Energy functionals of Kirchhoff-type problems having multiple global minima. (March 2015)

• Record Type: Journal Article
• Authors: Ricceri, Biagio
• Is Part Of: Nonlinear analysis. Volume 115(2015)
• Page Start: 130
• Page End: 136
• Publication Date: 2015-03
• Subjects: Kirchhoff-type problem -- Energy functional -- Global minimum -- Variational methods -- Strict minimax inequality -- Multiplicity; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2014.12.012
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, using the theory developed in Ricceri (2012), we obtain some results of a totally new type about a class of non-local problems. Here is a sample: Let Ω ⊂ R n be a smooth bounded domain, with n ≥ 4, let a, b, ν ∈ R, with a ≥ 0 and b > 0, and let p ∈ ] 0, n + 2 n − 2 [ . Then, for each λ > 0 large enough and for each convex set C ⊆ L 2 ( Ω ) whose closure in L 2 ( Ω ) contains H 0 1 ( Ω ), there exists v ∗ ∈ C such that the problem { − ( a + b ∫ Ω | ∇ u ( x ) | 2 d x ) Δ u = ν | u | p − 1 u + λ ( u − v ∗ ( x ) ) in Ω u = 0 on ∂ Ω has at least three weak solutions, two of which are global minima in H 0 1 ( Ω ) of the corresponding energy functional.