Multiplicity and concentration of solutions for Kirchhoff equations with exponential growth

• Published in: Bulletin of mathematical sciences 2024-12, Vol.14 (3)
• Main Authors: Sun, Xueqi, Fu, Yongqiang, Liang, Sihua
• Format: Article
• Language: English
• Subjects: critical exponential growth; fractional -Laplace; Kirchhoff equations; Lyusternik–Schnirelmann theory; Mountain Pass Theorem; variational method
• ISSN: 1664-3607; 1664-3615
• DOI: 10.1142/S1664360724500048

In this paper, we deal with fractional p -Laplace Kirchhoff equations with exponential growth of the form p s ( a + b [ u ] s , p p ) ( − Δ ) p s u + Z ( x ) | u | p − 2 u = h ( u ) in  ℝ N , where is a positive parameter, a , b > 0 , s ∈ ( 0 , 1 ) and p = N s ≥ 2 . Under some appropriate conditions for the nonlinear function h and potential function Z , and with the help of penalization method and Lyusternik–Schnirelmann theory, we establish the existence, multiplicity and concentration of solutions. To some extent, we fill in the gaps in [W. Chen and H. Pan, Multiplicity and concentration of solutions for a fractional p -Kirchhoff type equation, Discrete Contin. Dyn. Syst. 43 (2023) 2576–2607; G. Figueiredo and J. Santos, Multiplicity and concentration behavior of positive solutions for a Schrödinger–Kirchhoff type problem via penalization method, ESAIM Control Optim. Calc. Var. 20 (2014) 389–415; X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in ℝ 3 , J. Differential Equations 252 (2012) 1813–1834; J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations 253 (2012) 2314–2351].