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

Abstract

Let be a bounded, smooth domain in R 2n , n ≥ 2. The well-known MoserTrudinger inequality ensures the nonlinear functional J ρ (u) is bounded from below if and only if ρ ≤ ρ 2n := 2 2n n!(n − 1)!ω 2n , whereand ω 2n is the area of the unit sphere S 2n−1 in R 2n . In this paper, we prove the inf u∈X J ρ (u) is always attained for ρ ≤ ρ 2n .The existence of minimizers of J ρ at the critical value ρ = ρ 2n 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 2n is in contrast to the case of two dimensions.

Asymptotic behavior of a fourth order mean field equation with Dirichlet boundary condition

“…Let Q 0 be such that R(Q 0 , Q 0 ) = max Q∈Ω R(Q, Q). Similar computations in [page 799, [27]] yield…”

“…The proof of Theorem 1.2 follows along the lines of Sections 3 and 4 of [27]: we just need to change the Navier boundary condition to Dirichlet boundary condition. Let us sketch the changes.…”

“…Assuming that Ω is convex, the corresponding property (P1) and (P2) are established in [38]. Later, Lin and Wei [27] considered the attainment of least energy solution and removed the convexity assumption of [38]. Therefore, property (P1) and (P2) are established for (1.4).…”

“…This poses a major difficulty in using the method of moving planes (as in [27]) to exclude the boundary bubbles. We overcome this by using the Pohozaev identity and by proving strong pointwise estimates for blowingup solutions to (1.1).…”

“…It follows from (1.12) that J ρ is bounded from below if and only if ρ ≤ 64π 2 (for the proof, see the appendix of [27]). Furthermore, if ρ < 64π 2 , the minimizer of J ρ actually exists, that is, there exists a…”