Asymptotic symmetry of singular solutions of semilinear elliptic equations
Resource
Journal of Differential Equations 245 (9): 2534-2550
Journal
Journal of Differential Equations
Journal Volume
245
Journal Issue
9
Pages
2534-2550
Date Issued
2008
Author(s)
Prajapat, J.V.
Abstract
For dimensions 3 ≤ n ≤ 6, we derive lower bound for positive solution ofΔ u - μ u + K (x) ufrac(n + 2, n - 2) = 0 in B2 {set minus} {0}, u ∈ C2 (B2 {set minus} {0}) in the neighbourhood of singularity. Here μ > 0. We prove that there exists a constant c1 > 0 depending on n, μ, | K |, | ∇ K | such thatu (x) > c1 | x |- frac(n - 2, 2) in B1 {set minus} {0} . For dimension 3, we assume that K is Hölder continuous with exponent θ with frac(1, 2) < θ ≤ 1 while for dimensions n = 4, 5, 6, assume that K ∈ C1 is bounded between two positive constantsc | x |θ - 1 ≤ | ∇ K (x) | ≤ C | x |θ - 1 for c, C > 0 and frac(n - 2, 2) ≤ θ ≤ n - 2. As an application, we derive the asymptotic symmetry and a priori estimates for the solutions. © 2008 Elsevier Inc. All rights reserved.
Type
journal article
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