A priori bounds for energy-bounded solutions of critical polyharmonic equations

We investigate critical polyharmonic equations of the following type: $$Lu=\mathrm{\mid }u{\mathrm{\mid }}^{2^{\mathrm{♯}}-2}u\text{ in }\mathrm{\Omega }$$ with Dirichlet boundary conditions, in a smooth bounded domain $\mathrm{\Omega }$ of ${\mathbb{R}}^n$. Here $L$ is an elliptic differential operator of even integer order $2\le 2k<n$ whose leading order term is $(-\mathrm{\Delta }{)}^k$ and $2^{\mathrm{♯}}=\frac{2n}{n-2k}$ is the critical Sobolev exponent. Our main result establishes, in large dimensions, uniform \emph{a priori} bounds on bounded-energy solutions of this problem under a coercivity assumption of sorts on the lower-order terms of $L$. Our results are sharp, at least when $k=1$. Our approach uses asymptotic analysis techniques and in the course of the proof we obtain in particular a new global pointwise description of bounded-energy blowing-up solutions for this problem, which is of independent interest.