摘要
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 =φ(r)u^p-1, u 〉 0 in R^N, u ∈ D^1,2(R^N), where N ≥ 3, x = (x^1,z) ∈ R^K×R^N-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 R^N. The asymptotic behavior of ground state solution Up is also given, which also deduces that the ground state solution is not cylindrically symmetric as p goes to 2^*.
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 =φ(r)u^p-1, u 〉 0 in R^N, u ∈ D^1,2(R^N), where N ≥ 3, x = (x^1,z) ∈ R^K×R^N-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 R^N. The asymptotic behavior of ground state solution Up is also given, which also deduces that the ground state solution is not cylindrically symmetric as p goes to 2^*.
基金
Supported by Special Funds for Major States Basic Research Projects of China(G1999075107) Knowledge Innovation Program of CAS in China.