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.

Daomin Cao (Author), Shuangjie Peng (Author), Shusen Yan (Author)

9781108836838, Cambridge University Press

Hardback, published 18 February 2021

262 pages
23.4 x 15.1 x 1.9 cm, 0.49 kg

'This book presents in a very nice and self-contained manner the main methods to find (or to construct) solutions, which exhibit a concentration property, to non-compact elliptic problems.' Lutz Recke, ZB Math Reviews

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.

1. Non-Compact Elliptic Problems
2. Perturbation Methods
3. Local Uniqueness of Solutions
4. Construction of Infinitely Many Solutions
5. A Compactness Theorem and Application
6. The Appendix.

Subject Areas: Numerical analysis [PBKS], Differential calculus & equations [PBKJ]