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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