The mountain pass theorem: variants, generalizations, and some applications

383 Pages·2003·2.119 MB·English

by  Youssef Jabri

P1:FCH/FFX P2:FCH/FFX QC:FCH/FFX T1:FCH
0521827213cInd-01 CB577/Jabri-v1.cls June27,2003 20:16
This page intentionally left blank
P1:FCH/FFX P2:FCH/FFX QC:FCH/FFX T1:FCH
CB577/Jabri-FM CB577/Jabri-v1.cls July24,2003 10:15
TheMountainPassTheorem
Variationalmethodsareverypowerfultechniquesinnonlinearanalysisandareextensively
usedinmanydisciplinesofpureandappliedmathematics(includingordinaryandpartial
differentialequations,mathematicalphysics,gaugetheory,andgeometricalanalysis).
Thisbookpresentsmin-maxmethodsthroughacomprehensivestudyofthedifferent
facesofthecelebratedMountainPassTheorem(MPT)ofAmbrosettiandRabinowitz.The
readerisgentlyledfromthemostaccessibleresultstotheforefrontofthetheory,andat
eachstepinthiswalkbetweenthehills,theauthorpresentstheextensionsandvariantsof
theMPTinacompleteandunifiedway.Coverageincludesstandardtopics:theclassicaland
dualMPT;second-orderinformationfrom(PS)sequences;symmetryandtopologicalindex
theory;perturbationsfromsymmetry;convexity;andmore.Butitalsocoversothertopics
covered nowhere else in book form: the nonsmooth MPT; the geometrically constrained
MPT; numerical approaches to the MPT; and even more exotic variants. Each chapter
hasasectionwithsupplementarycommentsandbibliographicalnotes,andthereisarich
bibliographyandadetailedindextoaidthereader.Thebookissuitableforresearchersand
graduatestudents.Nevertheless,thestyleandthechoiceofthematerialmakeitaccessible
toallnewcomerstothefield.
i
P1:FCH/FFX P2:FCH/FFX QC:FCH/FFX T1:FCH
CB577/Jabri-FM CB577/Jabri-v1.cls July24,2003 10:15
ii
P1:FCH/FFX P2:FCH/FFX QC:FCH/FFX T1:FCH
CB577/Jabri-FM CB577/Jabri-v1.cls July24,2003 10:15
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
FOUNDINGEDITORG.-C.ROTA
EditorialBoard
R.Doran,P.Flajolet,M.Ismail,T.-Y.Lam,E.Lutwak
95 TheMountainPassTheorem
40 N.White(ed.) MatroidApplications
41 S.Sakai OperatorAlgebrasinDynamicalSystems
42 W.Hodges BasicModelTheory
43 H.StahlandV.Totik GeneralOrthogonalPolynomials
45 G.DaPratoandJ.Zabczyk StochasticEquationsinInfiniteDimensions
46 A.Bjo¨rneretal. OrientedMatroids
47 G.EdgarandL.Sucheston StoppingTimesandDirectedProcesses
48 C.Sims ComputationwithFinitelyPresentedGroups
49 T.Palmer BanachAlgebrasandtheGeneralTheoryof*-AlgebrasI
50 F.Borceux HandbookofCategoricalAlgebraI
51 F.Borceux HandbookofCategoricalAlgebraII
52 F.Borceux HandbookofCategoricalAlgebraIII
53 V.F.Kolchin RandomGraphs
54 A.KatokandB.Hasselblatt IntroductiontotheModernTheoryofDynamicalSystems
55 V.N.Sachkov CombinatorialMethodsinDiscreteMathematics
56 V.N.Sachkov ProbabilisticMethodsinDiscreteMathematics
57 P.M.Cohn SkewFields
58 R.Gardner GeometricTomography
59 G.A.Baker,Jr.,andP.Graves-Morris PadeApproximants,2ed
60 J.Krajicek BoundedArithmetic,PropositionalLogicandComplexityTheory
61 H.Groemer GeometricApplicationsofFourierSeriesandSphericalHarmonics
62 H.O.Fattorini InfiniteDimensionalOptimizationandControlTheory
63 A.C.Thompson MinkowskiGeometry
64 R.B.BapatandT.E.S.Raghavan NonnegativeMatriceswithApplications
65 K.Engel SpernerTheory
66 D.Cvetkovic,P.Rowlinson,andS.Simic EigenspacesofGraphs
67 F.Bergeron,G.Labelle,andP.Leroux CombinatorialSpeciesandTree-LikeStructures
68 R.GoodmanandN.Wallach RepresentationsandInvariantsoftheClassicalGroups
69 T.Beth,D.Jungnickel,andH.Lenz DesignTheoryI,2ed
70 A.PietschandJ.Wenzel OrthonormalSystemsforBanachSpaceGeometry
71 G.E.Andrews,R.Askey,andR.Roy SpecialFunctions
72 R.Ticciati QuantumFieldTheoryforMathematicians
73 M.Stern SemimodularLattices
74 I.LasieckaandR.Triggiani ControlTheoryforPartialDifferentialEquationsI
75 I.LasieckaandR.Triggiani ControlTheoryforPartialDifferentialEquationsII
76 A.A.Ivanov GeometryofSporadicGroups1
77 A.Schinzel PolynomialswithSpecialRegardtoReducibility
78 H.Lenz,T.Beth,andD.Jungnickel DesignTheoryII,2ed
79 T.Palmer BanachAlgebrasandtheGeneralTheoryof*-AlgebrasII
80 O.Stormark Lie’sStructuralApproachtoPDESystems
81 C.F.DunklandY.Xu OrthogonalPolynomialsofSeveralVariables
82 J.P.Mayberry TheFoundationsofMathematicsintheTheoryofSets
83 C.Foiasetal. Navier–StokesEquationsandTurbulence
84 B.PolsterandG.Steinke GeometriesonSurfaces
85 R.B.ParisandD.Kaminski AsymptoticsandMellin–BarnesIntegrals
86 R.McEliece TheTheoryofInformationandCoding,2ed
87 B.Magurn AlgebraicIntroductiontoK-Theory
88 T.Mora SystemsofPolynomialEquationsI
89 K.Bichteler StochasticIntegrationwithJumps
90 M.Lothaire AlgebraicCombinatoricsonWords
91 A.A.IvanovandS.V.Shpectorov GeometryofSporadicGroupsII
92 P.McMullenandE.Schulte AbstractRegularPolytopes
93 G.Gierzetal. ContinuousLatticesandDomains
iii
P1:FCH/FFX P2:FCH/FFX QC:FCH/FFX T1:FCH
CB577/Jabri-FM CB577/Jabri-v1.cls July24,2003 10:15
iv
P1:FCH/FFX P2:FCH/FFX QC:FCH/FFX T1:FCH
CB577/Jabri-FM CB577/Jabri-v1.cls July24,2003 10:15
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
The Mountain Pass Theorem
Variants,GeneralizationsandSomeApplications
YOUSSEF JABRI
UniversityofOujda,Morocco
v
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , United Kingdom
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521827218
© Youssef Jabri 2003
This book is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2003
- 
isbn-13 978-0-511-06267-4 eBook (NetLibrary)
- 
isbn-10 0-511-06267-2 eBook (NetLibrary)
- 
isbn-13 978-0-521-82721-8 hardback
- 
isbn-10 0-521-82721-3 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of
s for external or third-party internet websites referred to in this book, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
P1:FCH/FFX P2:FCH/FFX QC:FCH/FFX T1:FCH
CB577/Jabri-FM CB577/Jabri-v1.cls July24,2003 10:15
Tothememoryofmymother
vii
P1:FCH/FFX P2:FCH/FFX QC:FCH/FFX T1:FCH
CB577/Jabri-FM CB577/Jabri-v1.cls July24,2003 10:15
viii