Calculus of Variations and Geometric Measure Theory

L. Martinazzi - M. Struwe

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

created by martinazz on 05 Mar 2010
modified on 17 Jul 2018


Accepted Paper

Inserted: 5 mar 2010
Last Updated: 17 jul 2018

Journal: Math. Z.
Year: 2010

ArXiv: 1003.1329 PDF


On a smoothly bounded domain $\Omega\subset\R{2m}$ we consider a sequence of positive solutions $u_k\stackrel{w}{\rightharpoondown} 0$ in $H^m(\Omega)$ to the equation $(-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2}$ subject to Dirichlet boundary conditions, where $0<\lambda_k\to 0$. Assuming that $$\Lambda:=\lim{k\to\infty}\int\Omega uk(-\Delta)m uk dx<\infty,$$ we prove that $\Lambda$ is an integer multiple of $\Lambda_1:=(2m-1)!\vol(S^{2m})$, the total $Q$-curvature of the standard $2m$-dimensional sphere.