On maximum principles for radial solutions to nonlinear elliptic PDE's via Opial-type inequalities

TL;DR

This paper investigates qualitative properties of radial solutions to a class of degenerated nonlinear elliptic PDEs using Opial-type inequalities, establishing maximum principles, monotonicity, and nonexistence results.

Contribution

It introduces a novel approach employing Opial-type inequalities to analyze maximum principles and qualitative behavior of solutions to nonlinear elliptic PDEs.

Findings

Established maximum principles for radial solutions.

Proved monotonicity properties of solutions.

Demonstrated nonexistence of nontrivial solutions under certain conditions.

Abstract

We consider degenerated nonlinear PDE of elliptic type: $$ - \mathrm{div}(a(|x|)|\nabla w(x)|^{p-2} \nabla w(x)) + h(|x|,w(x),\langle\nabla w(x),\frac{x}{|x|}\rangle)=\phi(w(x)), $$ where $x$ belongs to the ball in $\bf{R}^n$. Using the argument based on Opial-type inequalities, we investigate qualitative properties of their radial solutions, like e.g. maximum principles, monotonicity, as well as nonexistence of the nontrivial solutions.