Journal of Inequalities and Applications 
Volume 1 (1997), Issue 1, Pages 47-71
doi:10.1155/S1025583497000040

Radial solutions of equations and inequalities involving the p-Laplacian

Wolfgang Reichel and Wolfgang Walter

 Mathematisches Institut I, Universität Karlsruhe, Karlsruhe D-76128, Germany
Received 31 July 1996

Abstract

Several problems for the differential equation Lpαu=g(r,u)  with  Lpαu=r−α(rα|u′|p−2u′)′ are considered. For α=N−1, the operator Lpα is the radially symmetric p-Laplacian in ℝN. For the initial value problem with given data u(r0)=u0,u′(r0)=u′0 various uniqueness conditions and counterexamples to uniqueness are given. For the case where g is increasing in u, a sharp comparison theorem is established; it leads to maximal solutions, nonuniqueness and uniqueness results, among others. Using these results, a strong comparison principle for the boundary value problem and a number of properties of blow-up solutions are proved under weak assumptions on the nonlinearity g(r,u).