Existence of Multiple Solutions for Nonlinear Dirichlet Problems with a Nonhomogeneous Differential Operator

Abstract

Abstract We consider nonlinear elliptic problems driven by a nonhomogeneous nonlinear differential operator. Using variational methods combined with Morse theory (critical groups), we prove two multiplicity results establishing three nontrivial smooth solutions. For the semilinear problem (linear differential operator), we produce four nontrivial smooth solutions. In the special case of the p-Laplacian differential operator, our framework of analysis incorporates equations which are resonant at infinity with respect to the principal eigenvalue.