Nonlinear Elliptic Partial Differential Equations: Workshop in Celebration of Jean-pierre Gossez's 65th Birthday September 2-4, 2009, Universite Libre De Bruxelles, Belgium PDF
Preview Nonlinear Elliptic Partial Differential Equations: Workshop in Celebration of Jean-pierre Gossez's 65th Birthday September 2-4, 2009, Universite Libre De Bruxelles, Belgium
C ONTEMPORARY M ATHEMATICS 540 Nonlinear Elliptic Partial Differential Equations Workshop in Celebration of Jean-Pierre Gossez's 65th Birthday September 2–4, 2009 Université Libre de Bruxelles, Belgium Denis Bonheure Mabel Cuesta Enrique J. Lami Dozo Peter Takácˇ Jean Van Schaftingen Michel Willem Editors American Mathematical Society Nonlinear Elliptic Partial Differential Equations This page intentionally left blank C ONTEMPORARY M ATHEMATICS 540 Nonlinear Elliptic Partial Differential Equations Workshop in Celebration of Jean-Pierre Gossez's 65th Birthday September 2–4, 2009 Université Libre de Bruxelles, Belgium Denis Bonheure Mabel Cuesta Enrique J. Lami Dozo Peter Takácˇ Jean Van Schaftingen Michel Willem Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Abel Klein Martin J. Strauss 2000 Mathematics Subject Classification. Primary 35B50, 35J15, 35J20, 35J25, 35J60, 35J70, 35P30, 46E35,49Q20, 58J05. Library of Congress Cataloging-in-Publication Data Workshop in Nonlinear Elliptic Partial Differential Equations (2009 : Universit´e Libre de Bruxelles) Nonlinear elliptic partial differential equations : Workshop in Nonlinear Elliptic Partial DifferentialEquationsincelebrationofJean-PierreGossez’s65thbirthday,September2–4,2009, Universit´eLibredeBruxelles,Belgium/DenisBonheure...[etal.],editors. p.cm. —(Contemporarymathematics;v.540) Includesbibliographicalreferencesandindex. ISBN978-0-8218-4907-1(alk.paper) 1.Differentialequations,Elliptic–Congresses. 2.Differentialequations,Nonlinear–Congresses. I.Gossez,J.-P.,1943– II.Bonheure,Denis. III.Title. QA377.W675 2009 515(cid:2).3533—dc22 2010049092 Copying and reprinting. Materialinthisbookmaybereproducedbyanymeansfor edu- cationaland scientific purposes without fee or permissionwith the exception ofreproduction by servicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercialuseofmaterialshouldbeaddressedtotheAcquisitionsDepartment,AmericanMath- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can [email protected]. Excludedfromtheseprovisionsismaterialinarticlesforwhichtheauthorholdscopyright. In suchcases,requestsforpermissiontouseorreprintshouldbeaddresseddirectlytotheauthor(s). (Copyrightownershipisindicatedinthenoticeinthelowerright-handcornerofthefirstpageof eacharticle.) (cid:2)c 2011bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. Copyrightofindividualarticlesmayreverttothepublicdomain28years afterpublication. ContacttheAMSforcopyrightstatusofindividualarticles. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 161514131211 Contents Preface vii Jean-Pierre Gossez’s short bio ix List of Participants xi Partial differential equations also have principles : maximum and antimaximum Jean Mawhin 1 On the Fuˇc´ık Spectrum for equations with symmetries Bernhard Ruf 15 Variations on the p-Laplacian Bernd Kawohl 35 Extremal functions in Poincar´e–Sobolev inequalities for functions of bounded variation Vincent Bouchez and Jean Van Schaftingen 47 An elementary proof of an inequality of Maz’ya involving L1 vector fields Pierre Bousquet and Petru Mironescu 59 Homoclinic type solutions for a class of differential equations with periodic coefficients David G. Costa and Chengyue Li 65 Quasilinear and singular systems: the cooperative case Jacques Giacomoni, Jesu´s Herna´ndez, and Abdelkrim Moussaoui 79 Manifolds of critical points in a quasilinear model for phase transitions Pavel Dra´bek, Rau´l F. Mana´sevich, and Peter Taka´cˇ 95 Weighted asymmetric problems for an indefinite elliptic operator Liamidi Leadi and Humberto Ramos Quoirin 135 Multiple non-trivial solutions of the Dirichlet problem for the prescribed mean curvature equation Franco Obersnel and Pierpaolo Omari 165 Limits as p(x) → ∞ of p(x)-harmonic functions with non-homogeneous Neumann boundary conditions Mayte Perez-LLanos and Julio D. Rossi 187 v vi CONTENTS Bifurcation and decay of solutions for a class of elliptic equations on RN C. A. Stuart 203 Existence of nodal solutions for some nonlinear elliptic problems S´ebastien de Valeriola and Michel Willem 231 Admissible Q−curvatures under isometries for the conformal GJMS operators Fr´ed´eric Robert 241 Preface A workshop celebrating Jean-Pierre Gossez’s 65th anniversary was organized onSeptember2,3and4,2009. Thismeetingwasattendedbyaround80youngand seniorresearchers. Asusualinsuchoccasions,itwasmainly agoodopportunityto gatheroldfriendsandfortheyoungesttomakenewcontacts. Theatmospherewas warm and very friendly, fruitful for an excellent workshop with beautiful lectures and many interesting and inspiring scientific discussions. The 16 lecturers were HenriBerestycki,DjairoG.deFigueiredo, Fran¸coisdeTh´elin,PavelDra´bek,Jesu´s Hern´andez, Bernd Kawohl, Pierre-Louis Lions, Jean Mawhin, Petru Mironescu, Mark Peletier, Fr´ed´eric Robert, Julio Daniel Rossi, Bernhard Ruf, Michael Struwe, Charles Stuart and Juan Luis V´azquez. We would like to express our gratitude to the participants and the lecturers, to the persons who helped in the organization of the meeting and to the follow- ing institutions that supported the conference : Fonds de la recherche scientifique (F.R.S.-FNRS), Gouvernement de la R´egion de Bruxelles-Capitale, Acad´emie Uni- versitaire Wallonie-Bruxelles, European Science Foundation, International Mathe- maticalUnion,InternationalSolvayInstitutesBrussels, Facult´eSOCO,Facult´edes Sciences and Solvay Brussels School of Economics and Management. We offer our thanks to the contributors to this volume and to the AMS for giving us the possibility to publish it in the Contemporary Mathematics Series. The efficient help of Christine Thivierge was greatly appreciated. The Editors vii This page intentionally left blank Jean-Pierre Gossez’s short bio Jean-PierreGossezwasborninMons(Belgium)the28thofNovember1943. He graduatedattheUniversit´eLibredeBruxelles(ULB)in1965inMathematicalSci- ences, thenspent threeyearsinParis underthe supervisionof Jacques-Louis Lions and took his Ph.D. at the ULB in 1969 under the direction of Th´eophile Lepage. Later he was successively postdoctoral researcher at the Fonds National de la Recherche Scientifique, research associate at the University of Chicago and charg´e de cours at the ULB. Since 1979, he has been Professeur ordinaire at the ULB where he taught analysis and general mathematics to undergraduate students and the theory of partial differential equations at the graduate level. He is Professeur ´em´erite at the ULB since October 1st, 2009. His research is mainly concerned with the theory of elliptic PDEs. At first, he studied monotone operators in Orlicz-Sobolev spaces and their applications to nonlinear boundary value problems. Then he took an active interest in nonlinear resonanceproblems. Heconsiderednonresonanceconditionsatvariouseigenvalues, problems with asymmetric nonlinearities, the Fuˇcik spectrum,... for the Laplacian and the p-Laplacian. Fifteen years ago, he got interested in the antimaximum principle, first for the p-Laplacian, then for semilinear and quasilinear problems withweights. Hisresearchontheantimaximumprincipleleadhimtoconsidersome minimax formulas for the principal eigenvalues, and more recently he investigated non selfadjoint elliptic problems. HehasbeeninvolvedincooperationprogrammeswithNorth-AfricaandLatin- America. He is member of the Commission for Development and Exchange and of the committee “Developping Countries Strategic Group” of the International Mathematical Union. ix
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