MINIMIZATION METHODS AND EXISTENCE OF SOLUTIONS FOR NONLINEAR DIFFERENTIAL EQUATIONS

The purpose of this lecture is to demonstrate the power of variational methods in the study of nonlinear partial differential equations to people working in other field of mathematics.

The equations considered here are variational, namely there is an energy functional ˆ defined on certain Sobolev spaces, whose critical points are the solutions of the equations. Depending on the nonlinearities in the equations, the functional ˆ can behave differently.

• For the case that ˆ is bounded from below, we look for the global minimizers of ˆ, which are solutions of the original equations.
•  If ˆ is unbounded from below, we will perform a constrained minimization procedure and find the constrained minimizers, which are also solutions of the equations.

We will use these techniques to study nonlinear elliptic boundary value problems setting on a bounded domain, so that the compact Sobole embedding can be applied. We will also consider stand-
ing waves of nonlinear Schrödinger equations, then we will encounter nonlinear elliptic equations setting on unbounded domain RN and the Sobolev embedding is no longer compact. For this case we will show how the concentration lemma of P.L. Lions saves us.