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

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 −Δ u = ø( 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) < p < 2 * = 2 N/( 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 u p has a unique maximum point x p = ( x′ p, z p ) and as p → 2 *, ∣ x′ p ∣ → r 0 which attains the maximum of ø 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 *.