On a variant of the maximum principle involving radial $p$-Laplacian with applications to nonlinear eigenvalue problems and nonexistence results

We obtain the variant of maximum principle for radial solutions of $p$-harmonic equation $-a\Delta_p(w)=\phi(w)$. As a consequence of this result we prove monotonicity of constant sign solutions, analyze the support of the solutions and study their oscillations. The results are applied to various type nonlinear eigenvalue problems and nonexistence theorems.