一类椭圆方程与椭圆系统的解的存在性

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^YjUYdAbstractThevariationalmethodisanapproximatecalculationmethodwhichbasedonthevariationalprinciple.Itisoneoftheimportantmethodsincomputationalmechanics,andaneffectiv

3、etooltosolvetheproblemofmechanicsandotherareas.ThestudyobjectofVariationalisextremumproblemofthefunctional.Thevariationalprinci-pleisavariationalformrepresentationsofphysicallaws,anditdoesnotcontainnewphysicalcontent.Itsvalueismainlyusedasthenewstartingpointforthestructure

4、ofclassicalmechanicstheory.Inotherwords,inallmotionstateofsatisfycertaincon-straints,actualmotionstateshouldmakesomephysicalquantitytheextremevalue.Inmechanics,optics,quantummechanicsandothersubjects,ithasthecorrespondingvariationalprinciple.Therearesomeimportantvariationa

5、lprinciplessuchasFermat’sprinciple,Hamiltonianprinciple,theprincipleofminimumenergyandminimumresid-ualenergyprinciple.Thevariationalmethodisapproximationcalculationmethodinthefieldofmechanics,physicsandotherdisciplines.Inthispaper,thedeformationofthefountaintheoremandfounta

6、intheorem,westudiedwithsuperlinearandasymptoticallylinearellipticequationsandsuperquadraicellipticsystemsandgivesomesufficientconditionsofexistencesofsolutions.Meanwhileweapplythemainresultstotheexistenceofsolutionsforboundaryvalueproblems.Thethesisisdividedintothreechapters

7、accordingtocontents.Inchapter1,itintroducetheoriginandsettingofvariationalmethod.Inchapter2,basedonavariantfountaintheoremestablishedbyW.Zou,westudythefollowingellipticboundaryvalueproblem−∆u=µf(x,u)inΩ,u=0on∂Ω,whereµ∈(0,∞),Ω⊂RN(N>2)isboundeddomainwithsmoothboundary,fisodd

8、inuandcontinuous.Forsuperlinearcase,insharpcontrasttotheexistingresultsintheliterature,wedonotmakethewell-knownAmbrosetti-Rabinowitztypetechnicalconditionnearinfinityandnotmakeanyassumptionsnearzeroonthebehaviorsofthenonlinearityf,weobtaintheexistenceandmultiplicityofsoluti

9、ons.Wealsoconsiderthesituationwherefisasymptoticallylinearwithassumptionsdifferentfromthos