The theory of nonlinear elliptic equations is currently one of the most actively developing branches of the theory of partial differential equations. This book investigates boundary value problems for nonlinear elliptic equations of arbitrary order. In addition to monotone operator methods, a broad range of applications of topological methods to nonlinear differential equations is presented: solvability, estimation of the number of solutions, and the branching of solutions of nonlinear equations. Skrypnik establishes, by various procedures, a priori estimates and the regularity of solutions of nonlinear elliptic equations of arbitrary order. Also covered are methods of homogenization of nonlinear elliptic problems in perforated domains. The book is suitable for use in graduate courses in differential equations and nonlinear functional analysis.

Readership

Researchers, mathematicians, and graduate students studying differential equations and nonlinear functional analysis.

Reviews

"This is a nice and useful book representing a systematic approach to nonlinear elliptic partial differential equations with several interesting examples and applications."

-- Zentralblatt MATH

Table of Contents

• Mappings of monotone type and solvability of quasilinear boundary value problems
• Degree of generalized monotone mappings
• Topological characteristics of nonlinear elliptic boundary value problems
• Solvability and behavior of solutions of nonlinear elliptic boundary value problems
• Solvability of the nonlinear Dirichlet problem in a narrow strip
• Solvability of semilinear boundary value problems
• A priori estimates and regularity of solutions of higher order quasilinear elliptic equations
• Behavior of solutions of quasilinear elliptic equations near the boundary
• Nonlinear elliptic problems in domains with fine-grained boundary
• Homogenization of nonlinear Dirichlet problems in domains with channels
• References