研究领域

研究方向：非线性泛函分析，变分方法及临界点理论，偏微分方程

主要工作： （1）建立C^1光滑的Poincare-Hopf定理；对无穷维形式在下述方面予以改进：（i）将有界Gromoll-Meyer对替换为泛函水平集对；（ii）去掉Morse不等式关于泛函水平集之间只有孤立临界点的限制；(iii)拓广著名的Marino-Prodi扰动定理； （2）摆脱经典隐函数定理在开集上可微的限制，证明开集稠子集上的隐函数定理，这一进展导致两方面突破，一是利用它建立全新的分支定理（包含经典的Crandall-Rabinbowitz分支定理），二是利用它得到弱光滑条件下局部Morse理论，特别是减弱著名数学家HelmutHofer相关工作的泛函光滑性条件。 （3）研究高维情况下Fucík谱基本性质，解决人们长期关注的从Fucík谱分出分支曲面的问题。特别是给出Fucík谱(II)型区域存在性的一个充分条件；还证明了Conley同伦指标平凡与临界群平凡之间的等价关系，这一等价关系对跳跃非线性问题可解性起至关重要的作用。 （4）建立一类与Nehari流形有关的新型环绕定理并发现超线性问题的第四个解与椭圆算子谱间隙的紧密联系。

近期论文

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1.ChongLi,ShujieLi,GapsofconsecutiveeigenvaluesofLaplaceoperatorandtheexistenceofmultiplesolutionsforsuperlinearellipticproblem,JournalofFunctionalAnalysis,Vo.271,2016,245-263. 2.ChongLi,GeneralizedPoincaré-Hopftheoremandapplicationtononlinearellipticproblem,JournalofFunctionalAnalysis,Vo.267,2014,3783-3814. 3.ChongLi,ShujieLi,Zhaoli,Liu,BifurcationsurfacesstemmingfromtheFucikspectrum，JournalofFunctionalAnalysis,Vo.263,Issue12(2012),4059-4080. 4.ChongLi,ShujieLi,JiaquanLiu,Splittingtheorem,Poincare-Hopftheoremandjumpingnonlinearproblems,JournalofFunctionalAnalysis,Vo.221,2005,No.2,439-455. 5.ChongLi,ShujieLi,ZhaoliLiu,JianzhongPan,OntheFucikspectrum,JournalofDifferentialEquations,Vo.244,2008,2498-2528. 6.ChongLi,ShujieLi,ZhaoliLiu,Existenceoftype(II)regionandconvexityandconcavityofpotentialfunctionalcorrespondingjumpingnonlinearproblems,CalculusofVariationsandPartialDifferentialEquations,Vo.32,2008,237-251. 7.ChongLi,ThequantitativeanalysisontheFucikspectrumandsolvabilityofjumpingnonlinearproblems,CommunicationsinContemporaryMathematics,Vo.13,No.2(2011),293-308. 8.ChongLi,Someremarksonthecriticalpointtheory,TopologicalMethodsinNonlinearAnalysis,Vo.30,No.2,2007. 9.ChongLi,TheexistenceofsolutionsofellipticequationswithNeumannboundaryconditionforsuperlinearproblems,ActaMathematicaSinica,EnglishSeries,Dec,2004,Vo.20,No.6,965-976. 10.ChongLi,TheexistenceofinfinitelymanysolutionsofaclassofnonlinearellipticequationswithNeumannboundaryconditionforbothresonanceandoscillationproblems,NonlinearAnalysis,54(2003),431-443. 11.ChongLi,ShujieLi,MultiplesolutionsandsignchangingsolutionsofaclassofnonlinearellipticequationswithNeumannboundarycondition,JMAA,298(2004),No.1,14-32. 12.李翀，李树杰，关于临界点理论的几个注记，中国科学，Vo.46,No.5,1-9,2016. 13.ChongLi,YanhengDing,ShujieLi,Multiplesolutionsofnonlinearellipticequationsforoscillationproblems,JMAA,Vo.303,2005,477-485. 14.ChongLi,ShiboLiu,Homologyofsaddlepointreductionandapplicationstoresonantellipticsystems,NonlinearAnalysis,Vo.81(2013),236-246. 15.LiMa,ChongLi,ZhaoLin,Monotonesolutionstoaclassofellipticanddiffusionequations,CommunicationonPureandAppliedAnalysis，Vo.6,2007,No.1,237-246. 16.JingZhang,ChongLi,Xiaoping,Xue,NontrivialSolutionsforaClassofQuasilinearProblemswithJumpingNonlinearitiesviaaCo-homologicalLocalSplitting,NonlinearAnalysis,Vo.75,Issue13(2012),4896–4903. 17.AixiaQian,ChongLi,InfinitelyManySolutionsforaRobinBoundaryValueProblem,InternationalJournalofDifferentialEquations,Vo.2010,ArticleID548702,9pages.