A priori bounds for energy-bounded solutions of critical polyharmonic equations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910269390716928 |
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| author | Carletti, Lorenzo Premoselli, Bruno |
| author_facet | Carletti, Lorenzo Premoselli, Bruno |
| contents | We investigate critical polyharmonic equations of the following type: $$ Lu = |u|^{2^\sharp-2} u \quad \text{ in } Ω$$ with Dirichlet boundary conditions, in a smooth bounded domain $Ω$ of $\mathbb{R}^n$. Here $L$ is an elliptic differential operator of even integer order $2 \le 2k < n$ whose leading order term is $(-Δ)^k$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_29690 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A priori bounds for energy-bounded solutions of critical polyharmonic equations Carletti, Lorenzo Premoselli, Bruno Analysis of PDEs 35B33, 35B44, 35B45, 35J30, 35J40, 35J91, 35G30, We investigate critical polyharmonic equations of the following type: $$ Lu = |u|^{2^\sharp-2} u \quad \text{ in } Ω$$ with Dirichlet boundary conditions, in a smooth bounded domain $Ω$ of $\mathbb{R}^n$. Here $L$ is an elliptic differential operator of even integer order $2 \le 2k < n$ whose leading order term is $(-Δ)^k$ 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. |
| title | A priori bounds for energy-bounded solutions of critical polyharmonic equations |
| topic | Analysis of PDEs 35B33, 35B44, 35B45, 35J30, 35J40, 35J91, 35G30, |
| url | https://arxiv.org/abs/2605.29690 |