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Contributions to Nonlinear Analysis: A Tribute to D.G. de Figueiredo on the Occasion of his 70th Birthday PDF

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Preview Contributions to Nonlinear Analysis: A Tribute to D.G. de Figueiredo on the Occasion of his 70th Birthday

Progress in Nonlinear Differential Equations and Their Applications Volume 66 Editor Haim Brezis Université Pierre et Marie Curie Paris and Rutgers University New Brunswick, N.J. Editorial Board Antonio Ambrosetti, Scuola Internazionale Superiore di Studi Avanzati, Trieste 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 Mahwin, Université Catholique de Louvain Louis Nirenberg, New York University Lambertus Peletier, University of Leiden Paul Rabinowitz, University of Wisconsin, Madison John Toland, University of Bath Contributions to Nonlinear Analysis A Tribute to D.G. de Figueiredo on the Occasion of his 70th Birthday Thierry Cazenave David Costa Orlando Lopes Raúl Manásevich Paul Rabinowitz Bernhard Ruf Carlos Tomei Editors Birkhäuser Basel Boston Berlin (cid:404) (cid:404) Editors: Thierry Cazenave Paul Rabinowitz Laboratoire Jacques-Louis Lions University of Wisconsin-Madison B.C. 187 Mathematics Department Université Pierre et Marie Curie 480 Lincoln Dr 4, place Jussieu Madison WI 53706-1388, USA 75252 Paris Cedex 05, France e-mail: [email protected] e-mail: [email protected] Bernhard Ruf David Costa Dipartimento di Matematica Department of Mathematical Sciences Università degli Studi University of Nevada Via Saldini 50 Las Vegas, NV 89154-4020, USA 20133 Milano, Italy e-mail: [email protected] e-mail: [email protected] Orlando Lopes Carlos Tomei Instituto de Matemática Departamento de Matemática UNICAMP - IMECC PUC Rio Caixa Postal: 6065 Rua Marquês de São Vicente, 225 13083-859 Campinas, SP, Brasil Edifício Cardeal Leme e-mail: [email protected] Gávea - Rio de Janeiro, 22453-900, Brasil e-mail: [email protected] Raúl Manásevich Departamento de Ingenieria Matemática Facultad de Ciencias Fisicas y Matemáticas Universidad de Chile Casilla 170, Correo 3, Santiago, Chile. e-mail: [email protected] 2000 Mathematics Subject Classifi cation 35, 49, 34 A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografi e; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. ISBN 3-7643-7149-8 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifi cally the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfi lms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2006 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced of chlorine-free pulp. TCF ∞ Printed in Germany ISBN 10: 3-7643-7149-8 e-ISBN: 3-7643-7401-2 ISBN 13: 978-3-7643-7149-4 9 8 7 6 5 4 3 2 1 www.birkhauser.ch Contents Dedication....................................................................ix J. Palis On Djairo de Figueiredo. A Mathematician ..............................xi E.A.M. Abreu, P.C. Carria˜o and O.H. Miyagaki Remarks on a Class of Neumann Problems Involving Critical Exponents .1 C.O. Alves and M.A.S. Souto Existence of Solutions for a Class of Problems in IRN Involving the p(x)-Laplacian ......................................................17 V. Benci and D. Fortunato A Unitarian Approach to Classical Electrodynamics: The Semilinear Maxwell Equations .....................................33 V. Benci, C.R. Grisanti and A.M. Micheletti Existence of Solutions for the Nonlinear Schro¨dinger Equation with V(∞)=0 .........................................................53 R.C. Chara˜o, E. Bisognin, V. Bisognin and A.F. Pazoto Asymptotic Behavior of a Bernoulli–Euler Type Equation with Nonlinear Localized Damping ...........................................67 L. Boccardo T-minima ..............................................................93 S. Bolotin and P.H. Rabinowitz A Note on Heteroclinic Solutions of Mountain Pass Type for a Class of Nonlinear Elliptic PDE’s ................................105 Y. Bozhkov and E. Mitidieri Existence of Multiple Solutions for Quasilinear Equations via Fibering Method ..................................................115 D. Castorina and F. Pacella Symmetry of Solutions of a Semilinear Elliptic Problem in an Annulus .........................................................135 A. Castro and J. Cossio Construction of a Radial Solution to a Superlinear Dirichlet Problem that Changes Sign Exactly Once ......................................149 vi Contents M.M. Cavalcanti, V.N. Domingos Cavalcanti and J.A. Soriano Global Solvability and Asymptotic Stability for the Wave Equation with Nonlinear Boundary Damping and Source Term ..................161 T. Cazenave, F. Dickstein and F.B. Weissler Multiscale Asymptotic Behavior of a Solution of the Heat Equation on RN ................................................................185 F.J.S.A. Corrˆea and S.D.B. Menezes Positive Solutions for a Class of Nonlocal Elliptic Problems ............195 D.G. Costa and O.H. Miyagaki On a Class of Critical Elliptic Equations of Caffarelli-Kohn-NirenbergType .......................................207 Y. Ding and A. Szulkin Existence and Number of Solutions for a Class of Semilinear Schro¨dinger Equations .................................................221 J.M. do O´, S. Lorca and P. Ubilla Multiparameter Elliptic Equations in Annular Domains ................233 C.M. Doria Variational Principle for the Seiberg–Witten Equations ................247 P. Felmer and A. Quaas Some Recent Results on Equations Involving the Pucci’s Extremal Operators .............................................................263 J. Fleckinger-Pell´e, J.-P. Gossez and F. de Th´elin Principal Eigenvalue in an Unbounded Domain and a Weighted Poincar´eInequality ....................................................283 C.L. Frota and N.A. Larkin Uniform Stabilization for a Hyperbolic Equation with Acoustic Boundary Conditions in Simple Connected Domains ...................297 J.V. Goncalves and C.A. Santos Some Remarks on Semilinear Resonant Elliptic Problems ..............313 O. Kavian Remarks on Regularity Theorems for Solutions to Elliptic Equations via the Ultracontractivity of the Heat Semigroup ......................321 Contents vii F. Ammar Khodja and M.M. Santos 2d Ladyzhenskaya–SolonnikovProblem for Inhomogeneous Fluids ......351 Y.Y. Li and L. Nirenberg Generalization of a Well-known Inequality .............................365 D. Lupo, K.R. Payne and N.I. Popivanov Nonexistence of Nontrivial Solutions for Supercritical Equations of Mixed Elliptic-Hyperbolic Type .....................................371 E.S. Medeiros On the Shape of Least-Energy Solutions to a Quasilinear Elliptic Equation Involving Critical Sobolev Exponents ........................391 M. Montenegro and F.O.V. de Paiva A-priori Bounds and Positive Solutions to a Class of Quasilinear Elliptic Equations .....................................................407 A.S. do Nascimento and R.J. de Moura The Role of the Equal-Area Condition in Internal and Superficial Layered Solutions to Some Nonlinear Boundary Value Elliptic Problems ..............................................................415 R.H.L. Pedrosa Some Recent Results Regarding Symmetry and Symmetry-breaking Properties of Optimal Composite Membranes ..........................429 A.L. Pereira and M.C. Pereira Generic Simplicity for the Solutions of a Nonlinear Plate Equation .....443 J.D. Rossi An Estimate for the Blow-up Time in Terms of the Initial Data ........465 B. Ruf Lorentz Spaces and Nonlinear Elliptic Systems ........................471 N.C. Saldanha and C. Tomei The Topology of Critical Sets of Some Ordinary Differential Operators .491 P.N. Srikanth and S. Santra A Note on the Superlinear Ambrosetti–Prodi Type Problem in a Ball ..505 Dedication This volume is dedicated to Djairo G. de Figueiredo on the occasion of his 70th birthday. In January 2003 David Costa, Orlando Lopes and Carlos Tomei, colleagues and friends of Djairo, invited us to join the organizing committee for a Workshop on Nonlinear Differential Equations, sending us the following message: Djairo’scareerisaremarkableexamplefortheBrazilian community.Weareproud of his mathematical achievements and his ability to develop so many successors, through systematicdedication toresearch, advising activities and academic orches- tration. Djairo is always organizing seminars and conferences and is constantly willing to help individuals and the community. It is about time that he should enjoy a meeting without having to work for it. How true! Of course we all accepted with great enthusiasm. The workshop took place in Campinas, June 7–11, 2004. It was a wonderful conference, with the participation of over 100 mathematicians from all over the world. The wide range of research interests of Djairo is reflected by the articles in this volume.Throughtheircontributions,theauthorsexpresstheirappreciation,grat- itude and friendship to Djairo. We are happy that another eminent Brazilian mathematician, Jacob Palis from IMPA, has accepted our invitation to give an appreciation of Djairo’s warm per- sonality and his excelling work. The editors: Thierry Cazenave David Costa Rau´l Mana´sevich Orlando Lopes Paul Rabinowitz Bernhard Ruf Carlos Tomei

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