First, inspired by Zhang [20] and Santos [16], we will define functions
and
by

So, for each
, let
given by

where

It is easy to check that

and, as a consequence,

Moreover, it is also easy to verify.

Lemma 2.1.

Suppose that
and
hold. Then, for each
,

(i)

(ii)
,

(iii)

(iv)
,

(v)
,

(vi)

By Lemma 2.1(iii), (iv), and (2.2), the function
, given by

is well defined and continuous. Again, by using Lemma 2.1(i) and (ii),

Besides this,
, for each
, and using Lemma 2.1, it follows that
satisfies, for each
, the following.

Lemma 2.2.

Suppose that
and
hold. Then, for each
,

(i)
,

(ii)

(iii)

(iv)

And, in relation to
, we have the folowing.

Lemma 2.3.

Suppose that
and
hold. Then, for each
,

(i)
,

(ii)

Finally, we will define, for each
,
, by

So,
is a continuous function and we have (see proof in the appendix).

Lemma 2.4.

Suppose that
and
hold. Then,

(i)

(ii)

(iii)

(iv)

(v)

By Lemma 2.4(ii), there exists a
such that
, where by either (
) or (
)
, we have

So, by Lemma 2.4(v), there exists a
such that
. That is,

Let
by

where
,
is given by
where
is the unique positive and radially symmetric solution of problem

More specifically, by DiBenedetto [23],
, for some
. In fact,
satisfies

So, by (2.10), (2.11), and (2.13), we have for each
,

Hence, after some pattern calculations, we show that there is a
such that
and

As consequences of (2.9), (2.13) and (2.15), we have
and

and hence, by Lemma 2.2 (i), (2.7) and
, we obtain

that is, by using (2.2), we have

In particular, making
, we get from (2.15), Lemma 2.2(i) and
that
and satisfies (1.8), for each
. That is,
is an upper solution to (1.1).

To prove (1.9), first we observe, using Lemma 2.2(i) and (2.15), that

So, by definition of
,
and hypothesis (1.5), we have

Thus,

Recalling that
and using (1.5) again, we obtain

Thus by (2.9), (2.13), and
, there is one positive constant
such that (1.9) holds. This ends the proof of Theorem 1.3.