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

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 inℝ n . 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.