A Paneitz-type problem in pierced domains

Abstract

We study the critical problem {equation} {{array}{ll} Δ^{2}u=u^{\frac{N+4}{N-4}} & {in}Ω\setminus \bar{B(ξ_0,\varepsilon)},\medskip u>0&{in}Ω\setminus \bar{B(ξ_0,\varepsilon)},\medskip u=Δu=0 & {on}\partial (Ω\setminus \bar{B(ξ_0,\varepsilon)}),{array}. \tag{P${}_{\epsilon }$} {equation} where $\Omega$ is an open bounded domain in ${\mathbb{R}}^N$, $N\ge 5$, ${\xi }_0\in \Omega$ and $B({\xi }_0,\epsilon )$ is the ball centered at ${\xi }_0$ with radius $\epsilon >0$ small enough. We construct solutions of (P${}_{\epsilon }$) blowing-up at the center of the hole as the size of the hole goes to zero.

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• Singular limits in higher order Lioville-type equations