Quantization for an elliptic equation of order 2m 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 $H^m(\Omega )$ to the equation $(-\Delta {)}^m{u}_k={\lambda }_k{u}_k{e}^{m{u}_k^2}$ subject to Dirichlet boundary conditions, where $0<{\lambda }_k\to 0$. Assuming that $$\Lambda :=\underset{k\to \infty }{\lim }{\int }_{\Omega }{u}_k(-\Delta {)}^m{u}_{k}dx<\infty ,$$ we prove that $\Lambda$ is an integer multiple of $Λ_1:=(2m-1)!\vol(S^{2m})$, the total $Q$-curvature of the standard $2m$-dimensional sphere.

33 pages

References in corpus (3)

• Classification of solutions to the higher order Liouville's equation on R^{2m}
• A threshold phenomenon for embeddings of $H_0^m$ into Orlicz spaces
• Asymptotics and quantization for a mean-field equation of higher order

Cited by in corpus (3)

• Existence and asymptotics for solutions of a non-local Q-curvature equation in dimension three
• Improved Adams-type inequalities and their extremals in dimension 2m
• Blow-up behaviour of a fractional Adams-Moser-Trudinger type inequality in odd dimension