Multiplicity of concentrating positive solutions for nonlinear Kirchhoff type problems with critical growth. Issue 15 (18th November 2021) - Zhang, Hui; Zhang, Fubao; Xu, Junxiang
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.