EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS FOR A NONLINEAR ELLIPTIC EQUATION ARISING IN ASTROPHYSICS*

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 −Δu = ø(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 ø on ℝN. 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*.