Singularly Perturbed Methods for Nonlinear Elliptic Problems

A systematic introduction to the singularly perturbed methods in the study of concentration solutions for nonlinear elliptic problems.

This book introduces singularly perturbed methods in a self-contained manner by investigating two relatively simple but typical non-compact elliptic problems. Avoiding using too many sophisticated estimates, this book is written for PhD students and junior mathematicians who plan to do their research in the area of elliptic differential equations.This introduction to the singularly perturbed methods in the nonlinear elliptic partial differential equations emphasises the existence and local uniqueness of solutions exhibiting concentration property. The authors avoid using sophisticated estimates and explain the main techniques by thoroughly investigating two relatively simple but typical non-compact elliptic problems. Each chapter then progresses to other related problems to help the reader learn more about the general theories developed from singularly perturbed methods. Designed for PhD students and junior mathematicians intending to do their research in the area of elliptic differential equations, the text covers three main topics. The first is the compactness of the minimization sequences, or the Palais-Smale sequences, or a sequence of approximate solutions; the second is the construction of peak or bubbling solutions by using the Lyapunov-Schmidt reduction method; and the third is the local uniqueness of these solutions.