Maximum principles and nonexistence results for radial solutions to equations involving p ‐Laplacian

We obtain the variant of maximum principle for radial solutions of, possibly singular, p ‐harmonic equations of the form\documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}\begin{eqnarray*} -a(|x|)\Delta_{p}(w)+h \left(|x|, w, \nabla w(x)\cdot\frac{x}{|x|}\right) =\phi(w), \end{eqnarray*}\end{document} as well as for solutions of the related ODE. We show that for the considered class of equations local maxima of | w | form a monotone sequence in | x | and constant sign solutions are monotone. The results are applied to nonexistence and nonlinear eigenvalue problems. We generalize our previous work for the case h ≡0. Copyright © 2010 John Wiley & Sons, Ltd.