Multiplicity of concentrating positive solutions for nonlinear Kirchhoff type problems with critical growth. Issue 15 (18th November 2021)

• Record Type: Journal Article
• Authors: Zhang, Hui; Zhang, Fubao; Xu, Junxiang
• Is Part Of: Applicable analysis. Volume 100:Issue 15(2021)
• Page Start: 3276
• Page End: 3297
• Publication Date: 2021-11-18
• Subjects: Kirchhoff type equation -- critical growth -- Ljusternik–Schnirelmann category -- penalization method; 32J20 -- 35J60 -- 35J92; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2020.1716969
• Languages: English
• ISSNs: 0003-6811

Abstract

ABSTRACT: In this paper, we study the following semilinear Kirchhoff type equation: 1 − ϵ 2 a + ϵ b ∫ R 3 | ∇ u | 2 Δ u + V ( x ) u = f ( u ) + u 5, u > 0, u ∈ H 1 ( R 3 ), where ϵ > 0 is a parameter, a, b are positive constants, V and f are continuous functions. Under certain assumptions on V and f, we investigate the relation between the number of solutions and the topology of the set where V attains its local minimum for small ε . We also describe the concentration phenomena of solutions as ϵ → 0 . The proof is based on the method of Nehari manifold, penalization techniques and Ljusternik–Schnirelmann category theory.