1 Mathematical Background ................................................ 1 1.1 Sobolev Spaces ..................................................... 2 1.1.1 Basic Definitions and Properties ............................. 2 1.1.2 Embedding Theorems ........................................... 6 1.1.3 Poincare Inequality .......................................... 7 1.1.4 Dual Space ................................................... 8 1.1.5 Green Formula ................................................ 9 1.1.6 One Dimensional Sobolev Spaces .............................. 10 1.2 Set-Valued Analysis ............................................... 14 1.2.1 Upper and Lower Semicontinuity .............................. 15 1.2.2 h-Lower and h-Upper Semicontinuity .......................... 18 1.2.3 Measurability of Multifunctions ............................. 21 1.2.4 Measurable Selections ....................................... 22 1.2.5 Continuous Selections ....................................... 24 1.2.6 Convergence in the Kuratowski Sense ......................... 28 1.3 Nonsmooth Analysis ................................................ 32 1.3.1 Convex Functions ............................................ 32 1.3.2 Fenchel Transform ........................................... 38 1.3.3 Subdifferential of Convex Functions ......................... 42 1.3.4 Clarke Subdifferential ...................................... 48 1.3.5 Weak Slope .................................................. 62 1.4 Nonlinear Operators ............................................... 67 1.4.1 Compact Operators ........................................... 68 1.4.2 Maximal Monotone Operators .................................. 70 1.4.3 Yosida Approximation ........................................ 81 1.4.4 Pseudomonotone Operators .................................... 83 1.4.5 Nemytskii Operators ......................................... 87 1.4.6 Ekeland Variational Principle ............................... 89 1.5 Elliptic Differential Equations ................................... 93 1.5.1 Ordinary Differential Equations ............................. 93 1.5.2 Partial Differential Equations .............................. 99 1.5.3 Regularity Results ......................................... 112 1.6 Remarks .......................................................... 120 2 Critical Point Theory ................................................ 123 2.1 Locally Lipschitz Functionals .................................... 123 2.1.1 Compactness-Type Conditions ................................ 123 2.1.2 Critical Points and Deformation Theorem .................... 128 2.1.3 Linking Sets ............................................... 136 2.1.4 Minimax Principles ......................................... 138 2.1.5 Existence of Multiple Critical Points ...................... 145 2.2 Constrained Locally Lipschitz Punctionals ........................ 149 2.2.1 Critical Points of Constrained Functions ................... 149 2.2.2 Deformation Theorem ........................................ 151 2.2.3 Minimax Principles ......................................... 154 2.3 Perturbations of Locally Lipschitz Functionals ................... 159 2.3.1 Critical Points of Perturbed Functions ..................... 159 2.3.2 Generalized Deformation Theorem ............................ 162 2.3.3 Minimax Principles ......................................... 166 2.4 Local Linking and Extensions ..................................... 171 2.4.1 Local Linking .............................................. 171 2.4.2 Minimax Principles ......................................... 181 2.4.3 Palais-Smale-Type Conditions ............................... 184 2.5 Continuous Functionals ........................................... 187 2.5.1 Compactness-Type Conditions ................................ 187 2.5.2 Deformation Theorem ........................................ 188 2.5.3 Minimax Principles ......................................... 195 2.6 Multivalued Functionals .......................................... 197 2.6.1 Compactness-Type Conditions ................................ 197 2.6.2 Multivalued Deformation Theorem ............................ 199 2.6.3 Minimax Principles ......................................... 200 2.7 Remarks .......................................................... 203 3 Ordinary Differential Equations ...................................... 207 3.1 Dirichlet Problems ............................................... 208 3.1.1 Formulation of the Problem ................................. 208 3.1.2 Approximation of the Problem ............................... 212 3.1.3 Existence Results .......................................... 225 3.1.4 Problems with Non-Cbnvex Nonlinearities .................... 231 3.2 Periodic Problems ................................................ 233 3.2.1 Auxiliary Problems ......................................... 233 3.2.2 Formulation of the Problem ................................. 239 3.2.3 Approximation of the Problem ............................... 240 3.2.4 Existence Results .......................................... 249 3.2.5 Problems with Non-Convex Nonlinearities .................... 258 3.2.6 Scalar Problems ............................................ 259 3.3 Nonlinear Boundary Conditions .................................... 268 3.4 Variational Methods .............................................. 283 3.4.1 Existence Theorems ......................................... 284 3.4.2 Homoclinic Solutions ....................................... 308 3.4.3 Scalar Problems ............................................ 317 3.4.4 Multiple Periodic Solutions ................................ 343 3.4.5 Nonlinear Eigenvalue Problems .............................. 353 3.4.6 Problems with Nonlinear Boundary Conditions ................ 356 3.4.7 Multiple Solutions for "Smooth" Problems ................... 365 3.5 Method of Upper and Lower Solutions .............................. 372 3.6 Positive Solutions and Other Methods ............................. 389 3.6.1 Positive Solutions ......................................... 389 3.6.2 Method Based on Monotone Operators ......................... 403 3.7 Hamiltonian Inclusions ........................................... 414 3.8 Remarks .......................................................... 448 4 Elliptic Equations ................................................... 453 4.1 Problems at Resonance ............................................ 454 4.1.1 Semilinear Problems at Resonance ........................... 455 4.1.2 Nonlinear Problems at Resonance ............................ 474 4.1.3 Variational-Hemivariational Inequality at Resonance ........ 485 4.1.4 Strongly Resonant Problems ................................. 495 4.2 Neumann Problems ................................................. 500 4.2.1 Spectrum of ( -Δp, W1,p(Ω)) ................................. 501 4.2.2 Homogeneous Neumann Problems ............................... 516 4.2.3 Nonhomogeneous Neumann Problem ............................. 526 4.3 Problems with an Area-Type Term .................................. 536 4.4 Strongly Nonlinear Problems ...................................... 558 4.5 Method of Upper and Lower Solutions .............................. 586 4.5.1 Existence of Solutions ..................................... 587 4.5.2 Existence of Extremal Solutions ............................ 598 4.6 Multiplicity Results ............................................. 608 4.6.1 Semilinear Problems ........................................ 608 4.6.2 Nonlinear Problems ......................................... 640 4.7 Positive Solutions ............................................... 661 4.8 Problems with Discontinuous Nonlinearities ....................... 683 4.9 Remarks .......................................................... 700 A Appendix ............................................................. 707 A.1 Set Theory and Topology ............................................ 707 A.2 Measure Theory ..................................................... 714 A.3 Functional Analysis ................................................ 718 A.4 Nonlinear Analysis ................................................. 724 List of Symbols ........................................................ 727 References ............................................................. 735 Index .................................................................. 763