BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Concentrating solutions for a Hénon-type problem on general domains Faya, Jorge


We consider the problem% \begin{equation*} \qquad\left\{ \begin{array} [c]{ll}% -\Delta u = \beta(x)|u|^{p-1-\epsilon }u & \text{in }\Omega,\\ u=0 & \text{on }\partial\Omega, \end{array} \right. \end{equation*} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq3,$ $p:=\frac{N+2}{N-2}$ is the Sobolev critical exponent, $\epsilon$ is a small positive parameter and the function $\beta\in C^{1}(\overline{\Omega})$ is strictly positive on $\overline{\Omega}$. In this talk we shall present a recent result about the existence of positive and sign changing solutions whose asymptotic profile is a sum of $k$ bubbles which accumulate at a single point at the boundary as $\varepsilon$ tends to zero. This is joint work with professors Juan D\'avila and Fethi Mahmoudi.

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