52 pubblicazioni classificate nel seguente modo:

Existence of saddle-type solutions for a class of quasilinear problems in R^2
TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS
Autore/i: Alves, Claudianor O.; Isneri, Renan J. S.; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: The main goal of the present paper is to prove the existence of saddle-type solutions for the following class of quasilinear problems $$ -\Delta_{\Phi}u + V'(u)=0\quad \text{in }\mathbb{R}^2, $$% where $$ \Delta_{\Phi}u=\text{div}(\phi(|\nabla u|)\nabla u), $$% $\Phi\colon \mathbb{R}\rightarrow [0,+\infty)$ is an N-function and the potential $V$ satisfies some technical condition and we have as an example $ V(t)=\Phi(|t^2-1|)$.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/319791

Existence of heteroclinic and saddle-type solutions for a class of quasilinear problems in whole ℝ2
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
Autore/i: Alves, Claudianor O.; Isneri, Renan J. S.; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: In this work, we use variational methods to prove the existence of heteroclinic and saddle type solutions for a class of quasilinear elliptic equations of the form $-\Delta_\Phi(u)+A(x,y)V'(u)=0$, $(x,y)\in\R^2$, where $\Phi:\R\to[0,+\infty)$ is an N-function, $A:\R^2\to\R$ is a periodic positive function and $V:\R\to\R$ is modeled on the Ginzburg-Landau potential.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/312007

Gradient Lagrangian systems and semilinear PDE
MATHEMATICS IN ENGINEERING
Autore/i: Alessio, F. G.; Montecchiari, P.
Classificazione: 1 Contributo su Rivista
Abstract: We survey some results about multiplicity of certain classes of entire solutions to semilinear elliptic equations or systems of the form −Δu=Fu(x,u), including the Allen Cahn or the stationary Nonlinear Schr"odinger case. In connection with this kind of problems we study some metric separation properties of sublevels of the functional V(u)=12‖∇u‖2H1(ℝN)−1p+1‖u‖p+1Lp+1(ℝN) in relation to the value of the exponent p+1∈(2,2∗N)
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/285070

A nondegeneracy condition for a semilinear elliptic system and the existence of multibump solutions
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
Autore/i: Montecchiari, P.; Rabinowitz, P. H.
Classificazione: 1 Contributo su Rivista
Abstract: For a class of semilinear elliptic systems, the existence of a broader class of multibump solutions is established under considerably weaker conditions than in earlier works. The key tool is the use of variational methods.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/295103

A Variant of the Mountain Pass Theorem and Variational Gluing
MILAN JOURNAL OF MATHEMATICS
Autore/i: Montecchiari, Piero; Rabinowitz, Paul H.
Classificazione: 1 Contributo su Rivista
Abstract: This paper surveys some recent work on a variant of the Mountain Pass Theorem that is applicable to some classes of differential equations involving unbounded spatial or temporal domains. In particular its application to a system of semilinear elliptic PDEs on $R^n$ and to a family of Hamiltonian systems involving double well potentials will also be discussed.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/283752

A note on a class of double well potential problems
RENDICONTI DELL'ISTITUTO DI MATEMATICA DELL'UNIVERSITÀ DI TRIESTE
Autore/i: Montecchiari, Piero; Rabinowitz, Paul H.
Classificazione: 1 Contributo su Rivista
Abstract: It is well known that under appropriate conditions on a double well potential, the associated Hamiltonian system possesses a pair of heteroclinic solutions joining the minima of the potential in addition to infinitely many other homoclinics and heteroclinics that oscillate between these minima. This paper studies the effect on such solutions of replacing the temporal domain, R, by a finite but long time interval.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/285383

New Multimedia Technologies as Tools for a Modern Approach to Scientific Communication and Teaching of Mathematical Sciences
The First Outstanding 50 Years of “Università Politecnica delle Marche”
Autore/i: Alessio, F. G.; Brambilla, M. C.; Calamai, A.; de Fabritiis, C.; Demeio, L.; Franca, M.; Marcelli, C.; Marietti, M.; Montecchiari, P.; Papalini, F.; Petrini, M.; Telloni, A. I.
Editore: Springer
Classificazione: 2 Contributo in Volume
Abstract: This paper describes various technological tools and e-learning resources used by the authors in their work as university teachers and communicators of mathematics to a general public. As pointed out by recent developments in mathematics education research, a technology enhanced teaching may have significant potentialities at tertiary level: e-learning is an essential support to overcome some logistic obstacles such as the large number of students per teacher, the small number of hours of lesson available, the heterogeneity of the freshmen’s mathematical background. Moreover, a fine design of digital environments can foster an engaging, inclusive, flexible and meaningful learning of mathematical topics. In this paper we give a perspective on some educational experiments carried out with engineering students of Università Politecnica delle Marche through specific functionalities of the Moodle platform (quiz, forum, workshop) and the dynamic geometry software Geogebra. We also report shortly on our approach to the use of new technological devices in the dissemination of scientific knowledge to a broader audience.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/272790

SADDLE SOLUTIONS FOR A CLASS OF SYSTEMS OF PERIODIC AND REVERSIBLE SEMILINEAR ELLIPTIC EQUATIONS
NETWORKS AND HETEROGENEOUS MEDIA
Autore/i: Alessio, Francesca Gemma; Montecchiari, Piero; Sfecci, Andrea
Classificazione: 1 Contributo su Rivista
Abstract: We study systems of elliptic equations −∆u(x)+Fu(x, u) = 0 with potentials F ∈ C2(Rn,Rm) which are periodic and even in all their variables. We show that if F(x,u) has flip symmetry with respect to two of the compo- nents of x and if the minimal periodic solutions are not degenerate then the system has saddle type solutions on Rn
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/266625

On global non-degeneracy conditions for chaotic behavior for a class of dynamical systems
ANNALES DE L INSTITUT HENRI POINCARÉ. ANALYSE NON LINÉAIRE
Autore/i: Montecchiari, Piero; Rabinowitz Paul, H.
Classificazione: 1 Contributo su Rivista
Abstract: For about 25 years, global methods from the calculus of variations have been used to establish the existence of chaotic behavior for some classes of dynamical systems. Like the analytical approaches that were used earlier, these methods require nondegeneracy conditions, but of a weaker nature than their predecessors. Our goal here is study such a nondegeneracy condition that has proved useful in several contexts including some involving partial differential equations, and to show this condition has an equivalent formulation involving stable and unstable manifolds.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/259643

Sustainable Engineering for Resilient Built and Natural Environments
The First Outstanding 50 Years of “Università Politecnica delle Marche”
Autore/i: Alici, Antonello; Bocci, Maurizio; Bonvini, Paolo; Brocchini, Maurizio; Calamai, Alessandro; Canestrari, Francesco; Capozucca, Roberto; Carbonari, Alessandro; Carbonari, Sandro; Cardone, Fabrizio; Clementi, Francesco; Clini, Paolo; Cocchi, Giammichele; Corvaro, Sara; Darvini, Giovanna; Davì, Fabrizio; Dezi, Luigino; Di Giuseppe, Elisa; D’Orazio, Marco; Ferretti, Maddalena; Ferrotti, Gilda; Gara, Fabrizio; Giretti, Alberto; Graziani, Andrea; Lancioni, Giovanni; Lemma, Massimo; Lenci, Stefano; Lorenzoni, Carlo; Malinverni, Eva Savina; Mancinelli, Alessandro; Mariano, Fabio; Mentrasti, Lando; Mondaini, Gianluigi; Montecchiari, Piero; Munafò, Placido; Naticchia, Berardo; Postacchini, Matteo; Quagliarini, Enrico; Quattrini, Ramona; Ragni, Laura; Serpilli, Michele; Soldini, Luciano; Virgili, Amedeo; Zampini, Giovanni
Editore: Springer
Classificazione: 2 Contributo in Volume
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/272866

A nondegeneracy condition for a semilinear elliptic system and the existence of 1- bump solutions
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
Autore/i: Montecchiari, Piero; Rabinowitz, Paul H.
Classificazione: 1 Contributo su Rivista
Abstract: Combining situations originally considered in [7] - [8], a semilinear elliptic system is treated and a nondegeneracy condition leading to the existence of multibump solutions is considerably weakened
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/267135

Prescribed energy connecting orbits for gradient systems
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
Autore/i: Alessio, Francesca Gemma; Montecchiari, Piero; Zuniga, Andres
Classificazione: 1 Contributo su Rivista
Abstract: We are concerned with conservative systems q’’=∇V(q), q∈RN for a general class of potentials V∈C^1(R^N). Assuming that a given sublevel set {V≤c} splits in the disjoint union of two closed subsets V_{c−} and V_{c+}, for some c∈R, we establish the existence of bounded solutions qc to the above system with energy equal to −c whose trajectories connect V_{c−} and V_{c+}. The solutions are obtained through an energy constrained variational method, whenever mild coerciveness properties are present in the problem. The connecting orbits are classified into brake, heteroclinic or homoclinic type, depending on the behavior of ∇V on ∂V_{c± }. Next, we illustrate applications of the existence result to double-well potentials V, and for potentials associated to systems of duffing type and of multiple-pendulum type. In each of the above cases we prove some convergence results of the family of solutions (q_c).
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/266467

Solutions of mountain pass type for double well potential systems
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
Autore/i: Montecchiari, Piero; Rabinowitz, Paul H.
Classificazione: 1 Contributo su Rivista
Abstract: This paper studies a Hamiltonian system possessing a double well potential for which the existence of multitransition heteroclinic and homoclinic solutions that are local minimizers of an associated functional is known. Under an additional mild non-degeneracy condition on the set of all homoclinic and heteroclinic solutions, the existence of further heteroclinic and homoclinic solutions that are of mountain pass type is established. A key tool for the existence arguments is a variant of the Mountain Pass Theorem that is of independent interest.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/259201

Analisi Matematica 1 - Teoria con esercizi svolti
Autore/i: Alessio, Francesca Gemma; Montecchiari, Piero
Editore: Società Editrice Esculapio
Luogo di pubblicazione: Bologna
Classificazione: 3 Libro
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/254357

Brake orbit solutions for semilinear elliptic systems with asymmetric double well potential
JOURNAL OF FIXED POINT THEORY AND ITS APPLICATIONS
Autore/i: Alessio, FRANCESCA GEMMA; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: We consider a class of semilinear elliptic system of the form -Delta u(x,y)+ abla W(u(x,y))=0,quad (x,y)inR^{2}, where W:R^{2} oR is a double well potential with minima a_pminR^$. We show, via variational methods, that if the set of minimal heteroclinic solutions to the one dimensional system -ddot q(x)+ abla W(q(x))=0, xinR, up to translations, is finite and constituted by not degenerate functions, then the system has infinitely many solutions uin C^{2}(R^{2})^{2}, parametrized by an energy value, which are periodic in the variable y and satisfy lim_{x opminfty}u(x,y)=a_{pm} for any yinR.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/239750

MATEMATICA ZERO - Per i precorsi e i test d'ingresso a Ingegneria e Scienze
Autore/i: Alessio, FRANCESCA GEMMA; DE FABRITIIS, Chiara; Marcelli, Cristina; Montecchiari, Piero
Editore: Pearson Italia
Luogo di pubblicazione: Milano
Classificazione: 3 Libro
Abstract: Il testo è parte di un sistema didattico integrato (testo+piattaforma online con quiz interattivi) per la comprensione e l'utilizzo della Matematica, rivolto agli studenti che si iscrivono al primo anno di corsi di laurea tecnico-scientifici. Lo scopo è quello di fornire uno strumento completo per la preparazione ai test d'ingresso e per l'eventuale frequenza dei precorsi di Matematica su argomenti di base, generalmente trattati nelle scuole superiori e indispensabili per il superamento degli esami universitari di Matematica di primo livello.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/236176

On the existence of multi-transition solutions for a class of elliptic systems
ANNALES DE L INSTITUT HENRI POINCARÉ. ANALYSE NON LINÉAIRE
Autore/i: Montecchiari, Piero; Paul H., Rabinowitz
Classificazione: 1 Contributo su Rivista
Abstract: The existence of solutions undergoing multiple spatial transitions between isolated periodic solutions is studied for a class of systems of semilinear elliptic partial differential equations. A key tool is a new result on the possible behavior of the set of single transition solutions.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/215314

Saddle solutions to Allen-Cahn equations in doubly periodic media
INDIANA UNIVERSITY MATHEMATICS JOURNAL
Autore/i: Alessio, FRANCESCA GEMMA; C., Gui; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: We consider a class of periodic Allen-Cahn equations $ -Delta u(x,y)+a(x,y)W'(u(x,y))=0,quad (x,y)inR^{2}$ where $ain C(R^2)$ is an even, periodic, positive function represenitng a doubly periodic media and $W:R oR$ is a classical double well potential such as the Ginzburg-Landau potential $W(s)=(s^{2}-1^{2})^{2}$}. We show the existence and asymptotic behavior of a saddle solution on the entire plane which has odd symmetry with respect to both axises and even symmetry with respect to the line $x=y$. This result generalizes the classic result on saddle solutions of Allen-Cahn equation in a homogeneous media.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/184106

A DOUBLE WELL POTENTIAL SYSTEM
ANALYSIS & PDE
Autore/i: Jaeyoung, Byeon; Montecchiari, Piero; Rabinowitz, PAUL H.
Classificazione: 1 Contributo su Rivista
Abstract: A semilinear elliptic system of PDEs with a nonlinear term of double well potential type is studied in a cylindrical domain. The existence of solutions heteroclinic to the bottom of the wells as minima of the associated functional is established. Further applications are given, including the existence of multitransition solutions as local minima of the functional.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/239753

SADDLE TYPE SOLUTIONS FOR A CLASS OF REVERSIBLE ELLIPTIC EQUATIONS
ADVANCES IN DIFFERENTIAL EQUATIONS
Autore/i: Alessio, FRANCESCA GEMMA; Autuori, Giuseppina; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: This paper is concerned with the existence of saddle type solutions for a class of semilinear elliptic equations of the type −∆u(x)+Fu(x,u) = 0, x ∈ Rn, n ≥ 2, (PDE) where F is a periodic and symmetric nonlinearity. Under a non degen- eracy condition on the set of minimal periodic solutions, saddle type solutions of (PDE) are found by a renormalized variational procedure.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/228480

Multiplicity of layered solutions for Allen-Cahn systems with symmetric double well potential
JOURNAL OF DIFFERENTIAL EQUATIONS
Autore/i: Alessio, FRANCESCA GEMMA; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: We study the existence of solutions $u:\R^{3}\to\R^{2}$ for the semilinear elliptic systems \begin{equation}\label{eq:abs} -\Delta u(x,y,z)+\nabla W(u(x,y,z))=0, \end{equation} where $W:\R^{2}\to\R$ is a double well symmetric potential. We use variational methods to show, under generic non degenerate properties of the set of one dimensional heteroclinic connections between the two minima $\a_{\pm}$ of $W$, that (\ref{eq:abs}) has infinitely many geometrically distinct solutions $u\in C^{2}(\R^{3},\R^{2})$ which satisfy $u(x,y,z)\to \a_{\pm}$ as ${x\to\pm\infty}$ uniformly with respect to $(y,z)\in\R^{2}$ and which exhibit dihedral symmetries with respect to the variables $y$ and $z$. We also characterize the asymptotic behaviour of these solutions as $|(y,z)|\to +\infty$.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/124862

An energy constrained method for the existence of layered type solutions of NLS equations
ANNALES DE L INSTITUT HENRI POINCARÉ. ANALYSE NON LINÉAIRE
Autore/i: Alessio, FRANCESCA GEMMA; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: We study the existence of positive solutions on $\R^{N+1}$ to semilinear elliptic equation $-\Delta u+u=f(u)$ where $N\geq 1$ and $f$ is modeled on the power case $f(u)=|u|^{p-1}u$. Denoting with $c$ the mountain pass level of $\f(u)=\tfrac 12\|u\|^{2}_{H^{1}(\R^{N})}-\int_{\R^{N}}F(u)\, dx$, $u\in H^{1}(\R^{N})$ ($F(s)=\int_{0}^{s}f(t)\, dt$), we show that for any $b\in [0,c)$ there exists a positive bounded solution $v_{b}\in C^{2}(\R^{N+1})$ such that $E_{v_{b}}(y)=\tfrac 12\|\partial_{y}v_{b}(\cdot,y)\|^{2}_{L^{2}(\R^{N})}-V(v_{b}(\cdot,y))=-b$. We also characterize the monotonicity, symmetry and periodicity properties of $v_{b}$.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/83761

Saddle solutions for bistable symmetric semilinear elliptic equations
NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS
Autore/i: Alessio, FRANCESCA GEMMA; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: This paper concerns the existence and the asymptotic characterization of saddle solutions in $R^{3}$ for semilinear elliptic equations of the form egin{equation}label{eq:abs} -Delta u+W'(u)=0,quad (x,y,z)inR^{3} end{equation} where $WinCC^{3}(R)$ is a double well symmetric potential, i.e. it satisfies $W(-s)=W(s)$ for $sinR$, $W(s)> 0$ for $sin (-1,1)$, $W(pm 1)=0$ and $W''(pm 1)>0$. Denoted with $ heta_{2}$ the saddle planar solution of ( ef{eq:abs}), we show the existence of a unique solution $ heta_{3}in C^{2}(R^{3})$ which is odd with respect to each variable, symmetric with respect to the diagonal planes, verifies $0< heta_{3}(x,y,z)<1$ for $x,y,z>0$ and $ heta_{3}(x,y,z) o_{z o+infty} heta_{2}(x,y)$ uniformly with respect to $(x,y)inR^{2}$.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/83764

Layered solutions with multiple asymptotes for non autonomous Allen–Cahn equations in R^{3}
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
Autore/i: Alessio, FRANCESCA GEMMA; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: We consider a class of semilinear elliptic equations of the form \begin{equation}\label{eq:abs} -\Delta u(x,y,z)+a(x)W'(u(x,y,z))=0,\quad (x,y,z)\in\R^{3}, \end{equation} where $a:\R\to\R$ is a periodic, positive, even function and, in the simplest case, $W:\R\to\R$ is a double well even potential. Under non degeneracy conditions on the set of minimal solutions to the associated one dimensional heteroclinic problem we show, via variational methods the existence of infinitely many geometrically distinct solutions $u$ of (\ref{eq:abs}) verifying $u(x,y,z)\to\pm 1$ as $x\to\pm\infty$ uniformly with respect to $(y,z)\in\R^{2}$ and such that $\partial_{y}u\not\equiv0$, $\partial_{z}u\not\equiv0$ in $\R^{3}$.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/66072

Note di Analisi Matematica Uno
Progetto Leonardo
Autore/i: Alessio, FRANCESCA GEMMA; Montecchiari, Piero
Editore: SOCIETA' EDITRICE ESCULAPIO
Luogo di pubblicazione: Bologna
Classificazione: 3 Libro
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/86808

Saddle type solutions to a class of semilinear elliptic equations
ADVANCES IN DIFFERENTIAL EQUATIONS
Autore/i: Alessio, FRANCESCA GEMMA; Calamai, Alessandro; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: We consider a class of semilinear elliptic equations of the form $$ -\Delta u(x,y)+W'(u(x,y))=0,\quad (x,y)\in\R^{2} $$ where $W:\R\to\R$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^{2}-1)^{2}$. We show, via variational methods, that for any $j\geq 2$, the equation has a solution $v_{j}\in C^{2}(\R^{2})$ with $|v_{j}(x,y)|\leq 1$ for any $(x,y)\in\R^{2}$ satisfying the following symmetric and asymptotic conditions: setting $\tilde v_{j}(\rho,\theta)=v_{j}(\rho\cos( \theta),\rho\sin(\theta))$, there results $\tilde v_{j}(\rho,\frac{\pi}{2}+\theta)=-\tilde v_{j}(\rho,\frac{\pi}{2}-\theta)$, $\tilde v_{j}(\rho,\theta+\frac{\pi}{j})=-\tilde v_{j}(\rho,\theta)$, $\forall (\rho,\theta)\in \R^{+}\times\R$ and $\tilde v_{j}(\rho,\theta)\to 1$ as $\rho\to+\infty$ for any $\theta\in [\frac{\pi}2-\frac{\pi}{2j},\frac{\pi}2)$.}
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/34140

Brake orbits type solutions to some class of semilinear elliptic equations
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
Autore/i: Alessio, FRANCESCA GEMMA; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: We consider a class of semilinear elliptic equations of the form $$-\Delta u(x,y)+a(x)W'(u(x,y))=0,\quad (x,y)\in\R^{2}$$ where $a:\R\to\R$ is a periodic, positive function and $W:\R\to\R$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^{2}-1)^{2}$. We show, via variational methods, that if the set of solutions to the one dimensional heteroclinic problem $$-\ddot q(x)+a(x)W'(q(x))=0,\ x\in\R,\qquad q(\pm\infty)=\pm 1,$$ has a discrete structure, then the equation has infinitely many solutions periodic in the variable $y$ and verifying the asymptotic conditions $u(x,y)\to\pm 1$ as $x\to\pm\infty$ uniformly with respect to $y\in\R$.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/51894

Entire solutions in ℝ2 for a class of Allen-Cahn equations
ESAIM. COCV
Autore/i: Alessio, Francesca Gemma; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: We consider a class of semilinear elliptic equations of the form $$ -\e^{2}\Delta u(x,y)+a(x)W'(u(x,y))=0,\quad (x,y)\in\R^{2} $$ where $\e>0$, $a:\R\to\R$ is a periodic, positive function and $W:\R\to\R$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^{2}-1)^{2}$. We look for solutions which verify the asymptotic conditions $u(x,y)\to\pm 1$ as $x\to\pm\infty$ uniformly with respect to $y\in\R$. We show via variational methods that if $\e$ is sufficiently small then the equation admits infinitely many of such solutions, distinct up to periodic translations, which are not solutions to the associated ODE problem $$ -\e^{2}\ddot q(x)+a(x)W'(q(x))=0,\qquad \lim_{x\to\pm\infty}q(x)=\pm 1. $$
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/34067

Multiplicity of entire solutions for a class of almost periodic Allen-Cahn type equations
ADVANCED NONLINEAR STUDIES
Autore/i: Alessio, FRANCESCA GEMMA; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: We consider a class of semilinear elliptic equations of the form $$-\Delta u(x,y)+a(\varepsilon x)W'(u(x,y))=0,\quad (x,y)\in\R^{2}$$ where $\e>0$, $a:\R\to\R$ is an almost periodic, positive function and $W:\R\to\R$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^{2}-1)^{2}$. We show via variational methods that if $\e$ is sufficiently small and $a$ is not constant then the equation admits infinitely many two dimensional entire solutions verifying the asymptotic conditions $u(x,y)\to\pm 1$ as $x\to\pm\infty$ uniformly with respect to $y\in\R$.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/53379

Chaotic behaviour of rapidly oscillating Lagrangian systems
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
Autore/i: Alessio, FRANCESCA GEMMA; V., COTI ZELATI; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: In the paper we prove that the Lagrangian system $$\ddot{q} = \alpha(\omega t) V'(q), \quad t \in \R, \ q \in \R^{N},$$ has, for some classes of functions $\alpha$, a chaotic behavior. More precisely the system has multi-bump solutions for all $\omega$ large. These classes of functions include some quasi-periodic and some limit-periodic ones, but not any periodic function. We prove the result using global variational methods.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/52448

Existence of infinitely many stationary layered solutions in R^2 for a class of periodic Allen Cahn Equations.
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
Autore/i: Alessio, FRANCESCA GEMMA; L., Jeanjean; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: We consider a class of periodic Allen-Cahn equations \begin{equation}\tag{$1$} -\Delta u(x,y)+a(x,y)W'(u(x,y))=0,\quad (x,y)\in\R^{2} \end{equation} where $a:\R^2\to\R^2$ is an even, periodic, positive function and $W:\R\to\R$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^{2}-b^{2})^{2}$. We show, via variational methods, that there exist infinitely many solutions, distinct up to periodic translations, of $(1)$ asymptotic as $x\to\pm\infty$ to the pure states $\pm b$, i.e., solutions satisfying the boundary conditions \begin{equation}\label{eq:et}\tag{$2$} \lim_{x\to\pm\infty}u(x,y)=\pm b, \quad \hbox{uniformly in $y \in \R.$} \end{equation} In fact, we prove the existence of solutions of $(1)$-$(2)$ which are periodic in the $y$ variable and if such solutions are finite modulo periodic translations, we can prove the existence of infinitely many (modulo periodic translations) solutions of $(1)$-$(2)$ asymptotic to different periodic solutions as $y \to \pm\infty$.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/53290

Infinitely many solutions for a class of semilinear elliptic equations in R^N
BOLLETTINO DELL'UNIONE MATEMATICA ITALIANA. B
Autore/i: Alessio, FRANCESCA GEMMA; P., Caldiroli; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/52690

Stationary layered solutions in R^2 for a class of non autonomous Allen-Cahn equations
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
Autore/i: Alessio, FRANCESCA GEMMA; L., Jeanjean; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: We consider a class of non autonomous Allen-Cahn equations \begin{equation} -\Delta u(x,y)+a(x)W'(u(x,y))=0,\quad (x,y)\in\R^{2}, \end{equation} where $W\in\CC^{2}(\R,\R)$ is a multiple-well potential and $a\in\CC(\R,\R)$ is a periodic, positive, non-constant function. We look for solutions to (0.1) having uniform limits as $x\to\pm\infty$ corresponding to minima of $W$. We show, via variational methods, that under a nondegeneracy condition on the set of heteroclinic solutions of the associated ordinary differential equation $-\ddot q(x)+a(x)W'(q(x))=0,$ $x\in\R,$ the equation (0.1) has solutions which depends on both the variables $x$ and $y$. In contrast, when $a$ is constant such nondegeneracy condition is not satisfied and all such solutions are known to depend only on $x$.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/52850

Multibump solutions to possibly degenerate equilibria for almost periodic Lagrangian systems
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
Autore/i: Alessio, FRANCESCA GEMMA; Bertotti, M. L.; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: We study via variational methods some chaotic features of a class of almost periodic Lagrangian systems on a torus. In particular we show that slowly oscillating perturbations of such systems admit a multibump dynamics relative to possibly degenerate equilibria.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/52000

Multibump solutions for a class of Lagrangian systems slowly oscillating at infinity
ANNALES DE L INSTITUT HENRI POINCARÉ. ANALYSE NON LINÉAIRE
Autore/i: Alessio, FRANCESCA GEMMA; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: We prove the existence of infinitely many homoclinic solutions for a class of second order hamiltonian systems of the form $-\ddot u+u=\alpha(t)\nabla W(u)$ where $W$ is superquadratic and $\dot\alpha(t)\to 0$, $0<\liminf\alpha(t)<\limsup\alpha(t)$ as $t\to +\infty$. In fact we prove that such a kind of systems admit a ``multibump'' dynamics
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/52420

Genericity of the multibump dynamics for almost periodic Duffing-like systems
PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH. SECTION A. MATHEMATICS
Autore/i: Alessio, FRANCESCA GEMMA; Caldiroli, P.; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: In this paper we consider ``slowly'' oscillating perturbations of almost periodic Duffing-like systems, i.e., systems of the form $\ddot u=u-(a(t)+\a(\o t))W'(u)$, $t\in$\R, $u\in$\R$^N$, where $W\in C^{2N}($\R$^N$,\R$)$ is superquadratic and $a$ and $\a$ are positive and almost periodic. By variational methods, we prove that if $\o>0$ is small enough then the system admits a multibump dynamics. As a consequence we get that the system $\ddot u=u-a(t)W'(u)$, $t\in$\R, $u\in$\R$^N$, admits multibump solutions whenever $a$ belongs to an open dense subset of the set of positive almost periodic continuous functions.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/52565

A global condition for periodic Duffing-like equations
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
Autore/i: Montecchiari, Piero; M., Nolasco; S., Terracini
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/31029

Heteroclinic motions joining almost periodic solutions for a class of Lagrangian systems
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
Autore/i: Alessio, FRANCESCA GEMMA; C., Carminati; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: We regard second order systems of the form $\ddot q=\nabla_q W (q,t)$, $t\in\R$, $q\in\RN$, where $W(q,t)$ is $\ZN$ periodic in $q$ and almost periodic in $t$. Variational arguments are used to prove the existence of heteroclinic solutions joining almost periodic solutions to the system.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/52686

On the existence of homoclinic orbits for the asymptotically periodic Duffing Equation
TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS
Autore/i: Alessio, FRANCESCA GEMMA; Caldiroli, P.; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: Using variational methods, we show the existence of a homoclinic orbit for the Duffing equation $-ddot u+u=a(t)|u|^{p-1}u$, where $p>1$ and $ain L^infty(R)$ is a positive function of the form $a=a_0+a_infty$ with $a_infty$ periodic, and $a_0(t) o 0$ as $t opminfty$ satisfying suitable conditions. Under the same assumptions on $a$, we also prove that the perturbed equation $-ddot u+u=a(t)|u|^{p-1}u+alpha(t)g(u)$ admits a homoclinic orbit whenever $gin C^1(R)$ satisfies $g(u)=O(u)$ as $u o 0$ and $alphain L^infty(R)$, $alpha(t) o 0$ as $t opminfty$ and $|alpha|_{L^infty}$ is sufficiently small.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/53421

Almost periodic solutions for a class of Duffing like systems
DIFFERENTIAL AND INTEGRAL EQUATIONS
Autore/i: V., COTI ZELATI; Montecchiari, Piero; M., Nolasco
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/31030

Genericity of the existence of infinitely many solutions for a class of semilinear elliptic equations in R^N
ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE
Autore/i: Alessio, FRANCESCA GEMMA; Caldiroli, P.; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: We show, by variational methods, that there exists a set $\AA$ open and dense in $\{ a\in L^\infty(\R^N):\liminf_{|x|\to\infty}a(x)\geq 0\}$ such that if $a\in \AA$ then the problem $ -\Delta u+u=a(x)|u|^{p-1}u$, $u\in H^1(\R^N)$, with $p$ subcritical (or more general nonlinearities), admits infinitely many solutions.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/52987

Connecting orbits for some classes of almost periodic Lagrangian systems
JOURNAL OF DIFFERENTIAL EQUATIONS
Autore/i: M. L., Bertotti; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/30807

On the existence of infinitely many solutions for a class of semilinear elliptic equations in R^N
ATTI DELLA ACCADEMIA NAZIONALE DEI LINCEI. RENDICONTI LINCEI. MATEMATICA E APPLICAZIONI
Autore/i: Alessio, FRANCESCA GEMMA; P., Caldiroli; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Abstract: We show, by variational methods, that there exists a set $A$ open and dense in ${ ain L^infty(R^N)~:~ ageq 0}$ such that if $ain A$ then the problem $ -Delta u+u=a(x)|u|^{p-1}u$, $uin H^1(R^N)$, with $p$ subcritical (or more general nonlinearities), admits infinitely many solutions.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/34134

Multibump homoclinic solutions for a class of second order, almost periodic Hamiltonian systems
NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS
Autore/i: V., COTI ZELATI; Montecchiari, Piero; M., Nolasco
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/31027

Multiplicity of homoclinics for a class of time recurrent second order Hamiltonian systems
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
Autore/i: Montecchiari, Piero; M., Nolasco; S., Terracini
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/31028

Multibump solutions for perturbation of periodic second order Hamiltonian systems
NONLINEAR ANALYSIS
Autore/i: Montecchiari, Piero; M., Nolasco
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/30533

Multibump solutions for Duffing-like systems
RENDICONTI DELL'ISTITUTO DI MATEMATICA DELL'UNIVERSITÀ DI TRIESTE
Autore/i: S., Abenda; P., Caldiroli; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/30531

Multiplicity results for a class of Semilinear Elliptic Equations on $R^m$
RENDICONTI DEL SEMINARIO MATEMATICO DELL'UNIVERSITA' DI PADOVA
Autore/i: Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/30532

Asymptotic behaviour for a class of multibump solutions to Duffing-like systems
Variational and local methods in the study of Hamiltonian systems : proceedings of the workshop : International Centre for Theoretical Physics, Trieste, Italy, 24-28 October, 1994 editors, A. Ambrosetti, G.F. Dell'Antonio.
Autore/i: Caldiroli, P.; Montecchiari, P.; Nolasco, Margherita
Editore: World Scientific
Classificazione: 4 Contributo in Atti di Convegno (Proceeding)
Abstract: We consider a class of second order Hamiltonian systems $\ddot q=q-V'(t,q)$ where $V(t,q)$ is asymptotic at infinity to a time periodic and superquadratic function $V_+(t,q)$. We prove the existence of a class of multibump solutions whose $\omega$-limit is a suitable homoclinic orbit of the system at infinity $\ddot q=q-V'_+(t,q)$
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/257953

Existence and multiplicity of homoclinic solutions for a class of asymptotically periodic second order Hamiltonian systems
ANNALI DI MATEMATICA PURA ED APPLICATA
Autore/i: Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/31024

Homoclinic orbits for second order Hamiltonian systems with potential changing sign
COMMUNICATIONS ON APPLIED NONLINEAR ANALYSIS
Autore/i: Caldiroli, P.; Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/31026

Multiplicity of homoclinic solutions for a class of asymptotically periodic second order Hamiltonian systems
ATTI DELLA ACCADEMIA NAZIONALE DEI LINCEI. CLASSE DI SCIENZE FISICHE, MATEMATICHE E NATURALI. RENDICONTI LINCEI. SUPPLEMENTO
Autore/i: Montecchiari, Piero
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/31025