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Methods on nonlinear elliptic equations. (English) Zbl 1214.35023

AIMS Series on Differential Equations and Dynamical Systems 4. Springfield, MO: American Institute of Mathematical Sciences (AIMS) (ISBN 978-1-60133-006-2/hbk). xii, 299 p. (2010).
The monograph presents basic concepts as well as real research examples to young researchers interested in the field of non-linear analysis of partial differential equations (PDEs); in particular, the text focuses on the analysis of semi-linear elliptic PDEs. After some necessary preparations for basic knowledge, a series of typical methods in non-linear analysis are introduced, some of which are well known while others are relatively new. The authors firstly illustrate these ideas and techniques using simple examples, then lead the readers to the research front explaining how these methods can be applied to solve practical problems through careful analysis of a series of recent research articles.
In Chapter 1 basic concepts of Sobolev spaces and some commonly used inequalities are introduced.
Chapter 2 shows how to find weak solutions for some typical linear and semi-linear PDEs by using functional analysis methods, mainly, the calculus of variations and critical point theories, including the well-known Mountain Pass Lemma.
In Chapter 3, \(W^{2,p}\) a priori estimates and regularity are established. It is proved that in most cases weak solutions are actually differentiable and hence are classical solutions. There are also presented two Regularity Lifting Theorems. Example are used to show how these Theorems can be applied to systems of PDEs and integral equations including a fully nonlinear system of Wolff type.
Chapter 4 is a preparation to chapters 5 and 6 introducing Riemannian manifolds, curvatures, covariant derivatives, and Sobolev embedding on manifolds.
Chapter 5 deals with semi-linear elliptic equations arising from prescribing Gaussian curvature on both positively and negatively curved manifolds. It is shown the existence of weak solutions in both subcritical and critical cases via variational approaches. It is also introduced the method of lower and upper solutions.
Chapter 6 focuses on the well-known Yamabe problem and its generalization, prescribing scalar curvature on \(S^n\) for \(n\geq 3\). The latter is in the critical case where the corresponding variational functional is not compact at any level sets. To recover the compactness the authors construct a max-min variational scheme.
Chapter 7 is devoted to the study of various maximum principles, in particular, the ones based on comparisons.
In Chapter 8, the method of moving planes and its variant, the method of moving spheres, are introduced and their application to obtain the symmetry, monotonicity, a priori estimates, and even non-existence of solutions is illustrated. It is also introduced an integral form of the method of moving planes. Chapters 7 and 8 function as a self-contained group. Readers who are only interested in the maximum principles and the method of moving planes can skip the first six chapters and start directly with Chapter 7.
This book serves as a bridge between graduate textbooks and research articles in the area of nonlinear elliptic partial differential equations. It can be considered as a handy textbook for use in a topics course on nonlinear analysis.

MSC:

35J60 Nonlinear elliptic equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35J20 Variational methods for second-order elliptic equations
58J05 Elliptic equations on manifolds, general theory
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J65 Nonlinear boundary value problems for linear elliptic equations
35D30 Weak solutions to PDEs
35J93 Quasilinear elliptic equations with mean curvature operator
35B50 Maximum principles in context of PDEs
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