Nonlinear nonhomogeneous periodic problems

We consider a nonlinear periodic problem driven by a nonhomogeneous differential operator and a Carathéodory 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-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p-}$$\end{document}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)=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda > \widehat{\lambda}_1(p)=}$$\end{document} the first nonzero eigenvalue of the periodic scalar Laplacian. Finally, in the semilinear case (p=2),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(p=2),}$$\end{document} we show that there is a second nodal solution, for a total of four nontrivial solutions all with sign information.