In this paper we study the boundary value problem
By a solution of we understand a function of class with absolutely continuous, which satisfies
In most of the paper we shall ask to satisfy the following conditions:
It is well known that conditions and ensure that
is a homeomorphism from
The vector version of the Laplace operator, namely the case when for
When it can be checked that the function as given in
examples (1.1), (1.2) and (1.3) also satisfy the
As we said before the function is assumed to be Carathéodory. This means that satisfies the following conditions:
In case (mapping into ), we shall say is Carathéodory, if () and () are satisfied for each and the function in () can be chosen independently of , i.e.,
We state a piece of notations used in this paper. For we shall set The norm in and will be denoted by while the norm in by
This paper is organized as follows. In section 2 we extend the concept of Asymptotically Homogeneous functions from the scalar to the vector case, and study some of their properties. In particular it is seen how this family is related to the vector -Laplace function at infinity.
In section 3 we study the
eigenvalues of the weighted eigenvalue problem of the form
In section 4 we give our first existence result for a system of ode's, under Dirichlet boundary conditions, containing a quasilinear operator generated by a vector function which is Asymptotically Homogeneous at infinity.
Section 5 is dedicated to the study of a system of ode's whose quasilinear operator is not Asymptotically Homogeneous, but its components are. A simple example of this situation is given by the function which is not Asymptotically Homogeneous at infinity. We prove in this section our second existence result for a system of ode's, with Dirichlet boundary conditions, containing this type of quasilinear operator.
Finally in Section 6, we give some examples of vector functions which are Asymptotically Homogeneous at infinity.