We prove the existence of a positive solution to the BVP (\Phi(t)u'(t))'=f(t,u(t)),\,\,\,\,\,\,\,\,\,\,\,u'(0)=u(1)=0,
imposing some conditions on Φ and f. In particular, weassume \Phi(t)f(t,u)
to be decreasing in t. Our methodcombines variational and topological arguments and can be appliedto some elliptic problems in annular domains. An L_\infty
boundfor the solution is provided by the L_\infty
norm of any testfunction with negative energy.