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 ODEu″+λa(t)g(u)=0,λ>0, where g(x)g(x) is a positive function on R+R+, superlinear at zero and sublinear at infinity, and a(t)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)g(x) when λ is small. Then, using critical point theory, we prove the existence of at least two positive T-periodic solutions for λ large. Some examples are also provided.