Locating the peaks of solutions via the maximum principle II: A local version of the method of moving planes

Abstract

AbstractLet 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 ≤ ρ2n := 22nn!(n − 1)!ω2n, where in ${\cal X}:=H^n(\Omega)\cap \{ u, (-\Delta)^j u \in H^1_0(\Omega), j=1,\dots, [{n-1 \over 2}] \}$, and ω2n is the area of the unit sphere 𝕊2n − 1 in ℝ2n. In this paper, we prove the infu∈X 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 = ρ2n is in contrast to the case of two dimensions. © 2003 Wiley Periodicals, Inc.