Uniqueness results for some nonlinear initial and boundary value problems (Q1106996)

In the first part of this article a uniqueness theorem is presented for nonlinear initial value problems of the type (*) \(dx/dt=\phi (x)+p(t),\) \(t>0\); \(x(0)=0\); where p is a real-valued continuous function which may change sign near the origin. The theorem is subsequently applied to establish a result related to the uniqueness of nonnegative nontrivial radial solutions of nonlinear boundary value problems of the type (**) \(\nabla \cdot (A(| \nabla u|)\nabla u+f(u)=0,\) \(x\in {\mathbb{R}}^ n;\) \(\lim_{x\to \infty}u(x)=0\); where A is a positive continuous function such that the product pA(p) is monotone. The class of equations (**) includes the mean curvature equation.