Papers by Pietro d'Avenia
arXiv: Analysis of PDEs, Jan 29, 2010
We are interested in the existence of infinitely many positive solutions of the Schrödinger-Poiss... more We are interested in the existence of infinitely many positive solutions of the Schrödinger-Poisson system −∆u + u + V (|x|)φu = |u| p−1 u, x ∈ R 3 , −∆φ = V (|x|)u 2 , x ∈ R 3 , where V (|x|) is a positive bounded function, 1 < p < 5 and V (r), r = |x| has the following decay property: V (r) = a r m + O 1 r m+θ with a > 0, m > 3 2 , θ > 0. The solutions founded are non-radial.
Calculus of Variations and Partial Differential Equations
We prove a multiplicity result for −ε 2 ∆gu + ωu + q 2 φu = |u| p−2 u −∆gφ + a 2 ∆ 2 g φ + m 2 φ ... more We prove a multiplicity result for −ε 2 ∆gu + ωu + q 2 φu = |u| p−2 u −∆gφ + a 2 ∆ 2 g φ + m 2 φ = 4πu 2 in M, where (M, g) is a smooth and compact 3-dimensional Riemannian manifold without boundary, p ∈ (4, 6), a, m, q = 0, ε > 0 small enough. The proof of this result relies on Lusternik-Schnirellman category. We also provide a profile description for low energy solutions.
Journal of Differential Equations
In this paper we consider an Hartree-Fock type system made by two Schrödinger equations in presen... more In this paper we consider an Hartree-Fock type system made by two Schrödinger equations in presence of a Coulomb interacting term and a cooperative pure power and subcritical nonlinearity, driven by a suitable parameter β ≥ 0. We show the existence of semitrivial and vectorial ground states solutions depending on the parameters involved. The asymptotic behavior with respect to the parameter β of these solutions is also studied.
We study the following nonlinear Schrödinger equation with a forth order dispersion term \[ Δ^2u-... more We study the following nonlinear Schrödinger equation with a forth order dispersion term \[ Δ^2u-βΔu=g(u) \quad \text{in } \mathbb{R}^N \] in the positive and zero mass regimes: in the former, $N\geq 2$ and $β> -2\sqrt{m}$, where $m>0$ depends on $g$; in the latter, $N\geq 3$ and $β>0$. In either regimes, we find an infinite sequence of solutions under rather generic assumptions about $g$; if $N=2$ in the positive mass case, or $N=4$ in the zero mass case, we need to strengthen such assumptions. Our approach is variational.
arXiv: Analysis of PDEs, Jun 29, 2014
We investigate a class of nonlinear Schrödinger equations with a generalized Choquard nonlinearit... more We investigate a class of nonlinear Schrödinger equations with a generalized Choquard nonlinearity and fractional diffusion. We obtain regularity, existence, nonexistence, symmetry as well as decays properties.
Journal of Differential Equations
In this paper we consider an Hartree-Fock type system made by two Schrödinger equations in presen... more In this paper we consider an Hartree-Fock type system made by two Schrödinger equations in presence of a Coulomb interacting term and a cooperative pure power and subcritical nonlinearity, driven by a suitable parameter β ≥ 0. We show the existence of semitrivial and vectorial ground states solutions depending on the parameters involved. The asymptotic behavior with respect to the parameter β of these solutions is also studied.
We study the following nonlinear Schrödinger equation with a forth order dispersion term \[ Δ^2u-... more We study the following nonlinear Schrödinger equation with a forth order dispersion term \[ Δ^2u-βΔu=g(u) \quad \text{in } \mathbb{R}^N \] in the positive and zero mass regimes: in the former, $N\geq 2$ and $β> -2\sqrt{m}$, where $m>0$ depends on $g$; in the latter, $N\geq 3$ and $β>0$. In either regimes, we find an infinite sequence of solutions under rather generic assumptions about $g$; if $N=2$ in the positive mass case, or $N=4$ in the zero mass case, we need to strengthen such assumptions. Our approach is variational.
arXiv: Analysis of PDEs, Jun 29, 2014
We investigate a class of nonlinear Schrödinger equations with a generalized Choquard nonlinearit... more We investigate a class of nonlinear Schrödinger equations with a generalized Choquard nonlinearity and fractional diffusion. We obtain regularity, existence, nonexistence, symmetry as well as decays properties.
In this paper we prove the existence of a nontrivial non-negative radial solution for a quasiline... more In this paper we prove the existence of a nontrivial non-negative radial solution for a quasilinear elliptic problem. Our aim is to approach the problem variationally by using the tools of critical points theory in an Orlicz-Sobolev space. A multiplicity result is also given.
In this paper we prove the existence of a nontrivial non-negative radial solution for a quasiline... more In this paper we prove the existence of a nontrivial non-negative radial solution for a quasilinear elliptic problem. Our aim is to approach the problem variationally by using the tools of critical points theory in an Orlicz-Sobolev space. A multiplicity result is also given.
Journal of Mathematical Analysis and Applications, 2017
In this paper we study a system which is equivalent to a nonlocal version of the well known Brezi... more In this paper we study a system which is equivalent to a nonlocal version of the well known Brezis Nirenberg problem. The difficulties related with the lack of compactness are here emphasized by the nonlocal nature of the critical nonlinear term. We prove existence and nonexistence results of positive solutions when N = 3 and existence of solutions in both the resonance and the nonresonance case for higher dimensions. Contents 1. Introduction and statement of the main results 1 2. Preliminaries 3 2.1. The reduction method 3 2.2. The Palais-Smale condition 6 3. The dimension N = 3: positive solutions 8 3.1. An existence result 9 3.2. A nonexistence result 11 4. The dimensions N 4 12 4.1. Positive solutions 12 4.2. The geometrical properties for the Linking Theorem 13 4.3. Sign changing solutions: the nonresonance case 15 4.4. Sign changing solutions: the resonance case 16 References 17
ESAIM: Control, Optimisation and Calculus of Variations, 2017
We study a class of minimization problems for a nonlocal operator involving an external magnetic ... more We study a class of minimization problems for a nonlocal operator involving an external magnetic potential. The notions are physically justified and consistent with the case of absence of magnetic fields. Existence of solutions is obtained via concentration compactness.
Advanced Nonlinear Studies, 2002
In this paper we prove the existence of standing wave solutions of nonlinear Schrödinger equation... more In this paper we prove the existence of standing wave solutions of nonlinear Schrödinger equation coupled with Maxwell equations which are non-radially symmetric.
Mathematical Methods in the Applied Sciences, 2015
By means of non-smooth critical point theory we obtain existence of infinitely many weak solution... more By means of non-smooth critical point theory we obtain existence of infinitely many weak solutions of the fractional Schrödinger equation with logarithmic nonlinearity. We also investigate the Hölder regularity of the weak solutions.
Advances in Nonlinear Analysis, 2014
This paper deals with the Klein–Gordon–Maxwell system in a bounded spatial domain with a nonunifo... more This paper deals with the Klein–Gordon–Maxwell system in a bounded spatial domain with a nonuniform coupling. We discuss the existence of standing waves in equilibrium with a purely electrostatic field, assuming homogeneous Dirichlet boundary conditions on the matter field and nonhomogeneous Neumann boundary conditions on the electric potential. Under suitable conditions we prove existence and nonexistence results. Since the system is variational, we use Ljusternik–Schnirelmann theory.
Nonlinearity, 2021
We find radial and nonradial solutions to the following nonlocal problem − Δ u + ω u = I α * F ( ... more We find radial and nonradial solutions to the following nonlocal problem − Δ u + ω u = I α * F ( u ) f ( u ) − I β * G ( u ) g ( u ) in R N under general assumptions, in the spirit of Berestycki and Lions, imposed on f and g, where N ⩾ 3, 0 ⩽ β ⩽ α < N, ω ⩾ 0, f , g : R → R are continuous functions with corresponding primitives F, G, and I α , I β are the Riesz potentials. If β > 0, then we deal with two competing nonlocal terms modelling attractive and repulsive interaction potentials.
We investigate a class of nonlinear Schrodinger equations with a generalized Choquard nonlinearit... more We investigate a class of nonlinear Schrodinger equations with a generalized Choquard nonlinearity and fractional diffusion. We obtain regularity, existence, nonexistence, symmetry as well as decays properties.
Differential and Integral Equations, 2003
ABSTRACT This paper deals with a model of solitary waves, in three space dimensions, which are ch... more ABSTRACT This paper deals with a model of solitary waves, in three space dimensions, which are characterized by a topological invariant called charge; these waves behave as relativistic particles. We study the interaction with an electromagnetic field. The Lagrangian density of the system is the sum of three terms: the first is that of the free soliton, the second is the classical Lagrangian density of an electromagnetic field, the third, which is due to the interaction, is chosen so that the electric charge coincides with the topological charge. We prove the existence of a static solution for every fixed value of the charge. The energy functional is strongly unbounded from above, as from below; after a reduction argument, the critical points are found by means of the Principle of Symmetric Criticality.
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Papers by Pietro d'Avenia