The fibering method applied to the level sets of a family of functionals.

Abstract:  Given an one-parameter family of C1-functionals, Φμ : X →R, defined on an uniformly convex Banach space X, we describe a method that permit us find critical points of Φμ at some energy level c ∈ R. In fact, we show the existence of a sequence μ(n,c), n ∈N, such that Φμ(n,c) has a critical level at c ∈ R, for all n ∈ N. Moreover, we show some good properties of the curves μ(n,c), with respect to c (for example, they are Lipschitz), and as a consequence of this analysis, we recover many know results on the literature concerning bifurcations of elliptic partial differential equations. Furthermore we prove new results for a large class of elliptic partial differential equations, which includes, for example, Ouyang, Lane-Enden, Concave-Convex, Kirchhoff and Schrödinger-Bopp-Podolsky type equations.

Posted on Dec 10, 2021 in Differential Equations, Seminars