A sharp oscillation property involving the critical Sobolev exponent for a class of superlinear elliptic problems

This paper studies the asymptotic behaviour as := u(0) of the first zero R() of the radially symmetric solution of the semilinear equation -Delta u = vertical bar u vertical bar(beta-1) u + h in (n), n 1, where h > 0 and > 1. We establish that R() = O((-(-1)/2)) if n = 1, 2 or n 3 and < (n + 2)/(n - 2), and conjecture that lim inf R() > 0 if n 3 and > (n + 2)/(n - 2).