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# Towers of Nodal Bubbles for the Bahri-Coron Problem in Punctured Domains

Ponente: Jorge Faya
Institución: Universidad de Chile
Tipo de Evento: Investigación
Cuándo 24/08/2017 de 11:00 a 12:00 Por definir vCal iCal

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$ which contains a ball centered at the origin. Consider   problem
\label{1}
(\wp_{\delta})\qquad\left\{
\begin{array}
[c]{ll}%
-\Delta u=|u|^{ 2^{*}- 2}u & \text{in }\Omega_{\delta},\\[3mm]
u=0 & \text{on }\partial\Omega_{\delta},\\
\end{array}
\right.

here   $N\geq 3$,   $2^{*}=\frac{2N}{N-2}$ is the critical Sobolev exponent and $\Omega_{\delta}:=\{x\in\Omega: |x|>\delta \}$. In this talk we will discuss the existence of nodal solutions $(u_{m, \delta})$ for problem $(\wp_{\delta})$. Moreover, if $\Omega$ is starshaped,   we show that the  solutions $(u_{m, \delta})$  concentrate and blowup at  $0$, as $\delta\rightarrow0$, and their limit profile is a  tower of nodal bubbles, i.e., they are a sum of rescaled nonradial sign-changing solutions to the limit problem
\label{1}
\qquad\left\{
\begin{array}
[c]{ll}%
-\Delta u=|u|^{ 2^{*}- 2}u, & u\in D^{1,2}(\mathbb{R^{N}})\\
\end{array}
\right.

centered at the origin.

This is a joint work with M. Clapp and F. Pacella.

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