A remark on fractional p–Kirchhoff problems involving multiple zeros. Issue 10 (3rd October 2019)

• Record Type: Journal Article
• Authors: Xiang, Mingqi; Zhang, Binlin
• Is Part Of: Complex variables and elliptic equations. Volume 64:Issue 10(2019)
• Page Start: 1655
• Page End: 1665
• Publication Date: 2019-10-03
• Subjects: P. Pucci; Fractional p–Laplacian -- degenerate Kirchhoff problems -- multiplicity of solutions -- asymptotic behavior of solutions; 35D30 -- 35R11 -- 35A15 -- 47G20; Functions of complex variables -- Periodicals; Differential equations, Elliptic -- Periodicals; 515.905
• DOI: 10.1080/17476933.2019.1571052
• Languages: English
• ISSNs: 1747-6933

Abstract

ABSTRACT: In this paper, we study the multiplicity of solutions for a p –Kirchhoff type problem driven by a nonlocal integro-differential operator. As a particular case, we consider the following problem: M ∬ R 2 N | u ( x ) − u ( y ) | p | x − y | N + s p d x d y + ∫ R N V ( x ) | u | p d x ( ( − Δ ) p s u + V ( x ) | u | p − 2 u ) = f λ ( x, u ) in R N, where ( − Δ ) p s is the fractional p –Laplacian, 0 < s < 1 < p < ∞ with sp < N, p s ∗ = N p / ( N − s p ), M : [ 0, ∞ ) → [ 0, ∞ ) is a continuous function vanishing in many different points, V : R N → ( 0, ∞ ) is a continuous function, and f λ : R N × R → R is a Carathéodory function for each λ > 0 . Under some suitable assumptions, we obtain the multiplicity of solutions for the above problem by applying the mountain pass theorem. Moreover, the asymptotic behavior of solutions is also investigated. A distinguished feature of this paper is that the Kirchhoff function M has multiple zeros.