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.

The appendix contains various Pohozaev identities, some preliminary properties of Sobolev spaces, certain fundamental estimates on elliptic equations, the Kelvin transformation, the kernel of some linear operators and the estimate for the Green’s function.

Chapter 3 is devoted to studying the local uniqueness of peak/bubbling solutions. In the 1990s, local uniqueness problems were discussed by the classical degree counting methods, which rely crucially on the estimates of the second order derivatives of the solutions. In this chapter, such problems are addressed by using the local Pohozaev identities.Such methods simplify the classical degree counting methods considerable.Once again, to avoid many sophisticated estimates,we choosenonlinear Schrodinger equations with subcritical growth and the Brezis-Nirenberg problem to illustrate the main techniques.