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

Abstract

This paper studies the asymptotic behaviour as α := u(0) ↑∞ of the first zero R(α) of the radially symmetric solution of the semilinear equation

in ℝⁿ, n ≥ 1, where h > 0 and β > 1. We establish that R(α) = O(α^{−(β−1)/2)} if n = 1, 2 or n ≥ 3 and β < (n + 2)/(n − 2), and conjecture that lim inf α→∞R(α) > 0 if n ≥ 3 and β > (n + 2)/(n − 2).