L∞-norm and energy quantization for the planar Lane-Emden problem with large exponent

For any smooth bounded domain Omega subset of R-2, we consider positive solutions to {-Delta u = u(p) in Omega u = 0 on partial derivative Omega which satisfy the uniform energy bound p parallel to del u parallel to(infinity) <= C for p > 1. We prove convergence to root e as p -> +infinity of the L-infinity- 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).