Abstract:

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
-△u=\phi(r) u^p-1, u>0, in R^N, u ∈D^1,2(R^N),

where N≥3, x=(x',z) ∈R^K×R^N-K,2≤ K≤ N,r=|x'|. It is proved that for 2(N-s)/(N-2)*=2N/(N-2), 0*, 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 u_p has a unique maximum point x_p=(x'_p,z_p) and as p \to 2^*,|x'_p|→ r₀ which attains the maximum of $\phi$ on R^N. The asymptotic behavior of ground state solution u_p is also given, which also deduces that the ground state solution is not cylindrically symmetric as p goes to 2^*.

Key words: Cylindrical symmetry, asymptotic behavior, ground state solution