Quasilinear Elliptic Equations with Degenerations and Singularities

✍ Scribed by Pavel Drábek; Alois Kufner; Francesco Nicolosi

• Publisher: De Gruyter
• Year: 1997
• Tongue: English
• Leaves: 232
• Series: De Gruyter Series in Nonlinear Analysis and Applications; 5
• Category: Library

✦ Synopsis

"The book is very well written and will be a valuable source for specialists in differential equations, functional analysis, operator theory, and mathematical physics, as well as for non-specialists who want to get an idea of this fascinating field of contemporary nonlinear analysis. The authors have achieved a double goal. On the one hand, they provide a self-contained account which describes the state-of-the-art of nonlinear elliptic equations with degenerations and singularities; on the other, their book will certainly stimulate further research in this field. The reviewer regards the second aspect as at least as important as the first one: in fact, in his opinion a book should open a new field, rather than close it." Zentralblatt für Mathematik

✦ Table of Contents

List of symbols, theorems, definitions, assumptions, examples
List of symbols
List of theorems
List of definitions
List of assumptions
List of examples
0 Introduction
1 Preliminaries
1.1 The domain Ω
1.2 Function spaces
1.3 Caratheodory functions, Nemytskij (superposition) operators
1.4 Function spaces (continued)
1.5 Weighted Sobolev spaces
1.6 Leray-Lions theorem
1.7 Degree of mappings of monotone type
1.8 Harnack-type inequality, decay of solution, local regularity and interpolation inequality
1.9 Some technical lemmas
2 Solvability of nonlinear boundary value problems
2.1 Formulation of the problem
2.2 Second order equations (bounded domains)
2.3 Second order equations (proof of Theorem 2.1)
2.4 Second order equations (unbounded domains)
2.5 Higher order equations (growth conditions)
2.6 Higher order equations (operator representation)
2.7 Higher order equations (degree of the mapping T)
2.8 Higher order equations (existence results)
2.9 Examples, remarks, comments
3 The degenerated p-Laplacian on a bounded domain
3.1 Basic notation
3.2 Existence of the least eigenvalue of the homogeneous eigenvalue problem
3.3 Existence of the least eigenvalue of the nonhomogeneous eigenvalue problem
3.4 Maximum principle for degenerated (singular) equations
3.5 Positive solutions of degenerated (singular) BVP
3.6 Bifurcation from the least eigenvalue
4 The p-Laplacian in ℝN
4.1 Nonlinear eigenvalue problem
4.2 Bifurcation problem for the p-Laplacian in ℝN
4.3 Bifurcation problem for the perturbed p-Laplacian in ℝN
Bibliography
Index