Infinitely many sign-changing solutions for Kirchhoff type problems in R3. (September 2019)

• Record Type: Journal Article
• Authors: Sun, Jijiang; Li, Lin; Cencelj, Matija; Gabrovšek, Boštjan
• Is Part Of: Nonlinear analysis. Volume 186(2019)
• Page Start: 33
• Page End: 54
• Publication Date: 2019-09
• Subjects: Infinitely many sign-changing solutions -- Kirchhoff type problems -- Invariant sets -- Descending flow; 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.2018.10.007
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we consider the following nonlinear Kirchhoff type problem: − a + b ∫ R 3 | ∇ u | 2 Δ u + V ( x ) u = f ( u ), in R 3, u ∈ H 1 ( R 3 ), where a, b > 0 are constants, the nonlinearity f is superlinear at infinity with subcritical growth and V is continuous and coercive. For the case when f is odd in u we obtain infinitely many sign-changing solutions for the above problem by using a combination of invariant sets method and the Ljusternik–Schnirelman type minimax method. To the best of our knowledge, there are only few existence results for this problem. It is worth mentioning that the nonlinear term may not be 4-superlinear at infinity, in particular, it includes the power-type nonlinearity | u | p − 2 u with p ∈ ( 2, 4 ] .