

We investigate the existence and the properties of normalized ground states of a nonlinear Schr\"odinger equation on a quantum hybrid formed by two planes connected at a point. The nonlinearities are of power type and $L^2$-subcritical, while the matching condition between the two planes generates two point interactions of different strengths on each plane, together with a coupling condition between the two planes. We prove that ground states exist for every value of the mass and two different qualitative situations are possible depending on the matching condition: either ground states concentrate on one of the plane only, or ground states distribute on both the planes and are positive, radially symmetric, decreasing and present a logarithmic singularity at the origin of each plane. Moreover, we discuss how the mass distributes on the two planes and compare the strengths of the logarithmic singularities on the two planes when the parameters of the matching condition and the powers of the nonlinear terms vary.

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arXiv2024

The search for NLS ground states on a hybrid domain: Motivations, methods, and results

We discuss the problem of establishing the existence of the Ground States for the subcritical focusing Nonlinear Schr\"odinger energy on a domain made of a line and a plane intersecting at a point. The problem is physically motivated by the experimental realization of hybrid traps for Bose-Einstein Condensates, that are able to concentrate the system on structures close to the domain we consider. In fact, such a domain approximates the trap as the temperature approaches the absolute zero. The spirit of the paper is mainly pedagogical, so we focus on the formulation of the problem and on the explanation of the result, giving references for the technical points and for the proofs.

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arXiv2024

NLS ground states on a hybrid plane

We study existence, nonexistence, and qualitative properties of ground states for a focusing, subcritical Nonlinear Schr\"odinger Equation on a hybrid plane, consisting of a half-line attached to a plane. Ground states are normalized minimizers of the associated energy, given by Nonlinear Schr\"odinger energies with contact interactions on the half-line and on the plane, plus a quadratic coupling term. At fixed mass, existence holds if the contact interaction on the half-line is not too repulsive, or the interaction on the plane is sufficiently attractive, or the coupling is strong enough. Nonexistence occurs when both interactions are sufficiently repulsive and the coupling is weak. Moreover, we discuss how the coupling affects the support and the symmetry properties of such ground states. These are the first results for a Nonlinear Schr\"odinger Equation on a mixed-dimensional manifold.

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arXiv2023

Normalized ground states for Schr\"odinger equations on metric graphs with nonlinear point defects

We investigate the existence of normalized ground states for Schr\"odinger equations on noncompact metric graphs in presence of nonlinear point defects, described by nonlinear $\delta$-interactions at some of the vertices of the graph. For graphs with finitely many vertices, we show that ground states exist for every mass and every $L^2$-subcritical power. For graphs with infinitely many vertices, we focus on periodic graphs and, in particular, on $\mathbb{Z}$-periodic graphs and on a prototypical $\mathbb{Z}^2$-periodic graph, the two-dimensional square grid. We provide a set of results unravelling nontrivial threshold phenomena both on the mass and on the nonlinearity power, showing the strong dependence of the ground state problem on the interplay between the degree of periodicity of the graph, the total number of point defects and their dislocation in the graph.

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arXiv2023

Two dimensional NLS ground states with attractive Coulomb potential and point interaction

We study the existence and the properties of ground states at fixed mass for a focusing nonlinear Schr\"odinger equation in dimension two with a point interaction, an attractive Coulomb potential and a nonlinearity of power type. We prove that for any negative value of the Coulomb charge, for any positive value of the mass and for any L$^2$-subcritical power nonlinearity, such ground states exist and exhibit a logarithmic singularity where the interaction is placed. Moreover, up to multiplication by a phase factor, they are positive, radially symmetric and decreasing. An analogous result is obtained also for minimizers of the action restricted to the Nehari manifold, getting the existence also in the L$^2$-critical and supercritical cases.

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arXiv2022

NLS ground states on the half-line with point interactions

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 $L^2$-subcritical or $L^{2}$-critical, while the point interaction is either attractive or repulsive. In the $L^{2}$-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 $L^{2}$-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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arXiv2022

Existence, structure, and robustness of ground states of a NLSE in 3D with a point defect

We study the ground states for the Schr\"odinger equation with a focusing nonlinearity and a point interaction in dimension three. We establish that ground states exist for every value of the mass; moreover they are positive, radially symmetric, decreasing along the radial direction, and show a Coulombian singularity at the location of the point interaction. Remarkably, the existence of the ground states is independent of the attractive or repulsive character of the point interaction.

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arXiv2021

Doubly nonlinear Schr\"odinger ground states on metric graphs

We investigate the existence of ground states at prescribed mass on general metric graphs with half-lines for focusing doubly nonlinear Schr\"odinger 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 in [Adami et al. '15, Adami et al. '16]. On the one hand, we provide a complete topological characterization of the problem, identifying 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 exact interplay between these different topological properties.

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arXiv2021

Ground states for the planar NLSE with a point defect as minimizers of the constrained energy

We investigate the ground states for the focusing, subcritical nonlinear Schr\"odinger equation with a point defect in dimension two, defined as the minimizers of the energy functional at fixed mass. We prove that ground states exist for every positive mass and show a logarithmic singularity at the defect. Moreover, up to a multiplication by a constant phase, they are positive, radially symmetric, and decreasing along the radial direction. In order to overcome the obstacles arising from the uncommon structure of the energy space, that complicates the application of standard rearrangement theory, we move to the study of the minimizers of the action functional on the Nehari manifold and then establish a connection with the original problem. An ad hoc result on rearrangements is given to prove qualitative features of the ground states.

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arXiv2020

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

We investigate the existence of ground states for the nonlinear Schr\"odinger Equation on star graphs with two subcritical focusing nonlinear terms: a standard power nonlinearity, and a delta-type nonlinearity located at the vertex. We find that if the standard nonlinearity is stronger than the pointwise one, then ground states exist for small mass only. On the contrary, if the pointwise nonlinearity prevails, then ground states exist for large mass only. All ground states are radial, in the sense that their restriction to each half-line is always the same function, and coincides with a soliton tail. Finally, if the two nonlinearities are of the same size, then the existence of ground states is insensitive to the value of the mass, and holds only on graphs with a small number of half-lines. Furthermore, we establish the orbital stability of the branch of radial stationary states to which the ground states belong, also in the mass regimes in which there is no ground state.

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arXiv2020

Non-Kirchhoff Vertices and NLS Ground States on graphs

We review some recent results and announce some new ones on the problem of the existence of ground states for the Nonlinear Schr\"odinger Equation on graphs endowed with vertices where the matching condition, instead of being free (or Kirchhoff's), is non-trivially interacting. In this category fall Dirac's delta conditions, delta prime, F\"ul\"op-Tsutsui, and others.

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arXiv2019

Prescribed mass ground states for a doubly nonlinear Schr\"odinger equation in dimension one

We investigate the problem of existence and uniqueness of ground states at fixed mass for two families of focusing nonlinear Schr\"odinger equations on the line. The first family consists of NLS with power nonlinearities concentrated at a point. For such model, we prove existence and uniqueness of ground states at every mass when the nonlinearity power is $L^2-$subcritical and at a threshold value of the mass in the $L^2-$critical regime. The second family is obtained by adding a standard power nonlinearity to the previous setting. In this case, we prove existence and uniqueness at every mass in the doubly subcritical case, namely when both the powers related to the pointwise and the standard nonlinearity are subcritical. If only one power is critical, then existence and uniqueness hold only at masses lower than the critical mass associated to the critical nonlinearity. Finally, in the doubly critical case ground states exist only at critical mass, whose value results from a non--trivial interplay between the two nonlinearities.

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