Monotonicity results in half spaces for quasilinear elliptic equations involving a singular term

TL;DR

This paper proves that positive solutions to a class of quasilinear elliptic equations with singular terms in a half-space are monotone increasing in the direction orthogonal to the boundary, using a priori estimates and the moving plane method.

Contribution

It establishes monotonicity results for solutions to singular quasilinear elliptic equations in half-spaces, extending known techniques to singular terms.

Findings

Solutions are monotone increasing in the boundary-normal direction.

The approach combines a priori estimates with the moving plane technique.

Results apply to equations with singular terms involving inverse powers of the solution.

Abstract

We consider positive solutions to $\displaystyle -\Delta_p u=\frac{1}{u^\gamma}+f(u)$ under zero Dirichlet condition in the half space. Exploiting a prio-ri estimates and the moving plane technique, we prove that any solution is monotone increasing in the direction orthogonal to the boundary.