Nonlinear nonhomogeneous periodic problems

Abstract

We consider a nonlinear periodic problem driven by a nonhomogeneous differential operator and a Caratheodory 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 \({\lambda > \widehat{\lambda}_1(p)=}\) the first nonzero eigenvalue of the periodic scalar Laplacian. Finally, in the semilinear case \({(p=2),}\) we show that there is a second nodal solution, for a total of four nontrivial solutions all with sign information.