Let N≥2and 1uis a non-negative classical solution of the Lane–Emden equation, thenu≡0. The method of moving planes combined with the Kelvin transform provides an elegant proof of this theorem. A classical approach of Serrin and Zou, originally used for the Lane–Emden system, yields another proof but only in lower dimensions. Motivated by this, we further refine this approach to find an alternative proof of the Liouville type theorem in all dimensions.
Villavert, J. (2017). A Refined Approach for Non-Negative Entire Solutions of Δ u + up = 0 with Subcritical Sobolev Growth, Advanced Nonlinear Studies, 17(4), 691-703. doi: https://doi.org/10.1515/ans-2016-6024
Advanced Nonlinear Studies