Existence of exceptional growing-up solutions for a class of non-linear second order ordinary differential equations

We establish the existence of nontrivial global solutions for the ordinary differential equation u″+f(u)=g(u′), when f is at most linear and g is superlinear. We then provide a result of global uniqueness in certain classes of power-like functions for general autonomous second order ordinary differential equations. This result allows us to show the exceptional character of the global solutions for a class of equations including u″+u=|u′|^p−1u′, for p>1.