李翀（中科院下屬研究院副研究員）

研究方向

非線性泛函分析、變分方法及臨界點(diǎn)理論、偏微分方程

代表論文

Chong Li, Shujie Li, ? The Fu?ík spectrum of Schr?dinger operator and the existence of four solutions of Schr?dinger equations with jumping nonlinearities.

Chong Li, Shujie Li, Gaps of consecutive eigenvalues of Laplace operator and the existence of multiple solutions for superlinear elliptic problem, Journal of Functional Analysis, Vo.271, 2016, 245-263.

Chong Li, Generalized Poincaré-Hopf theorem and application to nonlinear elliptic problem, Journal of Functional Analysis, Vo.267, 2014, 3783-3814.

Chong Li, Shujie Li, Zhaoli, Liu, Bifurcation surfaces stemming from the Fucik spectrum，Journal of Functional Analysis, Vo.263, Issue 12(2012), 4059-4080.

Chong Li, Shujie Li, Jiaquan Liu, Splitting theorem, Poincare-Hopf theorem and jumping nonlinear problems, Journal of Functional Analysis, Vo.221

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Chong Li,Shujie Li, Zhaoli Liu, Jianzhong Pan, On the Fucik spectrum, Journal of Differential Equations, Vo.244, 2008, 2498-2528.

Chong Li, Shujie Li, Zhaoli Liu, Existence of type(II) region and convexity and concavity of potential functional corresponding jumping nonlinear problems, Calculus of Variations and Partial Differential Equations, Vo.32, 2008, 237-251.

Chong Li, The quantitative analysis on the Fucik spectrum and solvability of jumping nonlinear problems, Communications in Contemporary Mathematics, Vo.13, No.2(2011), 293-308.

Chong Li, Some remarks on the critical point theory, Topological Methods in Nonlinear Analysis, Vo.30, No.2, 2007.

Chong Li, The existence of solutions of elliptic equations with Neumann boundary condition for superlinear problems, Acta Mathematica Sinica, English Series, Dec, 2004, Vo.20, No.6, 965-976.

Chong Li, The existence of infinitely many solutions of a class of nonlinear elliptic equations with Neumann boundary condition for both resonance and oscillation problems, Nonlinear Analysis, 54(2003), 431-443.

Chong Li, Shujie Li, Multiple solutions and sign changing solutions of a class of nonlinear elliptic equations with Neumann boundary condition, JMAA, 298(2004), No.1, 14-32.

李翀，李樹杰，關(guān)于臨界點(diǎn)理論的幾個(gè)注記，中國科學(xué)，Vo.46, No.5, 1-9, 2016.

Chong Li, Yanheng Ding, Shujie Li, Multiple solutions of nonlinear elliptic equations for oscillation problems, JMAA, Vo.303, 2005, 477-485.

Chong Li, Shibo Liu, Homology of saddle point reduction and applications to resonant elliptic systems, Nonlinear Analysis, Vo.81(2013), 236-246.

Li Ma, Chong Li, Zhao Lin, Monotone solutions to a class of elliptic and diffusion equations, Communication on Pure and Applied Analysis，Vo.6

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Jing Zhang, Chong Li, Xiaoping, Xue, Nontrivial Solutions for a Class of Quasilinear Problems with Jumping Nonlinearities via a Co-homological Local Splitting, Nonlinear Analysis, Vo.75, Issue 13(2012), 4896–4903.

Aixia Qian, Chong Li, Infinitely Many Solutions for a Robin Boundary Value Problem, International Journal of Differential Equations, Vo. 2010, Article ID 548702, 9 pages.

主要工作

（1）建立C^1光滑的Poincare-Hopf定理；對(duì)無窮維形式在下述方面予以改進(jìn)：（i）將有界Gromoll-Meyer對(duì)替換為泛函水平集對(duì)；（ii）去掉Morse不等式關(guān)于泛函水平集之間只有孤立臨界點(diǎn)的限制； (iii)拓廣著名的Marino-Prodi擾動(dòng)定理；

（2）擺脫經(jīng)典隱函數(shù)定理在開集上可微的限制，證明開集稠子集上的隱函數(shù)定理，這一進(jìn)展導(dǎo)致兩方面突破，一是利用它建立全新的分支定理（包含經(jīng)典的Crandall-Rabinbowitz分支定理），二是利用它得到弱光滑條件下局部Morse理論，特別是減弱著名數(shù)學(xué)家Helmut Hofer相關(guān)工作的泛函光滑性條件。

（3）研究高維情況下Fucík譜基本性質(zhì)，解決人們長期關(guān)注的從Fucík譜分出分支曲面的問題。特別是給出Fucík譜(II)型區(qū)域存在性的一個(gè)充分條件；還證明了Conley同倫指標(biāo)平凡與臨界群平凡之間的等價(jià)關(guān)系，這一等價(jià)關(guān)系對(duì)跳躍非線性問題可解性起至關(guān)重要的作用。

（4）建立一類與Nehari流形有關(guān)的新型環(huán)繞定理并發(fā)現(xiàn)超線性問題的第四個(gè)解與橢圓算子譜間隙的緊密聯(lián)系。

學(xué)術(shù)報(bào)告

7月20日上午，中科院數(shù)學(xué)與系統(tǒng)科學(xué)研究院研究員李樹杰、副研究員李翀?jiān)谖以呵笾獦菙?shù)學(xué)系會(huì)議室作了關(guān)于統(tǒng)計(jì)專業(yè)如何建設(shè)、專業(yè)教師如何開展高水平科學(xué)研究以及Morse理論的精彩報(bào)告。哈爾濱師范大學(xué)數(shù)學(xué)科學(xué)學(xué)院王玉文教授出席并主持了報(bào)告會(huì)，數(shù)學(xué)系全體教師及優(yōu)秀學(xué)生代表參加了報(bào)告會(huì)。

報(bào)告結(jié)束后，李樹杰研究員、李翀副研究員就數(shù)學(xué)系專業(yè)建設(shè)，學(xué)生如何選擇繼續(xù)深造專業(yè)及教師現(xiàn)有的科研項(xiàng)目中存在的問題等方面與參會(huì)教師及學(xué)生進(jìn)行了互動(dòng)交流并作了精彩的回答。兩位研究員的講座，對(duì)我院統(tǒng)計(jì)專業(yè)的建設(shè)，對(duì)我院教師增強(qiáng)科研意識(shí)，提高科研水平具有十分重要的指導(dǎo)意義。