A multiplicity theorem for problems with the p-Laplacian

摘要

We consider a nonlinear elliptic problem driven by the p-Laplacian, with a parameter, lambda is an element of R and a non-linearity exhibiting a superlinear behavior both at zero and at infinity. We show that if the parameter; is bigger than lambda(2) = the second eigenvalue of (-Delta(p), W-0(1.p) (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]. (c) 2006 Elsevier Inc. All rights reserved.