劉兆理教授長期從事臨界點理論及其套用研究，系統地發展了下降流不變集方法，率先將這一新方法成功地套用於非線性橢圓型微分方程理論，在解的存在性、解的個數、變號解的個數、變號解nodal域的個數等方面獲得了一批新結果，解決了多個疑難問題。出版專著一部，在Adv. Math., Comm. Math. Phy., J. Fun.Anal.，Math. Zeit. JDE., Ann. l'Inst. H. Poincaré-Anal. non Linéaire，Comm. PDE等SCI核心期刊上發表學術論文多篇，其研究成果已被他人引用200多次，國際上有多位數學家，包括多位科學院院士，在他們的工作中採用劉兆理教授的研究方法，並稱劉兆理教授的工作是開創性工作。
劉兆理教授長期從事臨界點理論及其套用研究，系統地發展了下降流不變集方法，率先將這一新方法成功地套用於非線性橢圓型微分方程理論，在解的存在性、解的個數、變號解的個數、變號解nodal域的個數等方面獲得了一批新結果，解決了多個疑難問題。出版專著一部，在Adv. Math., Comm. Math. Phy., J. Fun.Anal.，Math. Zeit. JDE., Ann. l'Inst. H. Poincaré-Anal. non Linéaire，Comm. PDE等SCI核心期刊上發表學術論文多篇，其研究成果已被他人引用200多次，國際上有多位數學家，包括多位科學院院士，在他們的工作中採用劉兆理教授的研究方法，並稱劉兆理教授的工作是開創性工作。
Positive solutions of a nonlinear Schrödinger system with nonconstant potentialsHaidong Liu · Zhaoli Liu
Existence and uniqueness of positive solutions of nonlinear Schrödinger systemsHaidong Liu · Zhaoli Liu · Jinyong Chang
Infinitely many solutions of p-sublinear p-Laplacian equationYongtao Jing · Zhaoli Liu
Multi-bump solutions and multi-tower solutions for equations on $/mathbb R^N$ (with Lishan Lin) Journal of Functional Analysis, to appear.
Solutions of elliptic problems with nonlinearities of linear growth (with Jiabao Su and Zhi-Qiang Wang) Calculus of Variations and Partial Differential Equations, to appear.
Multi-bump solutions for a semilinear Schroedinger equation (with Lishan Lin and Shaowei Chen) Indiana University Mathematics Journal, to appear.
Sign changing solutions with clustered layers near the origin for singularly perturbed semilinear elliptic problems on a ball (with Yihong Du, Angela Pistoia, and Shusen Yan) Methods and Applications of Analysis, 15(2008), 137-148. 獲獎
A twist condition and periodic solutions of Hamiltonian systems (with Jiabao Su and Zhi-Qiang Wang) Advances in Mathematics, 218(2008), 1895-1913.
Multiple bound states of nonlinear Schrodinger systems (with Zhi-Qiang Wang) Communications in Mathematical Physics, 282(2008), 721-731.
Nodal solutions of a perturbed elliptic problem (with Yi Li and Chun-Shan Zhao) Topological Methods in Nonlinear Analysis, 32(2008), 49-68.
On the Fucik spectrum (with Chong Li, Shujie Li, and Jianzhong Pan) Journal of Differential Equations, 244(2008), 2498-2528.
Existence of type (II) regions and convexity and concavity of potential functionals corresponding to jumping nonlinear problems (with Chong Li and Shujie Li) Calculus of Variations and Partial Differential Equations, 32(2008), 237-251.
Sign-changing solutions of nonlinear elliptic equations (with Zhi-Qiang Wang) Frontiers of Mathematics in China, 3(2008), 221-238
Bounded resonance problem for semilinear elliptic equations (with Jia-Bao Su) Discrete and Continuous Dynamical Systems, 19(2007), 431-445
Multiplicity of solutions for some elliptic equations involving critical and supercritical Sobolev exponents (with Shu-Jie Li) Topological Methods in Nonlinear Analysis, 28(2006), 235-262
Multiple solutions of nonlinear Schroedinger equations via flow invariance and Morse theory (with Zhi-Qiang Wang and Tobias Weth) Proceedings of the Royal Society of Edinburgh, 136A(2006), 945-969
Compactness results for Schroedinger equations with asymptotically linear terms (with Jia-Bao Su and Tobias Weth) Journal of Differential Equations, 231(2006) 501–512.
Nodal type bound states of Schroedinger equations via invariant set and minimax methods (with Francois A. van Heerden and Zhi-Qiang Wang) Journal of Differential Equations, 214(2005), 358-390.
Nodal solutions of a superlinear elliptic $p$-Laplacian equation (with Thomas Bartsch and Tobias Weth) Proceedings of the London Mathematical Society, 91(2005), 129-152.
Multi-bump type nodal solutions having a prescribed number of nodal domains: I (with Zhi-Qiang Wang) Annales de l'Institut Henri Poincare-Analyse Non Lineaire, 22(2005), 597-608.
Multi-bump type nodal solutions having a prescribed number of nodal domains: II (with Zhi-Qiang Wang) Annales de l'Institut Henri Poincare-Analyse Non Lineaire, 22(2005), 609-631.
An elliptic problem with jumping nonlinearities (with Jing-Xian Sun) Nonlinear Analysis, Theory, Method and Application, 63(2005), 1070-1082.
Schroedinger equations with concave and convex nonlinearities (with Zhi-Qiang Wang) Zeitschrift fur Angewandte Mathematik und Physik, 56(2005), 609-629.
Solutions of some nonlinear elliptic problems with perturbation terms of arbitrary growth (with Jia-Bao Su) Discrete and Continuous Dynamical Systems, 10(2004), 617-634.
Location and critical groups of critical points with an application to nonlinear eigenvalue problems (with Thomas Bartsch) Advances in Differential Equations, 9(2004), 645-676.
Multiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the $p$-Laplacian (with Thomas Bartsch) Communications in Contemporary Mathematics, 6(2004), 245-258.
On a superlinear elliptic $p$-Laplacian equation (with Thomas Bartsch) Journal of Differential Equations, 198(2004), 149-175.
Sign-changing solutions of superlinear Schr/"odinger equations (with Thomas Bartsch and Tobias Weth) Communications in Partial Differential Equations, 29(2004), 25-42.
Existence of a positive solution of an elliptic equation on $/mathbb R^N$ (with Zhi-Qiang Wang) Proceedings of the Royal Society of Edinburgh, 134A(2004), 191-200.
On the Ambrosetti-Rabinowitz superlinear condition (with Zhi-Qiang Wang) Advanced Nonlinear Studies, 4(2004), 563-574.
Nontrivial solutions for perturbed $p$-Laplacian on ${/cal R}^N$ (with Jia-Bao Su) Mathematische Nachrichten,248-249(2003), 190-199.
Contractibility of the level sets of the functionals associated with some elliptic boundary value problems and applications (with Shu-Jie Li) Nonlinear Differential Equations and Applications, 10(2003), 133-170.
Localized critical points in Banach spaces and sign changing solutions of nonlinear $p$-Laplacian equations in: Topology Methods, Variational Methods and Their Applications---ICM 2002 Satellite Conference on Nonlinear Functional Analysis, ed. H. Brezis, K.C. Chang, S.J. Li and P. Rabinowitz, World Scientific Co., New Jersey, 2003, pp 151-158.
Number of invariant sets of descending flow with applications in critical point theory (with Jing-Xian Sun) in: Morse Theory, Minimax Theory and Their Applications to Nonlinear Differential Equations-Lectures at Morningside Center of Mathematics, ed. H. Brezis, S.J. Li, J.Q. Liu, and P.H. Rabinowitz, International Press, New York, 2003, pp 139--156.
Multiplicity of solutions of semilinear elliptic boundary value problems with jumping nonlinearities at zero Nonlinear Analysis, Theory, Method and Application, 48(2002), 1051-1063.
Four versus two solutions of semilinear elliptic boundary value problems (with Jing-Xian Sun) Calculus of Variations and Partial Differential Equations, 14(2002), 319-327.
Positive solutions of a class of nonlinear elliptic eigenvalue problems Mathematische Zeitschrift, 242(2002), 663-686.
Perturbations from symmetric elliptic boundary value problems (with Shu-Jie Li) Journal of Differential Equations, 185(2002), 271-280.
A class of two-point boundary value problems (with Xue-Mei Zhang) Journal of Mathematical Analysis and Applications, 254(2001), 599-617.
Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations (with Jing-Xian Sun) Journal of Differential Equations, 172(2001), 257-299.
Properties of positive solutions of a nonlinear eigenvalue problem (with Yong-Qing Li) Zeitschrift fur Angewandte Mathematik und Physik, 52(2001), 433-447.
Exact number of solutions of a class of two-point boundary value problems involving concave and convex nonlinearities Nonlinear Analysis, Theory, Method and Application, 46(2001), 181-197.
Positive solutions of elliptic boundary value problems without (P.S.) type assumption (with Shu-Jie Li and Zhi-Qiang Wang) Indiana University Mathematical Journal, 50(2001), 1347-1369.
Solutions of an elliptic eigenvalue problem involving subcritical and critical exponents (with Yong-Qing Li) Communications in Partial Differential Equations, 26(2001), 2227-2248.
On the existence of positive solutions of a elliptic boundary value problem (with Fu-Yi Li) Chinese Annals of Mathematics, 21B(2000), 499-510.
On the existence of periodic solutions for a nonlinear system of ordinary differential equations, Acta Mathematica Sinica, English version: 16(2000), 505-514; Chinese version: 43(2000), 947-954.
Multiple and sign-changing solutions of an elliptic eigenvalue problem with constraint (with Yong-Qing Li) Sciences in China, Chinese version: 30(2000), 967-975. English version: 44(2001), 48-57.
Positive solutions of superlinear elliptic equations Journal of Functional Analysis, 167(1999), 370-398.
Multiple positive solutions of some nonlinear operator equations and applications (with Fu-Yi Li) Acta Mathematica Sinica, 41(1998), 97-102 (in Chinese).
指導學生
Multiple positive solutions of nonlinear two-point boundary value problems (with Fu-Yi Li) Journal of Mathematical Analysis and Applications, 203(1996), 610-625.
Periodic solutions for nonlinear $n$th-order ordinary differential equations Journal of Mathematical Analysis and Applications, 204(1996), 46-64.
Methods of Functional Analysis in Nonlinear Ordinary Differential Equations (with Da-Jun Guo, Jing-Xian Sun) Shandong Science and Technology Press, 1995, Jinan, 247pp (in Chinese).
Calculus of variations and super- and sub-solutions in reversed order (with Jing-Xian Sun) Acta Mathematica Sinica, 37(1994), 512-514, (in Chinese).
Chong Li, Shujie Li, Zhaoli Liu, Bifurcation surfaces stemming from the Fucik spectrum，Journal of Functional Analysis, Vo.263, Issue 12(2012), 4059-4080.
2011年11月3日上午，首都師範大學數學科學學院數學系主任、博士生導師、教育部長江學者特聘教授、國家傑出青年基金獲得者、北京市特聘教授劉兆理教授應邀到數信學院講學，為數學系部分教師和研究生作了一場精彩的學術報告。劉教授的報告題為《Multibump and multitower solutions of nonlinear Schrodinger equations》，主要介紹了橢圓型偏微分方程的多包解和多塔解的存在性，認為這類解在相應的極限方程中是不存在的。
獲獎
指導學生
