Critical Point Theory

Abstract

Many nonlinear problems can be reduced to the form Many nonlinear problems can be reduced to the form

$$G'(u) = 0,$$

((1.1.1))

where G is a C¹-functional on a Banach space E. In this case the problems can be attacked by specialized, important techniques which can produce results where other methods fail. The history of this approach can be traced back to the calculus of variations in which equations of the form (1.1.1) are the Euler-Lagrange equations of the functional G. The original method was to find maxima or minima of G by solving (1.1.1) and then show that some of the solutions are extrema. This approach worked well for one dimensional problems. In this case it is easier to solve (1.1.1) than it is to find a maximum or minimum of G. However, in higher dimensions it was realized quite early that it is easier to find maxima and minima of G than it is to solve (1.1.1). Consequently, the tables were turned, and critical point theory was devoted to finding extrema of G. This approach is called the direct method in the calculus of variations. If an extremum point of G can be identified, it will automatically be a solution of (1.1.1).