On Uniqueness Conditions for Decreasing Solutions of Semilinear Elliptic Equations

For I E C([O,no)) fl C'((O,00)) and b > 0, existence and uniqueness of radial solutions u = u(r) of the problem Au + f(u) = 0 in R' (n > 2), u(0) = b and u'(0) = 0 are well known. The uniqueness for the above problem with boundary conditions u(R) = 0 and u'(0) = 0 is less known beside the cases where limr_oo u(r) = 0. It is our goal to give some sufficient conditions for the uniqueness of the solutions of the problem D0 u + f(u+) = 0 (r > 0),u(p) = 0 and u'(0) = 0 based only on the evolution of the functions f(t) and