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

Abstract

We investigate critical polyharmonic equations of the type: $$Lu=|u{|}^{2^{♯}-2}u\text{ in }\Omega$$ with Dirichlet boundary conditions, in a smooth bounded domain $\Omega$ of ${\mathbb{R}}^n$. Here $L$ is an elliptic differential operator of integer order $2\le 2k<n$ whose leading order term is $(-\Delta {)}^k$ and $2^{♯}=\frac{2n}{n-2k}$ is the critical Sobolev exponent. Our main result establishes, in large dimensions, uniform \emph{a priori} bounds in $C^{2k}(\overline{\Omega })$ for bounded-energy solutions of this problem, that only depend on an upper bound on the energy. We prove this 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.

With respect to the V1 we corrected a few typos, slightly changed the abstract and we added one subsection in the introduction to discuss our results more clearly. Nothing has changed from Sections 2 onwards