Infinitely many bound state solutions for Kirchhoff type problems

Abstract

This paper is concerned with the following Kirchhoff type problems (0.1)−(a+b∫Ω|∇u|2dx)Δu=λ|u|q−2u+|u|p−2u,x∈Ω,u=0,x∈∂Ω,where constants a,b>0, the parameter λ≥0, 1<q<2<p<6 and Ω is a smooth bounded domain in R3. We consider two cases: the concave–convex nonlinearities (λ>0) and the superlinear nonlinearities (λ=0). By using the genus type min–max argument, we prove that (0.1) admits infinitely many bound state solutions for the above two cases.