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

TL;DR

This paper investigates the asymptotic behavior of solutions to a high-order elliptic PDE with critical exponential non-linearity, revealing a quantization phenomenon where the total curvature concentrates in integer multiples of a fundamental constant.

Contribution

It establishes a quantization result for the total $Q$-curvature of solutions to a critical exponential elliptic equation of order 2m, extending understanding of blow-up phenomena.

Findings

Total $Q$-curvature concentrates in integer multiples of a fundamental constant.

Solutions exhibit quantized energy levels related to the geometry of the sphere.

The limit of the integral of the energy density is an integer multiple of a specific constant.

Abstract

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 $$\Lambda:=\lim_{k\to\infty}\int_\Omega u_k(-\Delta)^m u_k dx<\infty,$$ 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.