Some results for equations containing a -Laplace like operator, a review
Raul Manasevich
CMM and DIM, Universidad de Chile

Abstract'' Let be a continuous function such that defined by is an increasing homeomorphism of . Let us also define the operator

that we call a -Laplace like operator. In this talk we will review some results for problems containing this nonlinear, nonhomogeneous operator which generalizes the -Laplace operator. In particular we will review some results for eigenvalue problems of the form

where is bounded domain and is a weight and extensions of these results to some systems.

Problems of this type, when properly formulated in the setting of Orlicz - Sobolev spaces, leads to several difficulties connected with the lack of homogeneity of and the structure of the corresponding spaces (in general they may not be reflexive). Thus, for example, the functional naturally associated to in problem , is in general neither everywhere defined nor a fortiori . This excludes the use of the standard Lagrange multiplier rule.