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

Abstract'' Let $ a:(0,\infty)\mapsto (0,\infty)$ be a continuous function such that $ \phi:\mathbb{R}\mapsto\mathbb{R}$ defined by $ \phi(s)=sa(s)$ is an increasing homeomorphism of $ \mathbb{R}$. Let us also define the operator

$\displaystyle -\Delta_\phi(u)={\rm div} (a(\vert\nabla u\vert)\nabla u ),

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

-\Delta_\phi(u) &= &\lambda\rho(x...
... \quad\mbox{ on }\partial\Omega,

where $ \Omega $ is bounded domain and $ \rho$ 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 $ \phi$ and the structure of the corresponding spaces (in general they may not be reflexive). Thus, for example, the functional naturally associated to $ -\Delta_\phi(u)$ in problem $ (P)$, is in general neither everywhere defined nor a fortiori $ C^1$. This excludes the use of the standard Lagrange multiplier rule.