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Towers of nodal bubbles for the Bahri-Coron problem in punctured domains

Banff International Research Station for Mathematical Innovation and Discovery

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^{N}$ which contains a ball of radius $R$ centered at the origin, $N\geq3.$ Under suitable symmetry assumptions, for each $\delta\in(0,R),$ we establish the existence of a sequence $(u_{m,\delta})$ of nodal solutions to the critical problem% \[ \left\{ \begin{array} [c]{ll}% -\Delta u=|u|^{2^{*}-2}u & \text{in }\Omega_{\delta}:=\{x\in\Omega :\left\vert x\right\vert >\delta\},\\ u=0 & \text{on }\partial\Omega_{\delta}, \end{array} \right. \] where $2^{*}:=\frac{2N}{N-2}$ is the critical Sobolev exponent. We show that, if $\Omega$ is strictly starshaped then, for each $m\in\mathbb{N},$ the solutions $u_{m,\delta}$ concentrate and blow up at $0,$ as $\delta \rightarrow0,$ and their limit profile is a tower of nodal bubbles, i.e., it is a sum of rescaled nonradial sign-changing solutions to the limit problem% \[ \left\{ \begin{array} [c]{c}% -\Delta u=|u|^{2^{*}-2}u,\\ u\in D^{1,2}(\mathbb{R}^{N}), \end{array} \right. \] centered at the origin. This is joint work with Jorge Faya (Universidad de Chile) and Filomena Pacella (Universit\`{a} "La Sapienza" di Roma).

Mathematics; Partial differential equations; Differential geometry

video/mp4

Author affiliation: Universidad Nacional Autónoma de México

2017-11-21

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63669

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/