Nonlinear Elliptic Partial Differential Equations: Workshop in Celebration of Jean-pierre Gossez's 65th Birthday September 2-4, 2009, Universite Libre De Bruxelles, Belgium

278 Pages·2011·3.817 MB·English

by  Denis Bonheure, Mabel Cuesta, Enrique J. Lami Dozo, Peter Takac, Jean Van Schaftingen (ed.)

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
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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
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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
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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