A second order non-autonomous problem on the half-line: a variational approach

We study the existence of positive solutions for the differential equation u″(x) = a(x)u(x)−g(u(x)), with u′(0) = u(+∞) = 0, where a is a positive function and g satisfies some growth hypotheses. The main motivation is to check that some well known results concerning the existence of homoclinics for the autonomous case (where a is constant) extend to the nonautonomous equation.