Concentration of ground states for quasilinear Kirchhoff type equations at critical growth

We are concerned with the existence and concentrating behavior of positive ground state solutions for a quasilinear Kirchhoff equation involving critical Sobolev exponent with competing potentials (Formula presented.) where a,b>0 are constants, ϵ>0 is a small parameter, and g is an even differential function related to the quasilinear term, such that G(t)=∫0tg(s)ds. Under some suitable assumptions on V, Q, K and h, we conclude that this equation admits a positive ground state solution for all sufficiently small ϵ>0 using variational methods, where the decay rate of the obtained solution as |x|→+∞ and its concentration on the set of minimal points of V and the sets of maximal points of Q and K as ϵ→0+ are also considered. In particular, we also investigate the nonexistence of ground state solutions.