Let us now introduce some weighted Sobolev spaces and their norms. Let

be a bounded domain in

with smooth boundary

. If

and

, we define

as being the subspace of

of the Lebesgue measurable function

, satisfying

If

and

, we define

(resp.,

as being the closure of

(resp.,

) with respect to the norm defined by

For the weighted Sobolev space
, we have the following compact imbedding theorem which is an extension of the classical Rellich-Kondrachov compact theorem.

Theorem 2.1 ((compact imbedding theorem) [13]).

Suppose that
is an open bounded domain with
boundary and
,
,
. Then, the imbedding
is compact if
,
.

We now consider the existence of positive solutions for problem (1.1). Our main tool will be the upper and lower solution method. This method, in the bounded domain situation, has been used by many authors, for instance, [10, 12, 13]. But for the unbounded domain, we need to establish this method and then to construct an upper solution and a lower solution for (1.1). We now give the definitions of upper and lower solutions.

Definition 2.2 (see [10, 12]).

A function

is said to be a weak lower solution of the equation

for any
,
.

Similarly, a function

is said to be a weak upper solution of (2.3) if

for any
and
in
.

A function
is said to be a weak solution of (2.3) if and only if
is a weak lower solution and weak upper solution of (2.3).

A function
is said to be less than or equal to
on
if
.

If

and

, we define the weighted Sobolev space

as being the closure of

with respect to the norm

defined by

The following lemma will be basic in our approach.

Lemma 2.3.

Let

be Lipschitz continuous and nondecreasing in

and locally Hölder continuous in

. Moreover, assume that there exist the functions

such that

Then, there exist a minimal weak solution

and a maximal weak solution

of (2.3) satisfying

and
.

Proof.

Denote

,

. Let

be a pair of upper and lower solutions of (2.3) with

, a.e. in

. We consider the boundary value problem

By Theorem 1.1 in [10], one concludes that there exists
which is a weak solution of (2.11) with
a.e. in
for
.

We define its extension by

Similarly, let

be a weak solution of the boundary value problem

and its extension is defined by

Since

, we have

. By Theorem 2.4 in [

12], we have

for

. In view of (2.15), the pointwise limits

exist and
in
.

Similar to the proof Theorem 1.1 in [10] and the proof of Theorem 7.5.1 in [23], it is not difficult to get from Theorem 2.1 that
is the maximal weak solution and
the minimal solution of (2.3), which satisfies (2.10) and
. This ends the proof of Lemma 2.3.

Our main results read as follows.

Theorem 2.4 (existence).

Let
,
. Assume the following.

The nonnegative functions
are Lipschitz continuous and nondecreasing,
. Additionally,
and
with
.

The nonnegative functions
,
are locally Hölder continuous. Let
. If

then there exists
, such that
, and the problem (1.1) admits a weak solution
.

Theorem 2.5 (nonexistence).

Let
,
. Assume that

;

there exist

such that

the functions
in
satisfy

where

and

has no nontrivial solution
.

Remark 2.6.

If assumption (2.19) holds, then

with
,
.

In fact, for this case, there exist

and

such that

for

. Therefore,

So, condition (2.19) implies (2.22).