Progress in Nonlinear Differential Equations and Their Applications Volume 43

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2 Progress in Nonlinear Differential Equations and Their Applications Volume 43 Editor Haim Brezis Universite Pierre et Marie Curie Paris and Rutgers University New Brunswick, N.J. Editorial Board Antonio Ambrosetti, Scuola Normale Superiore, Pisa A. Bahri, Rutgers University, New Brunswick Felix Browder, Rutgers University, New Brunswick Luis Cafarelli, Institute for Advanced Study, Princeton Lawrence C. Evans, University of California, Berkeley Mariano Giaquinta, University of Pisa David Kinderlehrer, Carnegie-Mellon University, Pittsburgh Sergiu Klainerman, Princeton University Robert Kohn, New York University P. L. Lions, University of Paris IX Jean Mawhin, Universite Catholique de Louvain Louis Nirenberg, New York University Lambertus Peletier, University of Leiden Paul Rabinowitz, University of Wisconsin, Madison John Toland, University of Bath

3 Nonlinear Analysis and its Applications to Differential Equations M.R. Grossinho M. Ramos C. Rebelo L. Sanchez Editors Springer Science+Business Media, LLC

4 M.R. Grossinho CMAF University of Lisbon Lisbon Portugal C. Rebelo CMAF University of Lisbon Lisbon Portugal M. Ramos CMAF University of Lisbon Lisbon Portugal L. Sanchez CMAF University of Lisbon Lisbon Portugal Library of Congress Cataloging-in-Publication Data Nonlinear analysis and its applications to differential equations / M.R. Grossinho..[et al.], editors. p. cm. - (Progress in nonlinear differential equations and their applications ; 43) Includes bibliographical references. ISBN ISBN (ebook) DOI / Differential equations. 2. Nonlinear functional analysis. I. Grossinho, M. R. (Maria do Rosano), II. Series. QA372.N '.35-dc CIP AMS Subject Classifications: Primary: 34B15, 34C25, 35J25, 47H10 Secondary: 35B40, 35Q30, 49J40, 58E05, 70G45 Printed on acid-free paper Springer Sdence+Business Media New York Originally published by Birkhäuser Boston in 2001 Softcover reprint of the hardcover 1st edition 2001 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN Reformatted from editors' files by TpXniques, Inc., Cambridge, MA

5 Contents Preface... ix Part 1. Short courses... 1 An Overview of the Method of Lower and Upper Solutions for ODEs C. De Coster, P. Habets... 3 On the Long-time Behaviour of Solutions to the Navier-Stokes Equations of Compressible Flow E. Feireisl Periodic Solutions of Systems with p-laplacian-like Operators J. Mawhin Mechanics on Riemannian Manifolds W.M. Oliva Twist Mappings, Invariant Curves and Periodic Differential Equations R. Ortega Variational Inequalities, Bifurcation and Applications K. Schmitt Part 2. Seminar papers Complex Dynamics in a Class of Reversible Equations F. Alessio, M. Calanchi, E. Serra Symmetry and Monotonicity Results for Solutions of Certain Elliptic PDEs on Manifolds L. Almeida, Y. Ge Nielsen Number and Multiplicity Results for Multivalued Boundary Value Problems J. Andres

6 vi Contents Bifurcation Theory and Application to Semilinear Problems near the Resonance Parameter D. Arcoya, J.L. Gamez Orientation and Degree for Fredholm Maps of Index Zero Between Banach Spaces P. Benevieri On the Method of Upper and Lower Solutions for First Order BVPs A. Cabada, E. Liz, R.L. Pouso Nonlinear Optimal Control Problems for Diffusive Elliptic Equations of Logistic Type A. Canada, J.L. Gamez, J.A. Montero On The Use of Time-Maps in Nonlinear Boundary Value Problems A. Capietto Some Aspects of Nonlinear Spectral Theory P. Drabek Asymmetric Nonlinear Oscillators C. Fabry, A. Fonda Hopf Bifurcation for a Delayed Predator-Prey Model and the Effect of Diffusion T. Faria Galerkin-Averaging Method in Infinite-Dimensional Spaces for Weakly Nonlinear Problems M. Feckan PBVPs for Ordinary Impulsive Differential Equations D. Franco, J.J. Nieto Homoclinic and Periodic Solutions for Some Classes of Second Order Differential Equations M.R. Grossinho, F. Minh6s, S. Tersian Global Bifurcation for Monge-Ampere Operators J. Jacobsen

7 Contents vii Remarks on Boundedness of Semilinear Oscillators M. Kunze The Dual Variational Method in Nonlocal Semilinear Tricomi Problems D. Lupo, K.R. Payne Symmetry Properties of Positive Solutions of Nonlinear Differential Equations Involving the p-laplace Operator F. Pacella A Maximum Principle with Applications to the Forced Sine-Gordon Equation A.M. Robles-Perez Lipschitzian Regularity Conditions for the Minimizing Trajectories of Optimal Control Problems A. V. Sarychev, D.F.M. Torres Abstract Concentration Compactness and Elliptic Equations on Unbounded Domains 1. Schindler, K. Tintarev

8 Preface In this book we present a significant part ofthe material given in an autumn school on "Nonlinear Analysis and Differential Equations," held at the CMAF (Centro de Matematica e Aplica<;6es Fundamentais), University of Lisbon, in September-October The school consisted of short courses and a seminar, thus offering participants a systematic approach to some classes of problems and an opportunity to be in contact with the most recent developments in several areas. Two of the courses presented here deal with ordinary differential equations: one updating the technique of upper and lower solutions, the other dealing with continuation theorems applied to quasi-linear systems (involving operators of p-laplacian type). Both ordinary and partial differential equations appear in the scope of a course on bifurcation for variational inequalities. Another course deals with a geometric approach to dynamical systems in the plane via twist theorems. Two other important and useful topics (although not in the mainstream of the volume) have also been included: asymptotic behavior and periodic solutions for N avier-stokes equations, and a mathematical formulation of mechanics in Riemannian manifolds. The papers originating from the seminar include a range of topics representative of the recent research of participants. With respect to the area of ordinary differential equations, we include papers on spectral theory for the one-dimensional p-laplacian, the use of the time-map in boundary value problems, and boundary value problems for impulsive and discontinuous equations. Nonlinear oscillations occur in the study of asymmetric systems, for which a notion of nonlinear resonance is proposed, and in variational approaches to homo clinics and heteroclinics to periodic solutions. The important problem of boundedness of solutions deserves some attention too. Partial differential equations is a recurrent theme. Thus for equations involving the n-dimensional p-laplacian operator, symmetries of positive solutions are studied. Symmetries are also considered for some PDEs on manifolds. Bifurcation theory is applied to nonlinear elliptic equations and to the Monge-Ampere operator. The dual method is used in the study of Tricomi problems. A telegraph equation is analyzed by means of a maximum principle. An abstract concentration-compactness method is proposed in connection with elliptic equations in n-dimensional space. Degree

9 x Preface and index theories are represented by studies on the Nielsen number, and the degree on Banach manifolds. The calculus of variations and optimal control appear explicitly both in the context of ODEs (Lipschitz regularity of minimizers) and PDEs (optimal control for some elliptic equations of logistic type). The dynamics of delayed systems appears in connection with asymptotic expansions and Hopf bifurcation. All papers have been refereed. It is our pleasure to express our thanks to all the invited lecturers and participants of the autumn school; their presence and their work at the CMAF was the main contribution to the success of the meeting. We believe they had a pleasant and scientifically rewarding time at the University of Lisbon. We warmly thank our colleagues Bento Louro and Luis Trabucho for their valuable help in the preparation of this volume. A special aknowledgement is due to Fundagiio para a Ciencia e a Tecnologia for its sponsorship (as part of a program funding afforded to CMAF). Our gratitude goes also to ClM (Centro lnternacional de Matematica) for its support of the event and publication of this volume. M.R. Grossinho M. Ramos C. Rebelo L. Sanchez Editors

10 Autumn School Nonlinear Analysis and Differential Equations Lisbon, 14 September - 23 October 1998 Scientific Committee: - Djairo DE FIGUEIREDO - Jean MAWHIN - Rafael ORTEGA - Luis SANCHEZ - Fabio ZANOLIN Organizing Committee: - Maria do Rosario GROSSINHO - Miguel RAMOS - Carlota REBELO - Luis SANCHEZ List of Participants: Luis ALMEIDA, CMLA-Cachan, France Doherty ANDRADE, Universidade Estadual de Maringa, Brazil Jan ANDRES, Palaky University, Czech Republic David ARCOYA, Universidad de Granada, Spain Pierluigi BENEVIERI, Universita di Firenze, Italy Bernard BROGLIATO, LAG, UMR CNR-INPG, France Antonio CANADA, Universidad de Granada, Spain Anna CAPIETTO, Universita di Torino, Italy Nicolai CHEMETOV, Universidade de Lisboa, Portugal Marielle CHERPION, Universite Catholique de Louvain, Belgium Monica CONTI, Istituto Politecnico di Milano, Italy Walter DAMBROSIO, Universita di Torino, Italy

11 xii Autumn School Proceedings Colette DE COSTER, Universite du Litoral, France Djairo DE FIGUEIREDO, Universidade de Campinas, Brazil Dan DOBROVOLSCHI, Institute of Meteorology and Hydrology, Romania Ana Rute DOMINGOS, Universidade de Lisboa, Portugal Pavel DRABEK, University of West Bohemia, Czech Republic Teresa FARIA, Universidade de Lisboa, Portugal Michal FECKAN, Comenius University, Slovakia Eduard FEIREISL, Institute of Mathematics, Czech Republic Alessandro FONDA, Universita di Trieste, Italy Daniel FRANCO, Universidad de Santiago de Compostela, Spain Pedro FREITAS, Universidade Tecnica de Lisboa, Portugal Fortios GIANNAKOPOULOS, University of Colonia, Germany Pedro GIRAO, Universidade Tecnica de Lisboa, Portugal Jean-Pierre GOSSEZ, Universite Libre de Bruxelles, Belgium Maria do Rosario GROSSINHO, Universidade Tecnica de Lisboa, Portugal Patrick HABETS, Universite Catholique de Louvain, Belgium Jon JACOBSEN, University of Utah, USA Markus KUNZE, Universitat Koln, Germany Susana LOIS, Universidad de Santiago - Lugo, Spain Rodrigo LOPEZ POUSO, Universidad de Santiago de Compostela, Spain Daniela Lupo, Istituto Politecnico di Milano, Italy To Fu MA, Universidade Estadual de Maringa, Brazil Manuel Monteiro MARQUES, Universidade de Lisboa, Portugal Rogerio MARTINS, Universidade Nova de Lisboa, Portugal Julia MATOS, Universidade de Lisboa, Portugal Jean MAWHlN, Universite Catholique de Louvain, Belgium Feliz MINHOS, Universidade de Evora, Portugal Daniel NUNEZ-LOPEZ, Universidad de Granada, Spain Walter OLIVA, Universidade Tecnica de Lisboa, Portugal Rafael ORTEGA, Universidad de Granada, Spain Filomena PACELLA, Universita degli Study di Roma La Sapienza, Italy Kevin PAYNE, University of Miami, USA Miguel RAMOS, Universidade de Lisboa, Portugal Carlota REBELO, Universidade de Lisboa, Portugal Aureliano ROBLES-PEREZ, Universidad de Granada, Spain Carlos ROCHA, Universidade Tecnica de Lisboa, Portugal Paula C. Patricio RODRIGUES, Universidade Nova de Lisboa, Portugal Luis SANCHEZ, Universidade de Lisboa, Portugal Andrei V. SARYCHEV, Universidade de Aveiro, Portugal Klauss SCHMITT, University of Utah, USA Enrico SERRA, Politecnico di Torino, Italy Boyan SIRAKOV, Universite de Paris 6, France Valentina TADDEI, Italy Massimo TARALLO, Universita degli Studi di Milano, Italy Susanna TERRACINI, Istituto Politecnico di Milano, Italy

12 Autumn School Proceedings xiii Stepan TERSIAN, University of Rousse, Bulgaria Kyril TINTAREV, Uppsala University, Sweden Laura TONEL, Universita di Torino, Italy Delfim TORRES, Universidade de Aveiro, Portugal Christophe TROESTLER, Universite de Mons-Hainaut, Belgium Salvador VILLEGAS, Universidad de Granada, Spain Michel WILLEM, Universite Catholique de Louvain, Belgium Fabio ZANOLIN, Universita di Udine, Italy