Infinitely many solutions for Kirchhoff problems with lack of compactness

Abstract In this paper, we consider the following Kirchhoff problem: − ( 1 + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( u ) , x ∈ R 3 , u ∈ H 1 ( R 3 ) , where b is a positive constant. Assume that f ( u ) is an odd function of u . Under some appropriate assumptions on V but without radial symmetry or compactness hypotheses, we establish the existence of infinitely many solutions to the above problem by using an approximation method employed by Sato and Shibata (2018) and some new arguments.