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RESEARCH PRODUCT

Lusternik-Schnirelmann Critical Values and Bifurcation Problems

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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.)