Quasilinear elliptic equations with degenerations and singularities. (English) Zbl 0894.35002

de Gruyter Series in Nonlinear Analysis and Applications. 5. Berlin: Walter de Gruyter. xii, 219 p. (1997).
Every calculus student knows that Sobolev spaces are an indispensable tool for studying elliptic equations, both linear and nonlinear, in a functional-analytic setting. Somewhat less known, however, is the prominant rôle of weighted Sobolev spaces which turn out to be quite useful whenever one encounters some kind of singularity or degeneracy (unbounded domains, domains with irregular boundary, “badly behaved” coefficients, critical growth of nonlinearities, and other phenomena of this type). Much information on the theory, methods, and applications of weighted Sobolev spaces may be found, for example, in the second author’s homonymous book [Weighted Sobolev spaces, J. Wiley, New York (1985; Zbl 0567.46009)].
After collecting the necessary notions and results from functional analysis and operator theory, in the second chapter the authors give a comprehensive study of nonlinear boundary value problems of elliptic type involving singularities or degenerations, with a particular emphasis on existence theorems for weak solutions. Here, one has to distinguish second-order equations from higher-order equations: for the first, the crucial tool is the Leray-Lions theorem, while for the second one needs a combination of the celebrated monotonicity method which goes back to G. Minty (who, surprisingly, is missing in the list of references) and the topological degree theory for special monotone mappings developed by I. V. Skrypnik, V. Mustonen, and others.
The third and fourth chapters deal with a somewhat different topic which, however, fits perfectly the title of the book, namely with spectral properties of nonlinear boundary value problems. Many important contributions are here due to the first author, see e.g. his book on “Solvability and bifurcations of nonlinear equations” [Longman, Harlow (1992; Zbl 0753.34002)]. A standard model is the so-called \(p\)-Laplacian which is defined for \(1<p<\infty\) by \(\Delta_p u=\text{div}(|\nabla u|^{p- 2}\nabla u)\) and reduces in case \(p=2\), of course, to the classical (linear) Laplace operator. While Chapter 3 is concerned with existence, uniqueness, and bifurcation of solutions to problems involving the \(p\)-Laplacian over bounded domains, in Chapter 4 the authors discuss problems over the whole space \(\mathbb{R}^N\). In the latter case, several new features occur (for example, noncompact imbeddings and resolvents) which are “hidden” in the case of bounded domains.
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.


35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35J70 Degenerate elliptic equations
47H99 Nonlinear operators and their properties
35J65 Nonlinear boundary value problems for linear elliptic equations