Zhitao ZHANG

Professor, Humboldt Research Fellow

     Office:

Room 523, Si Yuan Building 

     Telephone:

+86-10-62651296

     E-Mail:

zzt@math.ac.cn

    Research Interests：Critical point theory and nonlinear partial differential equations, Dynamics of

                                       Lotka-Volterra competition systems, Fixed points theory of nonlinear operators.

    Research Summary:

• Study dynamics of Lotka-Volterra competition systems with large interaction, prove there is no periodic solutions as the interaction is large.

• Construct the pseudo-gradient vector field in W01,p , and prove existence of at least one positive solution, one negative solution, and one sign-changing solution to p-Laplacian equations.

• Existence of multiple sulutions and sign-chaning solutions to asymptotically linear or superlinear problems, or under resonance with Fucik spectrum.

• Ambrosetti-Prodi problem in $R^2$ with one-sided critical growth (exponential growth).

• Existence and Non-existence of Solutions to Elliptic Equations related to the Caffarelli-Kohn-Nirenberg Inequalities.

• Prove a local minimizer of functional J in the C1-topology is still a local minimizer of J in W01,p-topology.

• Sign-changing and multiple solutions of Kirchhoff Type problems.

• Use Nehari manifold to extend the Anti-maximum principle of Laplacian operator to an existence theorem for p-Laplacian (p≠2).

• Use a combination of critical point theory and bifurcation arguments to obtain some exact multiplicity results as well as properties of solutions on each branch. Especially for super-linear problems, under some assumptions on f(u)，we get existence of 3 sign-changing solutions.

• Obtain existence and uniqueness theorems of fixed points for mixed monotone operators, some applications to nonlinear integral equations on unbounded regions and differential equations in Banach spaces are given.

• Normal cone is an important concept in Nonlinear Functional Analysis. A new necessary and sufficient condition is obtained for a normal cone.

• Use Coincide degree theory to impulsive differential equations, and get existence of periodic solutions.

  Representative Research Works:

• Zhitao Zhang, Shujie Li, On sign-changing and multiple solutions for p-Laplacian. Journal of Functional Analysis 197(2), 447-468, 2003.(SCI)

• T. Bartsch, S. Peng, Zhitao Zhang, Existence and Non-existence of Solutions to the Elliptic Equations Related to Caffarelli-Kohn-Nirenberg Inequalities. Calculus of Variations and PDE, 30, 113-136, 2007(SCI).

• Zhitao Zhang, B. Ruf, M. Calanchi, Elliptic equations in R2 with one-sided exponential growth. Comm. in Contemporary Mathematics 6(6), (2004), 947-971.(SCI)

• Zhitao Zhang, J.Chen, S. Li, Construction of pseudo-gradient vector field and sign-changing multiple solutions involving p-Laplacian, J. Differential Equations, 201(2), (2004), 287-303.(SCI)

• K. Perera, Zhitao Zhang, Nontrivial solutions of Kirchhoff type problems via the Yang index, J. Differential Equations, 221(1), (2006), 246-255. (SCI)

• E. N. Dancer, Zhitao Zhang, Dynamics of Lotka Volterra competition systems with large interaction. Journal of Differential Equations,182(2), (2002), 470--489 .(SCI)

• Zhitao Zhang, X. Li, Multiple solutions and sign-changing solutions for semilinear elliptic boundary problems with nonzero reaction at zero. Journal of Differential Equations, 178(2), (2002), 298-313.(SCI)

• Shujie Li, Zhitao Zhang, Fucik spectrum, sign-changing and multiple solutions theorems for semilinear elliptic boundary value problems with jumping nonlinearities at zero and infinity. Science in China (A) 44(7), (2001), 856-866. (SCI)

• Z. Guo, Zhitao Zhang, W 1,p versus C1 local minimizers and multiplicity results for quasilinear elliptic equations. J. Math. Anal. Appl. 286(1), (2003), 32-50.(SCI)

• Zhitao Zhang, New fixed points of mixed monotone operators and applications. J. Math. Anal. Appl., 204(1),1996, 307-319.(SCI)