Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight

We study the problem of the existence and multiplicity of positive periodic solutions to the scalar ODE u '' + lambda a(t)g(u) = 0, lambda > 0, where g(x) is a positive function on R(+), superlinear at zero and sublinear at infinity, and a(t) is a T-periodic and sign indefinite weight with negative mean value. We first show the nonexistence of solutions for some classes of nonlinearities g(x) when lambda is small. Then, using critical point theory, we prove the existence of at least two positive T-periodic solutions for lambda large. Some examples are also provided.