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2024 | Buch

Preliminaries Kapitel lesen Erstes Kapitel lesen

Nonlinear Second Order Elliptic Equations

verfasst von: Mingxin Wang, Peter Y. H. Pang

Verlag: Springer Nature Singapore

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book focuses on the following three topics in the theory of boundary value problems of nonlinear second order elliptic partial differential equations and systems: (i) eigenvalue problem, (ii) upper and lower solutions method, (iii) topological degree method, and deals with the existence of solutions, more specifically non-constant positive solutions, as well as the uniqueness, stability and asymptotic behavior of such solutions.

While not all-encompassing, these topics represent major approaches to the theory of partial differential equations and systems, and should be of significant interest to graduate students and researchers. Two appendices have been included to provide a good gauge of the prerequisites for this book and make it reasonably self-contained.

A notable strength of the book is that it contains a large number of substantial examples. Exercises for the reader are also included. Therefore, this book is suitable as a textbook for graduate students who havealready had an introductory course on PDE and some familiarity with functional analysis and nonlinear functional analysis, and as a reference for researchers.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Preliminaries
Abstract
In this chapter we first lay down some notations, conventions and basic assumptions. Then, for later applications, we briefly review some basic theories and results of the calculus in Banach spaces and unconditional local extrema, which are elementary knowledge of variational methods. Finally, we give two applications which will be used in Chap. 7.
Mingxin Wang, Peter Y. H. Pang
Chapter 2. Eigenvalue Problems of Second Order Linear Elliptic Operators
Abstract
Eigenvalue problems have a wide range of applications. In particular, the existence of positive solutions to second order semi-linear and quasi-linear elliptic equations and systems depends critically on the principal eigenvalue (the first or smallest eigenvalue) of a corresponding eigenvalue problem. In this chapter, we introduce the theory of eigenvalue problems for second order linear elliptic operators. These results will be used extensively in the later chapters. In the last chapter, we will also introduce the eigenvalue problem for the p-Laplace operator.
Mingxin Wang, Peter Y. H. Pang
Chapter 3. Upper and Lower Solutions Method for Single Equations

In this chapter we introduce the upper and lower solutions method for the boundary value problem of elliptic single equations.

Mingxin Wang, Peter Y. H. Pang
Chapter 4. Upper and Lower Solutions Method for Systems
Abstract
In this chapter we introduce the upper and lower solutions method for the boundary value problem of elliptic systems.
Mingxin Wang, Peter Y. H. Pang
Chapter 5. Theory of Topological Degree in Cones and Applications
Abstract
The theory of topological degree in finite dimensional as well as Banach spaces has been discussed in detail in many monographs. In this chapter, we only introduce the theory of topological degree in cones and its applications.
Mingxin Wang, Peter Y. H. Pang
Chapter 6. Systems with Homogeneous Neumann Boundary Conditions
Abstract
In this chapter, we will continue to focus on applications of methods developed in previous chapters. We shall do so by using examples arising from the modeling of chemical reactions and prey-predator population dynamics.
Mingxin Wang, Peter Y. H. Pang
Chapter 7. The p-Laplace Equations and Systems
Abstract
In this chapter we mainly focus on the properties of the operator
$$\displaystyle \begin{array}{@{}rcl@{}} \mathscr {L}_p^au:=-\varDelta _pu+a(x)|u|{ }^{p-2}u \end{array} $$
and the corresponding boundary value problems of equations and systems in \(\varOmega \), where
$$\displaystyle \varDelta _p u=\mathrm {div}\big (|\nabla u|{ }^{p-2}\nabla u\big ) $$
is the p-Laplacian of u, \(\varOmega \) is a bounded and smooth domain in \(\mathbb {R}^n\), \(1<p<\infty \), and \(a\in L^\infty (\varOmega )\). Throughout this chapter, we shall denote
$$\displaystyle p'=p/(p-1)\;\;\;\text{ for}\;\; 1<p<\infty . $$
Mingxin Wang, Peter Y. H. Pang
Backmatter
Metadaten
Titel
Nonlinear Second Order Elliptic Equations
verfasst von
Mingxin Wang
Peter Y. H. Pang
Copyright-Jahr
2024
Verlag
Springer Nature Singapore
Electronic ISBN
978-981-9986-92-7
Print ISBN
978-981-9986-91-0
DOI
https://doi.org/10.1007/978-981-99-8692-7

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