Concentration phenomena to a higher order Liouville equation

Abstract

We study blow-up and quantization phenomena for a sequence of solutions $(u_k)$ to the prescribed $Q$-curvature problem $$(-\Delta {)}^n{u}_k=Q_k{e}^{2n{u}_k}\text{in }\Omega \subset {\mathbb{R}}^{2n},{\int }_{\Omega }{e}^{2n{u}_k}dx\le C,$$ under natural assumptions on $Q_k$. It is well-known that, up to a subsequence, either $(u_k)$ is bounded in a suitable norm, or there exists ${\beta }_k\to \infty$ such that $u_k={\beta }_k(\phi +o(1))$ in $\Omega \setminus (S_1\cup S_{\phi })$ for some non-trivial non-positive $n$-harmonic function $\phi$ and for a finite set $S_1$, where $S_{\phi }$ is the zero set of $\phi$. We prove quantization of the total curvature ${\int }_{\tilde{\Omega }}{Q}_k{e}^{2n{u}_k}dx$ on the region $\tilde{\Omega }⋐(\Omega \setminus S_{\phi })$. We also consider a non-local case in dimension three.