Nonlinear nonhomogeneous periodic problems

We consider a nonlinear periodic problem driven by a nonhomogeneous differential operator and a Carath´eodory reaction. We show that it has at least three solutions, two of constant sign and the third nodal. In the particular case of the scalar p−Laplacian and with a parametric reaction of equidiffusive type, we show that three solutions with precise sign exist if the parameter λ >λ_1(p)= the first nonzero eigenvalue ofthe periodic scalar Laplacian. Finally, in the semilinear case (p = 2), weshow that there is a second nodal solution, for a total of four nontrivial solutions all with sign information.