Lusternik-Schnirelmann Critical Values and Bifurcation Problems

Abstract

We present a method to calculate bifurcation branches for nonlinear two point boundary value problems of the following type

$$ \{ _{u(a) = u(b) = 0,}^{ - u'' = \lambda G'(u)} $$

(1.1)

where G : R → R is a smooth mapping. This problem can be formulated equivalently as

$$ g' \left(u \right)= \mu u, $$

(1.2)

where

$$ g \left(u \right)= \overset{b} {\underset{a} {\int}} G \left(u \left(t \right) \right) dt $$

(1.3)

and μ = 1/λ. Solutions of this problem can be found by locating the critical points of the functional g : H → R on the spheres $S_r= \lbrace x \in H \mid \;\parallel x \parallel =r \rbrace, r >0.$ (The Lagrange multiplier theorem.)