Preface
Contents
List of symbols, theorems, definitions, assumptions, examples 9
List of symbols 9
List of theorems 11
List of definitions 11
List of assumptions 12
List of examples 12
0 Introduction 14
1 Preliminaries 26
1.1 The domain Omega 26
1.2 Function spaces 26
1.3 Caratheodory functions, Nemytskij (superposition) operators 28
1.4 Function spaces (continued) 29
1.5 Weighted Sobolev spaces 31
1.6 Leray-Lions theorem 40
1.7 Degree of mappings of monotone type 41
1.8 Hamack-type inequality, decay of solution, local regularity and interpolation inequality 44
1.9 Some technical lemmas 46
2 Solvability of nonlinear boundary value problems 48
2.1 Formulation of the problem 48
2.2 Second order equations (bounded domains) 50
2.3 Second order equations (proof of Theorem 2.1) 65
2.4 Second order equations (unbounded domains) 74
2.5 Higher order equations (growth conditions) 78
2.6 Higher order equations (operator representation) 82
2.7 Higher order equations (degree of the mapping T) 90
2.8 Higher order equations (existence results) 98
2.9 Examples, remarks, comments 103
3 The degenerated p-Laplacian on a bounded domain 119
3.1 Basic notation 119
3.2 Existence of the least eigenvalue of the homogeneous eigenvalue problem 121
3.3 Existence of the least eigenvalue of the nonhomogeneous eigenvalue problem 131
3.4 Maximum principle for degenerated (singular) equations 141
3.5 Positive solutions of degenerated (singular) BVP 144
3.6 Bifurcation from the least eigenvalue 152
4 The p-Laplacian in RN 169
4.1 Nonlinear eigenvalue problem 169
4.2 Bifurcation problem for the p-Laplacian in RN 177
4.3 Bifurcation problem for the perturbed p-Laplacian in RN 197
Bibliography 218
Index 224