Asymptotic symmetry and local behaviors of solutions to a class of anisotropic elliptic equations
Journal
Indiana University Mathematics Journal
Journal Volume
60
Journal Issue
5
Pages
1623-1653
Date Issued
2011
Author(s)
Abstract
Let 0 ≤ a < (N - 2)/2, a ≤ b < a + 1, p = 2N/(N - 2 + 2(b - a)), and B1 = B1(0) be the unit ball in ℝN, where N ≥ 3. We first prove that any positive solution u(x) ∈ C2(B1 \ {0}) of the equation (0.1) Equation Presented is asymptotically symmetric with respect to the origin, i.e., u(x) = u0(|x|)(1 + O(|x|ε)) as |x| → 0, where u 0(x) = u0(|x|) ∈ C2(ℝN \ {0}) is an entire solution of (0.1) and ε > 0. Equation (0.1) is arising from the celebrated Caffarelli-Kohn-Nirenberg inequality. For a = b < 0 and p = 2N/(N - 2), we show there is no positive solution of the equation (0.2)Equation Presented in D1,2a (Ω), where Ω ⊆ ℝN+ is a cone domain satisfying Ω = R+×ω with ω ⊂ SN-1 being star-shaped with respect to the north pole on SN-1. © 2011 Indiana University Mathematics Journal.
Type
journal article