Properties of solutions to some weighted $p$-Laplacian equation

TL;DR

This paper investigates qualitative properties of positive solutions to a class of weighted p-Laplacian equations with degenerate elliptic operators, considering various nonlinearities and weight functions.

Contribution

It establishes new qualitative results for solutions to weighted p-Laplacian equations with different nonlinearities and weight functions in smooth domains.

Findings

Positive solutions exhibit specific regularity properties.

Weight functions in the Muckenhoupt class influence solution behavior.

Results apply to a range of nonlinearities in degenerate elliptic equations.

Abstract

In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by \[-\operatorname{div}(w|\nabla u|^{p-2}\nabla u)=f(x,u);\;\;w\in \mathcal{A}_p\] on smooth domain and for varying nonlinearity $f$.