arXiv subjects
Publications and source records attributed to Alessio Pomponio.
In this paper we use variational methods to establish a Berestycki-Lions type result for a class of generalized Kadomtsev-Petviashvili equation in $\mathbb{R}^N$. The positive and zero mass cases are considered.
We consider a system arising from a nonrelativistic Chern-Simon-Higgs model, in which a charged field is coupled with a gauge field. We prove an existence result for small coupling constants.
This paper deals with the prescribed mean curvature equations both in the Euclidean case and in the Lorentz-Minkowski case in presence of a nonlinearity $g$ such that $g'(0)>0$. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. Moreover these solutions are periodic, if $N=1$, while they are radial symmetric and decay to zero at infinity with their derivatives, if $N\ge 2$.
We are interested in standing waves of a modified Schrödinger equation coupled with the Chern-Simons gauge theory. By applying a constraint minimization of Nehari-Pohozaev type, we prove the existence of radial ground state solutions. We also investigate the non-existence for nontrivial solutions.
In this paper we study a system which we propose as a model to describe the interaction between matter and electromagnetic field from a dualistic point of view. This system arises from a suitable coupling of the Schrödinger and the Born-Infeld lagrangians, this latter replacing the role that, classically, is played by the Maxwell lagrangian. We use a variational approach to find an electrostatic radial ground state solution by means of suitable estimates on the functional of the action.
This paper is devoted to the study, with variational technique, of (p,q)-Laplacian equations in presence of general nonlinearities. Especially we obtain the existence result for the zero mass case, which includes a large class of pure power nonlinearities. More general quasilinear problems of Born-Infeld type are also considered.
We look for three dimensional vortex-solutions, which have finite energy and are stationary solutions, of Klein-Gordon-Maxwell-Proca type systems of equations. We prove the existence of three dimensional cylindrically symmetric vortex-solutions having a least possible energy among all symmetric solutions. Moreover we show that, if the Proca mass disappears, then the solutions tends to a solution of the Klein-Gordon-Maxwell system.
In this paper, we deal with the electrostatic Born-Infeld equation \begin{equation}\label{eq:BI-abs} \tag{$\mathcal{BI}$} \left\{ \begin{array}{ll} -\operatorname{div}\left(\displaystyle\frac{\nabla ϕ}{\sqrt{1-|\nabla ϕ|^2}}\right)= ρ, & \hbox{in } \mathbb{R}^N, \\ \displaystyle\lim_{|x|\to \infty}ϕ(x)= 0, \end{array} \right. \end{equation} where $ρ$ is an assigned extended charge density. We are interested in the existence and uniqueness of the potential $ϕ$ and finiteness of the energy of the electrostatic field $-\nabla ϕ$. We first relax the problem and treat it with the direct method of the Calculus of Variations for a broad class of charge densities. Assuming $ρ$ is radially distributed, we recover the weak formulation of \eqref{eq:BI-abs} and the regularity of the solution of the Poisson equation (under the same smootheness assumptions). In the case of a locally bounded charge, we also recover the weak formulation without assuming any symmetry. The solution is even classical if $ρ$ is smooth. Then we analyze the case where the density $ρ$ is a superposition of point charges and discuss the results in [Kiessling, Comm. Math. Phys. 314 (2012), 509--523]. Other models are discussed, as for instance a system arising from the coupling of the nonlinear Klein-Gordon equation with the Born-Infeld theory.
This paper deals with semilinear elliptic problems of the type \[ \left\{ \begin{array}{ll} -Δu+α(x)u= β(x)|u|^{p-1}u \quad \hbox{in }\mathbb{R}^N, u(x)>0\quad\hbox{in } \mathbb{R}^N, \qquad u \in H^1(\mathbb{R}^N), \end{array} \right. \] where $p$ is superlinear but subcritical and the coefficients $α$ and $β$ are positive functions such that $α(x) \to a_\infty > 0$ and $β(x)\to b_\infty > 0$, as $|x| \to \infty$. Aim of this work is to describe some phenomena that can occur when the coefficients are "competing".
This paper is motivated by a gauged Schrödinger equation in dimension 2. We are concerned with radial stationary states under the presence of a vortex at the origin. Those states solve a nonlinear nonlocal PDE with a variational structure. We will study the global behavior of that functional, extending known results for the regular case.
In this paper we give a multiplicity result for the following Chern-Simons-Schrödinger equation \[ -Δu+2q u \int_{|x|}^{\infty}\frac{u^{2}(s)}{s}h_u(s)ds +q u\frac{h^{2}_u(|x|)}{|x|^{2}} = g(u), \quad\hbox{in }\mathbb{R}^2, \] where $\displaystyle h_u(s)=\int_0^s τu^2(τ) \ d τ$, under very general assumptions on the nonlinearity $g$. In particular, for every $n\in \mathbb N$, we prove the existence of (at least) $n$ distinct solutions, for every $q\in (0,q_{n})$, for a suitable $q_n$.
This paper is motivated by a gauged Schrodinger equation in dimension 2 including the so-called Chern-Simons term. The radially symmetric case leads to an elliptic problem with a nonlocal defocusing term, in competition with a local focusing nonlinearity. In this work we pose the equations in a ball under homogeneous Dirichlet boundary conditions. By using singular perturbation arguments we prove existence of solutions for large values of the radius. Those solutions are located close to the boundary and the limit profile is given.
This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $$ - Δu(x) + \left(ω+ \frac{h^2(|x|)}{|x|^2} + \int_{|x|}^{+\infty} \frac{h(s)}{s} u^2(s)\, ds \right) u(x) = |u(x)|^{p-1}u(x), $$ where $$ h(r)= \frac{1}{2}\int_0^{r} s u^2(s) \, ds. $$ This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for $p\in(1,3)$, the functional may be bounded from below or not, depending on $ω$. Quite surprisingly, the threshold value for $ω$ is explicit. From this study we prove existence and non-existence of positive solutions.
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 study semiclassical states for the problem $$ -\eps^2 Δu + V(x) u = f(u) \qquad \hbox{in} \RN,$$ where $f(u)$ is a superlinear nonlinear term. Under our hypotheses on $f$ a Lyapunov-Schmidt reduction is not possible. We use variational methods to prove the existence of spikes around saddle points of the potential $V(x)$.
We find infinitely many positive non-radial solutions for a nonlinear Schrodinger-Poisson system.
In this paper we prove a multiplicity result concerning the critical points of a class of functionals involving local and nonlocal nonlinearities. We apply our result to the nonlinear Schrodinger-Maxwell system and to the nonlinear elliptic Kirchhoff equation assuming on the local nonlinearity the general hypotheses introduced by Berestycki and Lions.
We study the class of nonlinear Klein-Gordon-Maxwell systems describing a standing wave (charged matter field) in equilibrium with a purely electrostatic field. We improve some previous existence results in the case of an homogeneous nonlinearity. Moreover, we deal with a limit case, namely when the frequency of the standing wave is equal to the mass of the charged field; this case shows analogous features of the well known "zero mass case" for scalar field equations.