Drábek P., Kufner A., Nicolosi F. Quasilinear Elliptic Equations with Degenerations and Singularities

Berlin: de Gruyter, 1997. — 232 p.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.List of symbols, theorems, definitions, assumptions, examples
List of symbols
List of theorems
List of definitions
List of assumptions
List of examples
Introduction
Preliminaries
The domain Ω
Function spaces
Caratheodory functions, Nemytskij (superposition) operators
Function spaces (continued)
Weighted Sobolev spaces
Leray-Lions theorem
Degree of mappings of monotone type
Harnack-type inequality, decay of solution, local regularity and interpolation inequality
Some technical lemmas
Solvability of nonlinear boundary value problems
Formulation of the problem
Second order equations (bounded domains)
Second order equations (proof of Theorem 2.1)
Second order equations (unbounded domains)
Higher order equations (growth conditions)
Higher order equations (operator representation)
Higher order equations (degree of the mapping T)
Higher order equations (existence results)
Examples, remarks, comments
The degenerated p-Laplacian on a bounded domain
Basic notation
Existence of the least eigenvalue of the homogeneous eigenvalue problem
Existence of the least eigenvalue of the nonhomogeneous eigenvalue problem
Maximum principle for degenerated (singular) equations
Positive solutions of degenerated (singular) BVP
Bifurcation from the least eigenvalue
The p-Laplacian in ℝN
Nonlinear eigenvalue problem
Bifurcation problem for the p-Laplacian in ℝN
Bifurcation problem for the perturbed p-Laplacian in ℝN
Bibliography
Index