$L^{\infty }$-norm and energy quantization for the planar Lane–Emden problem with large exponent

Abstract

For any smooth bounded domain $\Omega \subset {\mathbb {R}}^2$, we consider positive solutions to

$$\begin{aligned} \left\{ \begin{array}{lr}-\Delta u= u^p &{} \text{ in } \Omega \\ u=0 &{} \text{ on } \partial \Omega \end{array}\right. \end{aligned}$$

which satisfy the uniform energy bound

$$\begin{aligned}p\Vert \nabla u\Vert _{\infty }\le C\end{aligned}$$

for $p>1$. We prove convergence to $\sqrt{e}$ as $p\rightarrow +\infty$ of the $L^{\infty }$-norm of any solution. We further deduce quantization of the energy to multiples of $8\pi e$, thus completing the analysis performed in De Marchis et al. (J Fixed Point Theory Appl 19:889–916, 2017).