Quantization for an elliptic equation of order 2 m with critical exponential non-linearity

Abstract

On a smoothly bounded domain $${\Omega\subset\mathbb{R}^{2m}}$$ we consider a sequence of positive solutions $${u_k\stackrel{w}{\rightharpoondown}0}$$ in H m (Ω) to the equation $${(-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2}}$$ subject to Dirichlet boundary conditions, where 0<λ k → 0. Assuming that $$0 < \Lambda:=\lim_{k\to\infty}\int\limits_\Omega u_k(-\Delta)^m u_k dx < \infty,$$ we prove that Λ is an integer multiple of Λ1 :=(2m − 1)! vol(S 2m ), the total Q-curvature of the standard 2m-dimensional spher