Hardy不等式與非線性橢圓型雙調(diào)和奇性方程的Dirichlet問題.pdf

1、華南理工大學(xué)碩士學(xué)位論文Hardy不等式與非線性橢圓型雙調(diào)和奇性方程的Dirichlet問題姓名：熊輝申請學(xué)位級別：碩士專業(yè)：應(yīng)用數(shù)學(xué)指導(dǎo)教師：沈堯天20030501IIAbstractTherearethreemaintopicsstudiedinthepresentthesisThefirstistheHatdyinequalities，includingitsclassification，theproofoftheoptimali

2、tyoftheirconstantsandth出applicationsThesecondandthirdtopicsarerespectivelyweaklysingularproblemandstronglysingularproblemFollowinginproperorderandadvancinggraduallytheauthorfoundsseveralHardyinequalitiesfirst，whichcreate

3、theprerequisiteforexistenceandnonexistenceofnontrivialsolutionsofsingularproblemWithdiff口entsingularitytheauthorstudiestheappearanceanddisappearingofcriticaldimensionsforbiharmonicequationsTheHardyinequalitiesarefoundedi

4、nspace掣andclassifiedbytheorderoftheoperatorsWegiveeveryinequalityallaJileandprovetheoptimalEyoftheconstants，andweprovidethecorrespondingreachingfunctionsforcompletionFurthermore，weapplythemtononlinearellipticequationsWea

5、lsoanalysistheremaindertermsoftheoptimalHardyinequalities，andbyaddinglowordereven4linear”護(hù)norm，wemakethe；nequalitiesimproved，whichprovideamethodtodealwiththeproblemsofcriticaldimensionsandweaklycriticaldimensionsforpolyh

6、armonicoperatorsSolweknowitisonlytheoptimalconstantsthatcannotbeimprovedbutnottheinequalitiesthemselvesThetwosemilinearbiharmouicDirichletproblemstudiedaresolvedbymountainpasstheoremandEkeland’svariationalprincipleWhenPi

7、slowerthanthecriticalexponentand。superlinearweprovethere鋤atleasttwopositivesolutions缸1，t‘2∈W學(xué)p(Q)suchthatj(“1)0J(t上2)Astothesecondproblem，inthespaceW22(n)，withthehelpofinterp枷inequalityweprovethereexistsatlesstonenontriv

8、ialsolutionU∈W20(n)inthecaseofexactlylinearyWealsostudythreesemilinearsingularproblems，0neiswithsubcriticalexponentandcriticalpotential，anotheri8withsubcriticalpotentialandcriticalexponentlthethirdiswithsubcriticalexpone

9、ntandcriticalpotentialinvolvinglogarithmByusingPohozaevidentityforsingularsolutionsinweightedSobolevspace，wegetanonexistenceresult，Andbyusingconcentratecornpactnessprinciple，wegettwodifferentresultsofexistenceofpositives