A note on minimizing the Moser-Trudinger functional of high order derivatives

Let Omega be a bounded smooth domain in R-2n (n >= 2). In this note, we consider the functional J(p)(u) = 1/2 integral(Omega) vertical bar(-Delta)(n/2) u(x)vertical bar(2) dx - rho log integral(Omega) h(x)e(u(x)) dx. Suppose h(x) is a smooth function with 0 < a <= h(x) <= b. Then, using the idea of Lin and Wei [C. Lin, J. Wei, Locating the peaks of solutions via the maximum principle II: A local version of the method of moving planes, Comm. Pure Appl. Math. LV [(2003) 784-809], we prove the existence of minimizers of J(rho) for any rho <= rho 2n = 2(2n) n!(n-1)omega(2n) in a space of functions H = H-n (Omega) boolean AND {u, (-Delta)(j) u epsilon H-0(1)(Omega), j = 1,2,...,[n-1/2]} where omega(2n), is the area of the unit sphere S2n-1 in R-2n. (C) 2007 Elsevier Ltd. All rights reserved.Mathematics, AppliedMathematicsSCI(E)0ARTICLE41437-14446