Radial symmetry of solutions to anisotropic and weighted diffusion equations with discontinuous nonlinearities

TL;DR

This paper extends the symmetry results of solutions to anisotropic and weighted diffusion equations with discontinuous nonlinearities, generalizing classical isotropic results to more complex settings using integral inequalities.

Contribution

It provides the first proof of radial symmetry in anisotropic weighted problems, extending Gidas-Ni-Nirenberg results and introducing a new integral inequality approach for N>2.

Findings

Radial symmetry established for solutions in anisotropic weighted settings.

Extension of Gidas-Ni-Nirenberg symmetry results to anisotropic and weighted cases.

Introduction of a new integral inequality method for N>2.

Abstract

We prove radial symmetry for bounded nonnegative solutions of a weighted anisotropic problem. Given the anisotropic setting that we deal with, the term "radial" is understood in the Finsler framework. In the whole space, J. Serra obtained the symmetry result in the isotropic unweighted setting. In this case we provide the extension of his result to the anisotropic setting. This provides a generalization to the anisotropic setting of a celebrated result due to Gidas-Ni-Nirenberg and such a generalization is new even for in the case of linear operators whenever the dimension is greater than 2. In proper cones, the results presented are new even in the isotropic and unweighted setting for suitable nonlinear cases. Even for the previously known case of unweighted isotropic setting, the present paper provides an approach to the problem by exploiting integral (in)equalities which is new for $N>2$: this complements the corresponding symmetry result obtained via the moving planes method by Berestycki-Pacella.