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 Ɣ Ɣ Editors: Thierry Cazenave Laboratoire Jacques-Louis Lions B.C 187 Université Pierre et Marie Curie 4, place Jussieu 75252 Paris Cedex 05, France e-mail: cazenave@ccr.jussieu.fr David Costa Department of Mathematical Sciences University of Nevada Las Vegas, NV 89154-4020, USA e-mail: costa@unlv.edu Orlando Lopes Instituto de Matemática UNICAMP - IMECC Caixa Postal: 6065 13083-859 Campinas, SP, Brasil e-mail: lopes@ime.unicamp.br Raúl Manásevich Departamento de Ingenieria Matemática Facultad de Ciencias Fisicas y Matemáticas Universidad de Chile Casilla 170, Correo 3, Santiago, Chile e-mail: manasevi@dim.uchile.cl Paul Rabinowitz University of Wisconsin-Madison Mathematics Department 480 Lincoln Dr Madison WI 53706-1388, USA e-mail: rabinowi@math.wisc.edu Bernhard Ruf Dipartimento di Matematica Università degli Studi Via Saldini 50 20133 Milano, Italy e-mail: ruf@mat.unimi.it Carlos Tomei Departamento de Matemática PUC Rio Rua Marquês de São Vicente, 225 Edifício Cardeal Leme Gávea - Rio de Janeiro, 22453-900, Brasil e-mail: carlos@mat.puc-rio.br 2000 Mathematics Subject Classification 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 Nationalbibliografie; detailed bibliographic data is available in the Internet at 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, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms 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 987654321 www.birkhauser.ch Contents Dedication ix J Palis On Djairo de Figueiredo A Mathematician xi E.A.M Abreu, P.C Carri˜ ao and O.H Miyagaki Remarks on a Class of Neumann Problems Involving Critical Exponents 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 Schră odinger Equation with V () = 53 R.C Char˜ ao, 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-Nirenberg Type 207 Y Ding and A Szulkin Existence and Number of Solutions for a Class of Semilinear Schră odinger Equations 221 ´ S Lorca and P Ubilla J.M O, 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´e Inequality 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–Solonnikov Problem 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 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’s career is a remarkable example for the Brazilian community We are proud of his mathematical achievements and his ability to develop so many successors, through systematic dedication to research, advising activities and academic orchestration 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 Through their contributions, the authors express their appreciation, gratitude 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 personality and his excelling work The editors: Thierry Cazenave David Costa Ra´ ul Man´ asevich Orlando Lopes Paul Rabinowitz Bernhard Ruf Carlos Tomei ... choices: the dualistic standpoint and the unitarian standpoint In the dualistic standpoint the matter is described as a field ψ which is the source of the electromagnetic field (E, H) and it is itself... a world figure, suggested to Djairo to get a PhD in probability and statistics Actually, Elon Lima, also one of our world figures, tells me that he had the occasion to detect Djairo’s talent some... colleague Geraldo Avila, as advised by Elon Lima to the founder of it, Darcy Ribeiro In 1965 he returned to the United States This time, he went to the University of Wisconsin and right after to the University