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

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.