Nonlinear, Nonhomogeneous Periodic Problems with no Growth Control on the Reaction

J Dyn Control Syst. 2015;21(3):423-441. doi: 10.1007/s10883-014-9245-4. Epub 2014 Aug 30.


We consider a nonlinear periodic problem driven by a nonhomogeneous differential operator, which includes as a particular case the scalar p-Laplacian. We assume that the reaction is a Carathéodory function which admits time-dependent zeros of constant sign. No growth control near ±∞ is imposed on the reaction. Using variational methods coupled with suitable truncation and comparison techniques, we prove two multiplicity theorems providing sign information for all the solutions.

Keywords: Constant sign solutions; Mountain pass theorem; Nodal solutions; Nonhomogeneous differential operator; Nonlinear strong maximum principle; Second deformation theorem.