Contents
1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Sobolev Spaces and Embedding Theorems . . . . . . . . . . . . . 1
1.2 CriticalPoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Cone andPartialOrder . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 BrouwerDegree . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Compact Map and Leray–Schauder Degree . . . . . . . . . . . . . 13
1.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5.2 Properties of Compact Maps . . . . . . . . . . . . . . . . 14
1.5.3 The Leray–Schauder Degree . . . . . . . . . . . . . . . . 15
1.6 Fredholm Operators . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7 FixedPoint Index . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.8 Banach’s Contract Theorem, Implicit Functions Theorem . . . . . 20
1.9 Krein–Rutman Theorem . . . . . . . . . . . . . . . . . . . . . . . 20
1.10 Bifurcation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.11 Rearrangements of Sets and Functions . . . . . . . . . . . . . . . 23
1.12 Genus and Category . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.13 MaximumPrinciples andSymmetryofSolution . . . . . . . . . . 27
1.14 Comparison Theorems . . . . . . . . . . . . . . . . . . . . . . . . 28
2 Cone and Partial Order Methods . . . . . . . . . . . . . . . . . . . . 35
2.1 IncreasingOperators . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 DecreasingOperators . . . . . . . . . . . . . . . . . . . . . . . . 45
2.3 Mixed Monotone Operators . . . . . . . . . . . . . . . . . . . . . 54
2.4 Applications of Mixed Monotone Operators . . . . . . . . . . . . 74
2.5 Further Results on Cones and Partial Order Methods . . . . . . . . 84
3 Minimax Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.1 Mountain Pass Theorem and Minimax Principle . . . . . . . . . . 99
3.2 Linking Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.3 Local Linking Methods . . . . . . . . . . . . . . . . . . . . . . . 103
3.3.1 DeformationLemmas . . . . . . . . . . . . . . . . . . . . 104
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3.3.2 The Three Critical Points Theorem for Functionals
Bounded Below . . . . . . . . . . . . . . . . . . . . . . . 107
3.3.3 Super-quadratic Functionals . . . . . . . . . . . . . . . . . 110
3.3.4 Asymptotically Quadratic Functionals . . . . . . . . . . . 112
3.3.5 Applications to Elliptic Boundary Value Problems . . . . . 116
3.3.6 Local Linking and Critical Groups . . . . . . . . . . . . . 121
4 Bifurcation and Critical Point . . . . . . . . . . . . . . . . . . . . . . 131
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.2 MainResultswithParameter . . . . . . . . . . . . . . . . . . . . 133
4.3 Equations Without the Parameter . . . . . . . . . . . . . . . . . . 141
5 Solutions of a Class of Monge–Ampère Equations . . . . . . . . . . . 143
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.2 MovingPlaneArgument . . . . . . . . . . . . . . . . . . . . . . . 145
5.3 Existence and Non-existence Results . . . . . . . . . . . . . . . . 149
5.4 BifurcationandtheEquationwithaParameter . . . . . . . . . . . 153
5.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6 Topological Methods and Applications . . . . . . . . . . . . . . . . . 175
6.1 Superlinear System of Integral Equations and Applications . . . . 175
6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.1.2 Existence of Non-trivial Solutions . . . . . . . . . . . . . . 175
6.1.3 Application to Two-Point Boundary Value Problems . . . . 185
6.2 Existence of Positive Solutions for a Semilinear Elliptic System . . 186
6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.2.2 Existence of Positive Solutions . . . . . . . . . . . . . . . 189
7 Dancer–Fuˇcik Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.1 The Spectrum of a Self-adjoint Operator . . . . . . . . . . . . . . 199
7.2 Dancer–Fuˇcik Spectrum on Bounded Domains . . . . . . . . . . . 200
7.3 Dancer–Fuˇcik Point Spectrum on RN . . . . . . . . . . . . . . . . 204
7.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 204
7.3.2 The Trivial Part of the Fuˇcik Point Spectrum . . . . . . . . 205
7.3.3 Non-trivial Fuˇcik Eigenvalues by Minimax Methods . . . . 208
7.3.4 Some Properties of the First Curve and the Corresponding
Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . 212
7.4 Dancer–Fuˇcik Spectrum and Asymptotically Linear Elliptic
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
7.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 215
7.4.2 Proofs of Main Theorems . . . . . . . . . . . . . . . . . . 217
8 Sign-Changing Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.1 Sign-Changing Solutions for Superlinear Dirichlet Problems . . . . 221
8.1.1 Nehari Manifold and Sign-Changing Solutions . . . . . . . 221
8.1.2 Additional Properties of Sign-Changing Solutions to
Superlinear Elliptic Equations . . . . . . . . . . . . . . . . 226
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8.2 Sign-Changing Solutions for Jumping Nonlinear Problems . . . . . 231
8.2.1 On Limit Equation of Lotka–Volterra Competing System
with Two Species . . . . . . . . . . . . . . . . . . . . . . 231
8.2.2 On General Jumping Nonlinear Problems . . . . . . . . . . 235
8.2.3 Sign-Changing Solutions of p-LaplacianEquations . . . . 244
8.2.4 Sign-Changing Solutions of Schrödinger Equations . . . . 246
9 Extension of Brezis–Nirenberg’s Results and Quasilinear Problems . 249
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
9.2 W
1,p
0 () Versus C1
0 ( ¯ ) Local Minimizers . . . . . . . . . . . . . 251
9.3 Multiplicity Results for the Quasilinear Problems . . . . . . . . . 253
9.4 Uniqueness Results . . . . . . . . . . . . . . . . . . . . . . . . . 267
10 Nonlocal Kirchhoff Elliptic Problems . . . . . . . . . . . . . . . . . . 271
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
10.2 Yang Index and Critical Groups to Nonlocal Problems . . . . . . . 272
10.3 Variational Methods and Invariant Sets of Descent Flow . . . . . . 278
10.4 Uniqueness of Solution for a Class of Kirchhoff-Type Equations . . 282
11 Free Boundary Problems, System of Equations for Bose–Einstein
Condensate and Competing Species . . . . . . . . . . . . . . . . . . . 285
11.1 Competing System with Many Species . . . . . . . . . . . . . . . 285
11.1.1 Existence and Uniqueness of Positive Solution . . . . . . . 285
11.1.2 The Limit Spatial Segregation System of Competing
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
11.2 Optimal Partition Problems . . . . . . . . . . . . . . . . . . . . . 291
11.2.1 An Optimal Partition Problem Related to Nonlinear
Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . 291
11.2.2 An Optimal Partition Problem for Eigenvalues . . . . . . . 295
11.3 Schrödinger Systems from Bose–Einstein Condensate . . . . . . . 298
11.3.1 Existence of Solutions for Schrödinger Systems . . . . . . 300
11.3.2 The Limit State of Schrödinger Systems . . . . . . . . . . 310
11.3.3 Cα Estimateof theSolutionsofParabolicSystems . . . . . 316
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
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