Locating the Peaks of Solutions via the Maximum Principle II: A Local Version of the Method of Moving Planes
Journal
Communications on Pure and Applied Mathematics
Journal Volume
56
Journal Issue
6
Pages
784-809
Date Issued
2003
Author(s)
Wei, J.
Abstract
Let Ω be a bounded, smooth domain in ℝ2n, n ≥ 2. The well-known Moser-Trudinger inequality ensures the nonlinear functional Jρ(u) is bounded from below if and only if ρ ≤ ρ 2n:= 22nn!(n - 1)!ω2n, where J ρ(u) = 1/2 ∫Ω |(-Δ) n/2u|2 - ρlog ∫Ω eudx in χ:= Hn(Ω) ∩ {u, (-Δ)ju ∈ H01(Ω), j = 1, ..., [n-1/2]}, and ω 2n is the area of the unit sphere double-struck S2n-1 in ℝ2n. In this paper, we prove the infu∈χ Jρ(u) is always attained for ρ ≤ ρ2n. The existence of minimizers of Jρ at the critical value ρ = ρ2n is a delicate problem. The proof depends on the blowup analysis for a sequence of bubbling solutions. Here we develop a local version of the method of moving planes to exclude the boundary bubbling. The existence of minimizers for Jρ at the critical value ρ = ρ 2n is in contrast to the case of two dimensions. © 2003 Wiley Periodicals, Inc.
Type
journal article