The author first analyzes the existence of ground state solutions and cylindrically symmetric solutions and then the asymptotic behavior of the ground state solution of the equation -Au = φ(r)up-1, u ＞ 0 in RN, u ∈ D1,2(RN), where N ≥ 3, x =(x',z) ∈ RK × RN-K,2 ≤ K ≤ N,r = |x'|. It is proved that for 2(N-s)/(N-2) ＜p ＜ 2* = 2N/(N - 2), 0 ＜ s ＜ 2, the above equation has a ground state solution and a cylindrically symmetric solution. For p = 2*, the above equation does not have a ground state solution but a cylindrically symmetric.solution, and when p close to 2*, the ground state solutions are not cylindrically symmetric. On the other hand, it is proved that as p close to 2*, the ground state solution up has a unique maximum point xp = (x'p, zp) and as p → 2*, |x'p| → r0 which attains the maximum of φ on RN. The asymptotic behavior ofground state solution up is also given, which also deduces that the ground state solutionis not cylindrically symmetric as p goes to 2*.