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

Abstract

We consider degenerated nonlinear PDE of elliptic type: −div(a(|x|)|∇w(x)|p−2∇w(x))+h|x|,w(x),∇w(x),x|x|=ϕ(w(x)), a.e. in the ball in ℝn. Using the argument based on Opial‐type inequalities, we investigate qualitative properties of their radial solutions, like, for example, the a priori estimates, maximum principles, monotonicity, as well as nonexistence of the nontrivial solutions.