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

Abstract

We consider degenerated nonlinear PDE of elliptic type: $$-\mathrm{div}(a(|x|)|\nabla w(x){|}^{p-2}\nabla w(x))+h(|x|,w(x),⟨\nabla w(x),\frac{x}{|x|}⟩)=\varphi (w(x)),$$ where $x$ belongs to the ball in ${\mathbf{R}}^{\mathbf{n}}$. 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.