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
onto
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
property
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
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(1.5) |
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.