Solutions of second-order and fourth-order ODE's on the half-line

We start by studying the existence of positive solutions (on the positive half-line) for the differential equation u''(x)=a(x)u-g(u), under the condition u'(0)=0, and that u vanishes at infinity. The coefficient a(x) is positive, g satisfies suitable growth hypotheses or, in alternative, is bounded. Then we deal with the analogous fourth order problem, where the left-hand side of the equation is replaced by -u''''+cu'' (c>0), g(u)/u is a power of |u|, and the further condition u'''(0)=0 is required.