A study on the critical Kirchhoff problem in high-dimensional space

In this present paper, we consider the following critical Kirchhoff problem - (a+lambda integral(RN)vertical bar u vertical bar(2)dx)Delta u + mu = mu vertical bar u vertical bar(p-2)u + vertical bar u vertical bar(2*-2)u in R-N, where a, lambda is an element of R and m, mu is an element of R+ boolean OR{0}, N >= 3 and 20 and a <= 0. We obtain a series of fairly complete existence and multiplicity results and have a clear understand the solutions of this pure critical Kirchhoff problem. In particular, if N >= 5, a>0 and lambda>0 is suitable small, we obtain two positive solutions, in which one is a mountain pass solution and another one is a global (local) minimum solution. In the second part, the original perturbation problem with m,mu>0 has been considered and two positive solutions also have been obtained for N >= 5, which is rather different compared with the case that lambda=0.