A multiplicity theorem for problems with the p-Laplacian

We consider a nonlinear elliptic problem driven by the p-Laplacian, with a parameter λ ∈ R and a nonlinearity exhibiting a superlinear behavior both at zero and at infinity. We show that if the parameter λ is bigger than λ2 = the second eigenvalue of (− p,W 1,p 0 (Z)), then the problem has at least three nontrivial solutions. Our approach combines the method of upper–lower solutions with variational techniques involving the Second Deformation Theorem. The multiplicity result that we prove extends an earlier semilinear (i.e. p = 2) result due to Struwe [M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990]. © 2006 Elsevier Inc. All rights reserved.