Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

Solutions and multiple solutions for superlinear perturbations of the periodic scalar p-Laplacian

Sophia Th. Kyritsia1, Donal O'Regana2 and Nikolaos S. Papageorgioua3

a1 Department of Mathematics, Hellenic Naval Academy, Piraeus 18539, Greece (

a2 Department of Mathematics, National University of Ireland, Galway, Ireland (

a3 Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece (


We consider a nonlinear periodic problem driven by the scalar p-Laplacian and with a reaction term which exhibits a (p – 1)-superlinear growth near ±∞ but need not satisfy the Ambrosetti-Rabinowitz condition. Combining critical point theory with Morse theory we prove an existence theorem. Then, using variational methods together with truncation techniques, we prove a multiplicity theorem establishing the existence of at least five non-trivial solutions, with precise sign information for all of them (two positive solutions, two negative solutions and a nodal (sign changing) solution).

(Received September 04 2011)


  • scalar p-Laplacian;
  • critical groups;
  • mountain pass theorem;
  • C condition;
  • p-superlinearity;
  • AR condition

2010 Mathematics subject classification

  • Primary 34B15;
  • 34B18;
  • Secondary 34C25;
  • 58E05