arXiv subjects
Publications and source records attributed to Serena Rocci.
We show that the classical Brezis-Nirenberg problem $$Δu + |u|^{4 \over N-2} u + \varepsilon u = 0 ,\quad {\mbox {in}} \quad Ω, \quad u= 0 , \quad {\mbox {on}} \quad \partial Ω$$ admits nodal solutions clustering around a point on the boundary of $Ω$ as $\varepsilon \to 0$, for smooth bounded domains $Ω\subset \mathbb{R}^N $ in dimensions $N\geq 7$.
The problem \begin{equation} \label{bn} -Δu=|u|^{4\over n-2}u+λV u\ \hbox{in}\ Ω,\ u=0\ \hbox{on}\ \partialΩ \end{equation} where $Ω$ is a bounded regular domain in $\mathbb R^n$, $λ\in \mathbb R$ and $V\in C^0(\overline Ω),$ that was introduced by Brezis and Nirenberg in their famous paper, where they address the existence of positive solutions in the autonomous case, i.e. the potential $V$ is constant. Since then, a huge amount of work has been done. In the following we will make a brief history highlighting the results which are much closer to the problem we wish to study in the present paper.