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.

(1^{0})For each the problem

with boundary condition (1.2) has no solution on .

(2^{0})The equation

has no solution on .

(3^{0})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 (1^{0}) 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 (2^{0}), 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 (3^{0}), 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

(1^{0}) for all and all where satisfies

(2^{0}) for uniformly

(3^{0})for large enough , the equation

has no solution on , where

(4^{0})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 (3^{0}), (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 (4^{0}), 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