In this section, we will apply Leray-Schauder's degree to deal with the existence of solutions for (1.1)-(1.2), when
satisfies sub-(
) growth condition. Moreover, the asymptotic behavior has been discussed.
Theorem 3.1.
Assume that
is an open bounded set in
such that the following conditions hold.
(10)For each
the problem
with boundary condition (1.2) has no solution on
.
(20)The equation
has no solution on
.
(30)The Brouwer degree
.
Then problems (1.1)-(1.2) have a solution on
.
Proof.
Let us consider the following equation with boundary value condition (1.2):
For any
observe that if
is a solution to (3.1) with (1.2) or
is a solution to (3.3) with (1.2), we have necessarily
It means that (3.1) with (1.2) and (3.3) with (1.2) have the same solutions for 
We denote
defined by
where
is defined by (2.29). Let
and the fixed point of
is a solution for (3.3) with (1.2). Also problem (3.3) with (1.2) can be written in the equivalent form
Since
is Caratheodory, it is easy to see that
is continuous and sends bounded sets into equi-integrable sets. It is easy to see that
is compact continuous. According to Lemmas 2.2 and 2.3, we can conclude that
is continuous and compact from
to
for any
. We assume that for
, (3.7) does not have a solution on
; otherwise we complete the proof. Now from hypothesis (10) it follows that (3.7) has no solutions for
. For
(3.3) is equivalent to the problem
and if
is a solution to this problem, we must have
Hence
where
is a constant. From Lemma 2.5, there exist
such that
,
Hence
, it holds
, a constant. Thus by (3.9)
which together with hypothesis (20), implies that
Thus we have proved that (3.7) has no solution
on
then we get that for each
, the Leray-Schauder degree
is well defined for
, and from the properties of that degree, we have
Now it is clear that the problem
is equivalent to problem (1.1)-(1.2), and (3.12) tells us that problem (3.13) will have a solution if we can show that
Since
then
From Lemma 2.2, we have
. By the properties of the Leray-Schauder degree, we have
where the function
is defined in (3.2) and
denotes the Brouwer degree. By hypothesis (30), this last degree is different from zero. This completes the proof.
Our next theorem is a consequence of Theorem 3.1. As an application of Theorem 3.1, let us consider the following equation with (1.2)
where
is Caratheodory,
is continuous and Caratheodory, and for any fixed
if
then
.
Theorem 3.2.
Assume that the following conditions hold
(10)
for all
and all
where
satisfies 
(20)
for
uniformly
(30)for large enough
, the equation
has no solution on
, where 
(40)the Brouwer degree
for large enough
, where 
Then problem (3.18) with (1.2) has at least one solution.
Proof.
Denote
At first, we consider the following problem:
According to the proof of Theorem 3.1, we know that (3.21) with (1.2) has the same solution of
where 
We claim that all the solutions of (3.21) are uniformly bounded for
. In fact, if it is false, we can find a sequence of solutions
for (3.21) with (1.2) such that
as
, and
for any
.
Since
are solutions of (3.21) with (1.2), so
. According to Lemma 2.5, there exist
such that
, then
where
means the function which is uniformly convergent to 0 (as
). According to the property of
and (3.23), then there exists a positive constant
such that
then we have
Denote
, then
Thus
Since
, from (3.27) we have
Denote
, then
and
, then
possesses a convergent subsequence (which denoted by
), and then there exists a vector
such that
Without loss of generality, we assume that
. Since
, there exist
such that
and then from (3.27) we have
Since
(as
),
and
, we have
From (3.28)–(3.32), we have
So we get
where
satisfies 
Since
from(1.2) and (3.34), we have
Since
, according to the continuity of
we have
and it is a contradiction to (3.35). This implies that there exists a big enough
such that all the solutions of (3.21) with (1.2) belong to
, and then we have
If we prove that
, then we obtain the existence of solutions (3.18) with (1.2).
Now we consider the following equation with: (1.2)
where 
We denote
defined by
Similar to the proof of Theorem 3.1, we know that (3.38) with (1.2) has the same solution of
Similar to the discussions of the above, for any
all the solutions of (3.38) with (1.2) are uniformly bounded.
If
is a solution of the following equation with (1.2):
then we have
Since
we have
and it means that
is a solution of
according to hypothesis (30), (3.38) has no solutions
on
then we get that for each
, the Leray-Schauder degree
is well defined, and from the properties of that degree, we have
Now it is clear that
So
If we prove that
, then we obtain the existence of solutions (3.18) with (1.2). By the properties of the Leray-Schauder degree, we have
By hypothesis (40), this last degree is different from zero. We obtain that (3.18) with (1.2) has at least one solution. This completes the proof.
Corollary 3.3.
If
is Caratheodory, which satisfies the conditions of Theorem 3.2,
where
are positive functions, and satisfies
then (3.18) with (1.2) has at least one solution.
Proof.
Since
then
has only one solution
, and
and according to Theorem 3.2, we get that (3.18) with (1.2) has at least a solution. This completes the proof.
Now let us consider the boundary asymptotic behavior of solutions of system (1.1)-(1.2).
Theorem 3.4.
If
is a solution of (1.1)-(1.2) which is given in Theorem 3.2, then
(i)
(ii)
as 
(iii)
as 
Proof.
Since
exists and
,
and
both exist and equal, we can conclude that
. Since
we have
Thus
(i)
(ii)
as
(iii)
as
This completes the proof.
Corollary 3.5.
Assume that
exists,
, and
then
(i)
(ii)
as 
(iii)
as 