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

Let Ω be a bounded, smooth domain in ℝ 2 n , n ≥ 2. The well‐known Moser‐Trudinger inequality ensures the nonlinear functional J ρ ( u ) is bounded from below if and only if ρ ≤ ρ 2 n := 2 2 n n !( n − 1)!ω 2 n , where$$J_{\rho}(u) = {1 \over 2} \int_{\Omega} |(-\Delta)^{n/2} u|^2 - \rho \log \int_{\Omega} e^u \,dx$$ in ${\cal X}:=H^n(\Omega)\cap \{ u, (-\Delta)^j u \in H^1_0(\Omega), j=1,\dots, [{n-1 \over 2}] \}$ , and ω 2 n is the area of the unit sphere 2 n − 1 in ℝ 2 n . In this paper, we prove the inf u ∈ X J ρ ( u ) is always attained for ρ ≤ ρ 2 n . The existence of minimizers of J ρ at the critical value ρ = ρ 2 n 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 J ρ at the critical value ρ = ρ 2 n is in contrast to the case of two dimensions. © 2003 Wiley Periodicals, Inc.