Asymptotic profile of positive solutions of Lane-Emden problems in dimension two

We consider families $u_p$ of solutions to the problem \begin{equation}\label{problemAbstract} \left\{\begin{array}{lr}-Δu= u^p & \mbox{ in }Ω\\ u>0 & \mbox{ in }Ω\\ u=0 & \mbox{ on }\partial Ω\end{array}\right.\tag{$\mathcal E_p$} \end{equation} where $p>1$ and $Ω$ is a smooth bounded domain of $\mathbb R^2$. We give a complete description of the asymptotic behavior of $u_p$ as $p\rightarrow +\infty$, under the condition \[p\int_Ω |\nabla u_p|^2\,dx\rightarrow β\in\mathbb R\qquad\mbox{ as $p\rightarrow +\infty$}.\]

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