Items related to Variational, Topological, and Partial Order Methods...
Synopsis
1 Preliminaries.- Sobolev spaces and embedding theorems.- Critical point.- Cone and partial order.- Brouwer Degree.- Compact map and Leray-Schauder Degree.- Fredholm operators.- Fixed point index.- Banach's Contract Theorem, Implicit Functions Theorem.- Krein-Rutman theorem.- Bifurcation theory.- Rearrangements of sets and functions.- Genus and Category.- Maximum principles and symmetry of solution.- Comparison theorems.- 2 Cone and Partial Order Methods.- Increasing operators.- Decreasing operators.- Mixed monotone operators.- Applications of mixed monotone operators.- Further results on cones and partial order methods.- 3 Minimax Methods.- Mountain Pass Theorem and Minimax Principle.- Linking Methods.- Local linking Methods.- 4 Bifurcation and Critical Point.- Introduction.- Main results with parameter.- Equations without the parameter.- 5 Solutions of a Class of Monge-Ampère Equations.- Introduction.- Moving plane argument.- Existence and non-existence results.- Bifurcation and the equation with a parameter.- Appendix.- 6 Topological Methods and Applications.- Superlinear system of integral equations and applications.- Existence of positive solutions for a semilinear elliptic system.- 7 Dancer-Fučik Spectrum.- The spectrum of a self-adjoint operator.- Dancer-Fučik Spectrum on bounded domains.- Dancer-Fučik point spectrum on RN.- Dancer-Fučik spectrum and asymptotically linear elliptic problems.- 8 Sign-changing Solutions.- Sign-changing solutions for superlinear Dirichlet problems.- Sign-changing solutions for jumping nonlinear problems.- 9 Extension of Brezis-Nirenberg's Results and Quasilinear Problems.- Introduction.- W01,p(Ω) versus C01(Ω) local minimizers.- Multiplicity results for the quasilinear problems.- Uniqueness results.- 10 Nonlocal Kirchhoff Elliptic Problems.- Introduction.- Yang index and critical groups to nonlocal problems.- Variational methods and invariant sets of descent flow.- Uniqueness of solution for a class of Kirchhoff-type equations.- 11 Free Boundary Problems, System of equations for Bose-Einstein Condensate and Competing Species.- Competing system with many species.- Optimal partition problems.- Schrödinger systems from Bose-Einstein condensate.- Bibliography.
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About the Author
Zhitao Zhang is a professor at Academy of Mathematics and Systems Science. He was a Humboldt Research Fellow at Giessen University from June 2004 to June 2006. In 2011, he received a excellent advisor award of the Chinese Academy of Sciences. He mainly studies Nonlinear Analysis, Partial differential equations, Integral equations etc. He has published almost 50 papers in very important international journals such as Ann. Inst. H. Poincare Anal. Non Lineaire, J. Funct. Anal., Transactions of American Mathematical Society, J. Differential Equations etc. He has invited to write Chapter 13 for the Handbook of Nonconvex Analysis and Applications, International Press, 2010. His researches have been supported by Key program of National Natural Foundation of China. His papers have been cited by others more than 500 times. He was invited many times to present lectures at important international conferences.
"About this title" may belong to another edition of this title.
- PublisherSpringer
- Publication date2012
- ISBN 10 3642307108
- ISBN 13 9783642307102
- BindingPaperback
- LanguageEnglish
- Number of pages344
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