Ljusternik-Schnirelman theory in partially ordered Hilbert spaces

Abstract

Abstract. We present several variants of Ljusternik-Schnirelman type theorems in partially ordered Hilbert spaces, which assert the locations of the critical points constructed by the minimax method in terms of the order structures. These results are applied to nonlinear Dirichlet boundary value problems to obtain the multiplicity of sign-changing solutions.

Sign-changing solutions for some nonlinear problems with strong resonance

“…Then, similar to the proof of Theorem 2.9 [11], where K c is replaced by K c ∩S and A by A ∩ S, we have…”

“…Let J C 1 (H, ℝ). In the article [10], those authors construct the pseudo-gradient flow s for J, and have the same definition as [11]. Definition 3.1 Let W ⊂ X be an invariant set under s. W is said to be an admissible invariant set for J if (a) W is the closure of an open set in X;…”

Infinitely many sign-changing solutions for a Schrö dinger equation

“…If v ≢ 0, because ||J'(u n )|| ||u n || 0, as the similar proof in Lemma 6.22 of [2] 5) which contradicts (3.3). This proves that J satisfies the (w*-PS) condition.…”

A Local Minimax-Newton Method for Finding Multiple Saddle Points with Symmetries

“…time translation, etc. Symmetries described by compact group actions in variational problems have been used in the literature to prove the existence of multiple critical points, typically, in the Ljusternik-Schnirelman theory (see, e.g., [14] and others). It is known that symmetries in a nonlinear variational problem can lead to the existence of many solutions of saddle type and can also cause (symmetric) degeneracy.…”