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:

- ()
- For any
- ()
- There exists a function
such that
as
and
for all

It is well known that conditions and ensure that
is a homeomorphism from
onto
The vector version of the Laplace operator, namely the case when for

as well as the cases

satisfy conditions and Further examples of functions satisfying these conditions can be found in [8].

When it can be checked that the function as given in
examples (1.1), (1.2) and (1.3) also satisfy the
property

In the scalar case, functions satisfying (1.4) have been called asymptotically homogeneous functions, see [1], [4], [5] and [6], where they were used in connection with the existence of solutions to quasilinear elliptic problems. They form an important class of non-homogeneous functions satisfying a suitable homogeneous behavior at infinity (or zero) without being necessarily asymptotic to any power at infinity or zero.

As we said before the function is assumed to be Carathéodory. This means that satisfies the following conditions:

- ()
- for almost every the function is continuous;
- ()
- for each the function is measurable on ;
- ()
- for each there is
such that, for almost every and every
with
one has

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.,

for a.e all and all

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

(1.5) |

where is positive a.e. and belongs to This is a key result to be used in later sections. Together with this result we also show in this section that an associated initial value problem has a unique solution.

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.