Decay rate and radial symmetry of the exponential elliptic equation

2012/09/04 by Kim, Sunghoon, Hui, Kin Ming
#35B06 #35B40 #35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1209.0544

Abstract

Let n≥ 3, α, β∈ℝ, and let v be a solution Δv+αe^v+βx⋅∇ e^v=0 in ℝⁿ, which satisfies the conditions lim_R→∞(1)/(log R)∫₁^Rρ^1-n (∫_Bρe^v dx)dρ∈ (0,∞) and |x|²e^v(x)≤ A₁ in \Rⁿ. We prove that (v(x))/(log |x|)→ -2 as |x|→∞ and α>2β. As a consequence under a mild condition on v we prove that the solution is radially symmetric about the origin.

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