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2 results

  • 2 - Perturbation Methods

  • Daomin Cao, Chinese Academy of Sciences, Beijing, Shuangjie Peng, Shusen Yan
  • Book:
    Singularly Perturbed Methods for Nonlinear Elliptic Problems
    Published online:
    28 January 2021
    Print publication:
    18 February 2021, pp 54-128

  • Summary

    The Lyapunov-Schmidt reduction method and its variants have been widely used to construct peak solutions or bubbling solutions for singularly perturbed elliptic problems. In Chapter 2, we discuss the existence of such solutions, as well as the necessary condition for the location of the concentration points of the solutions.As illustrations of the main idea, we study two typical singularly perturbed elliptic problems in Chapter 2, and they are the nonlinear Schrodinger equations with subcritical growth and the Brezis-Nirenberg problem. Problems have been chosen so that many sophisticated estimates can be avoided.

Singularly Perturbed Methods for Nonlinear Elliptic Problems
  • Daomin Cao, Shuangjie Peng, Shusen Yan
  • Published online:
    28 January 2021
    Print publication:
    18 February 2021
  • 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.