Concentrating bounded states for a class of singularly perturbed Kirchhoff type equations with a general nonlinearity

We are concerned with the following Kirchhoff type equation:{−(ε2a+εb∫R3|∇u|2)Δu+V(x)u=f(u) in R3,u>0,u∈H1(R3), where ε is a small positive parameter, a,b>0. Under general conditions on f due to Zhang et al. (2014) [37], we construct a family of positive solutions uε∈H1(R3) which concentrates around the local minima of V as ε→0. Our result completes the study made in Figueiredo et al. (2014) [10] in the sense that, only the subcritical growth was considered.