Multiple critical points of saddle geometry functionals

Abstract

We study the multiplicity of critical points for continuously differentiable functionals on real Banach spaces. We prove that a functional which satisfies the assumptions of the Saddle Point Theorem and moreover is bounded from below has at least three critical points. Apparently, there is a global minimizer and a saddle point and we show the existence of a third critical point. The idea of the proof is based on the minus-gradient flow. This result is closely related to the three critical points theorem of H. Brezis and L. Nirenberg which assumes a local linking. Finally, we apply the result on the Dirichlet problem for semilinear stationary PDEs. The analysis includes, for example, the existence of multiple stationary solutions of bistable (or Allen-Cahn) equation and semipositone problems.

Nonuniqueness of implicit lattice Nagumo equation

“…Then there exists δ > 0 such that for every b ∈ R N satisfying b 2 < δ the functional F given by (2.8) has at least three critical points. P r o o f. The statement follows from [28], Theorem 1.1. This theorem states the following.…”

“…Since all the assumptions of [28], Theorem 1.1 are satisfied, there exist at least three critical points of the functional F provided b 2 < δ. Theorem 3.11. Let λ 1 < .…”