Abstract:We investigate critical polyharmonic equations of the following type: $$ Lu = |u|^{2^\sharp-2} u \quad \text{ in } \Omega $$ with Dirichlet boundary conditions, in a smooth bounded domain $\Omega$ of $\mathbb{R}^n$. Here $L$ is an elliptic differential operator of even integer order $2 \le 2k < n$ whose leading order term is $(-\Delta)^k$ and $2^\sharp = \frac{2n}{n-2k}$ 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.
From: Bruno Premoselli [view email]
[v1]
Thu, 28 May 2026 09:52:37 UTC (55 KB)
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