Sturm type theorems for general quasi-variational nonlinear problems

Abstract

Consider the general equation \( [G_p(u^{'})] + f(t,u) = Q (t,u, u^{'}), t \in (a,b) \) associated to the initial-value problem \( u(a) \neq {0} , u^{'}(a) = u(b) = 0 \) where \( f \) is a restoring force and Q represents a nonlinear damping term. Through an analysis of the equation, we give precise estimates of b in terms of the initial data that extend some results derived from Sturm comparison theorems for linear differential equations of second order. In particular, we show some important theorems of non existence of radial solutions of Dirichlet problems in \( {\Bbb R}^n \) that either significantly improve the former ones, with the m-Laplacian operator, or cover cases never appeared before, with the mean curvature operator.