In the first part of the paper a variational characterisation of the periodic eigenvalues (the so-called Fučik spectrum) of a semilinear, positive homogeneous Sturm–Liouville equation is given. The proof relies on the S1-invariance of the equation.
In the second part a nonlinear Sturm–Liouville equation with, typically, an exponential nonlinearity is considered. It is proved that under certain conditions this equation is solvable for arbitrary forcing terms. The proof uses a comparison of the minimax levels of the functional associated to this equation with suitable values in the Fučik spectrum.