This monograph presents fundamental methods and topics in nonlinear analysis and their efficient application to nonlinear boundary value problems for elliptic equa- tions. The book is divided into 12 chapters, with9chapters covering the theoretical material-Sobolev spaces, nonlinear operators,nonsmooth analysis,degree theory, variational principles and critical point theory,Morse theory,bifurcation theory,reg- ularity theorems and maximum principles, and spectrum of differential operators- followed by three chapters containing applications to ordinary differential equations and nonlinear elliptic equations with Dirichlet or Neumann boundary conditions. The last three chapters,but not only those, consist to a large extent of original results due to the authors, and many of these results appear here in a novel form, with significant improvements and developments. We emphasize that the first nine chapters devoted to general theories are notjust acollection of relevant tools tostudy the nonlinear boundary value problems considered in the last three chapters. They offer broad and essential insight into powerful abstract theories. Major objectives for us have been to make a self-contained presentation for every treated subject and show that it applies to different types of problems.
This book originated in the collaboration of the three authors that gave rise during a period of about 10years to a series of research papersstudying nonlinearboundary value problems with Dirichlet and Neumann boundary conditions and having in the differential part Laplacian,p-Laplacian,or, more generally,even nonhomogeneous differential operators. These papers are reflected in our book,although the initial results are mostly rewritten, revised, and sharpened in the text here.A distinct feature of our work is that we combine various methods such as nonlinear operator theories,degree theory, lower and upper solutions, variational methods, Morse theory, regularity, maximum principles, and spectral theory. For instance,this can be seen in the study of multiple solutions, where every solution is usually obtained through a different approach and method.The material in our book mainly addresses researchers in pure and applied mathematics, physics, mechanics,and engineering. It is also accessible to graduate students in mathematical and applied sciences, who will find updated information and a systematic exposition of important parts of modern mathematics.
The authors are deeply indebted to Dr. Lucas Fresse for his immense and generous help related to the whole work for the present book. We have decisively benefited from his brilliant ideas and insight. For instance,the version of the first deformation theorem and its application to the limit case in the minimax principle,Lemma 6.65, which is helpful in our presentation of Morse theory, the general version of the Moser iteration procedure,and the unified formulation of the local minimizer principle given in Theorem 12.18 are due to him.His outstanding conributions in improving every chapter of our book are gratefully acknowledged. The authors express their gratitude to Springer Science+Business Media,LLC, NewYork,for its highly professional assistance,and first of all we thankour editors Vaishali Damle,Eve Mayer,and Marc Strauss for strong moral support and kind understanding.
The author Viorica Venera Motreanu acknowledges with thanks the support of Marie Curie Intra-European Fellowship for Career Development within the European Community's Seventh Framework Programme(Grant Agreement No. PIEF-GA-2010-274519). Dumitru Motreanu Perpignan, France

1Sobolev Spaces 1

1.1Sobolev Spaces 1

1.2Remarks 13

2Nonlinear Operators 15

2.1Compact Operators 15

2.2 Operators of Monotone Type 23

2.3 Nemytskii Operators 41

2.4 Remarks 43

Nonsmooth Analysis 45

3.1Convex Analysis. 45

3.2Locally Lipschitz Functions50

3.3Remarks 58

4 Degree Theory 61

4.1Brouwer's Degree 61

4.2 Leray-Schauder Degree 70

4.3 Degree for Operators of Monotone Type 76

4.4 Remarks 96

5Variational Principles and Critical Point Theory 97

5.1Ekeland Variational Principle 97

5.2 Critical Points and Deformation Theorems 104

5.3 Minimax Theorems for Critical Points 116

5.4 Critical Points for Functionals with Symmetries 127

5.5 Generalizations 132

5.6 Remarks 138

6Morse Theory 141

6.1Elements of Algebraic Topology 141

6.2Critical Groups 152

6.3 Morse Relations 156

6.4Computation of Critical Groups 167

6.5Remarks 178

7Bfurcation Theory 181

7.1Bifurcation Theory 181

7.2Remarks200

8 Regularity Theorems and Maximum Principles 201

8.1Regularity of Solutions 201

8.2 Maximum Principles and Comparison Results 210

8.3Remarks 222

9Spectrum of Differential Operators 223

9.1Spectrum of the Laplacian 232

9.2Spectrum of p-Laplacian 234

9.3 Spectrum of p-Laplacian Plus an Indefinite Potential 254

9.4FuČík Spectrum 264

9.5Remarks 269

10Ordinary Differential Equations 271

10.1 Nonlinear Periodic Problems 271

10.2Nonsmooth Periodic Systems 285

10.3Remarks 301

11Nonlinear Elliptic Equations with Dirichlet Boundary Conditions 303

11.1 Nonlinear Dirichlet Problems Using Degree Theory 303

11.2 Nonlinear Dirichlet Problems Using Variational Methods 321

11.3 Nonlinear Dirichlet Problems Using Morse Theory 347

11.4 Nonlinear Dirichlet Problems Using Nonlinear Operator Theory 366

11.5 Remark 383

12 Nonlinear Eliptic Equations with Neumann Boundary Conditions 387

12.1Nonlinear Neumann Problems Using Variational Methods387

12.2 Nonlinear Neumann Problems with Nonhomogeneous

Differential Operators 403

12.3 Sublinear and Superlinear Neumann Problems 418

12.4 RemarkS 434

List of Symbols 437

Reference 441

Index 457

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