Least energy solutions for fractional Kirchhoff problems with logarithmic nonlinearity. (September 2020)

• Record Type: Journal Article
• Authors: Xiang, Mingqi; Hu, Die; Yang, Di
• Is Part Of: Nonlinear analysis. Volume 198(2020)
• Publication Date: 2020-09
• Subjects: 35K55 -- 35R11 -- 47G20; Fractional Kirchhoff problems -- Least energy solutions -- Logarithmic nonlinearity -- Nehari manifold; 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.2020.111899
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we study the existence of least energy solutions to the following fractional Kirchhoff problem with logarithmic nonlinearity M ( [ u ] s, p p ) ( − Δ ) p s u = h ( x ) | u | θ p − 2 u ln | u | + λ | u | q − 2 u x ∈ Ω, u = 0 x ∈ R N ∖ Ω, where s ∈ (0, 1), 1 < p < N ∕ s, Ω ⊂ R N is a bounded domain with Lipschitz boundary, M ( [ u ] s, p p ) = [ u ] s, p ( θ − 1 ) p with θ ≥ 1 and [ u ] s, p is the Gagliardo seminorm of u, h ∈ C ( Ω ¯ ) may change sign, λ > 0 is a parameter, q ∈ ( 1, p s ∗ ) and ( − Δ ) p s is the fractional p − Laplacian. When θ p < q < p s ∗ and h is a positive function on Ω, the existence of least energy solutions is obtained by restricting the discussion on Nehari manifold. When 1 < q < θ p and h is a sign-changing function on Ω, two local least energy solutions are obtained by using the Nehari manifold approach.