Positive solutions for semi-linear elliptic equations in exterior domains

We prove the existence of a solution, decaying to zero at infinity, for the second order differential equation $$ frac{1}{A(t)}(A(t)u'(t))'+phi(t)+f(t,u(t))=0,quad tin (a,infty). $$ Then we give a simple proof, under some sufficient conditions which unify and generalize most of those given in the bibliography, for the existence of a positive solution for the semilinear second order elliptic equation $$ Delta u+varphi(x,u)+g( |x|) x. abla u =0, $$ in an exterior domain of the Euclidean space ${mathbb{R}}^{n},ngeq 3$