Abstract
Keywords:
Laplacian; LeraySchauder degree; fixed point1 Introduction
In this paper, we consider the existence of solutions for the following system:
where , (); is called Laplacian; , ; , () and , ; , they are both nonnegative, , ; ; and are nonnegative constants; and are positive parameters.
The study of differential equations and variational problems with variable exponent growth conditions has attracted more and more attention in recent years. Many results have been obtained on these problems, for example, [116]. We refer to [3,12,16] for the applied background of these problems. If (a constant), becomes the wellknown pLaplacian. If is a general function, represents a nonhomogeneity and possesses more nonlinearity, thus is more complicated than (see [7]).
In recent years, because of the wide mathematical and physical background (see [1719]), the existence of positive solutions for the pLaplacian equation group has received extensive attention. Especially, when , the existence of positive solutions for the equation group boundary value problems has been obtained (see [2025]). On the integral boundary value problems, we refer to [2630]. But as for the Laplacian equation group, there are few papers dealing with the existence of solutions, especially the existence of solutions for the systems with multipoint and integral boundary value problems. Therefore, when is a general function, this paper mainly investigates the existence of solutions for a class of Laplacian differential systems with multipoint and integral boundary value conditions. Moreover, we discuss the existence of nonnegative solutions.
Let and , the function , () is assumed to be Carathéodory, by which we mean:
(i) For almost every , the function is continuous;
(ii) For each , the function is measurable on J;
(iii) For each , there are such that, for almost every and every with , , , , one has
Throughout the paper, we denote
The inner product in will be denoted by , will denote the absolute value and the Euclidean norm on . For , we set , ; . For any , we denote , and . For any , we denote . Spaces C, and W will be equipped with the norm , and , respectively. Then , and are Banach spaces. Denote with the norm .
We say a function is a solution of (P) if satisfies the differential equation in (P) a.e. on J and the boundary value conditions.
In this paper, we always use to denote positive constants if this does not lead to confusion. Denote
We say () satisfies a sub growth condition if satisfies
where , and . We say satisfies a general growth condition if does not satisfy a sub growth condition.
We will discuss the existence of solutions for (P) in the following two cases:
(i) satisfies a sub growth condition for ;
(ii) satisfies a general growth condition for .
This paper is organized as follows. In Section 2, we do some preparation. In Section 3, we discuss the existence of solutions of (P). Finally, in Section 4, we discuss the existence of nonnegative solutions for (P).
2 Preliminary
For any , denote (). Obviously, has the following properties.
Lemma 2.1 (see [5])
is a continuous function and satisfies the following:
(i) For any, is strictly monotone, that is,
(ii) There exists a function, as, such that
It is well known that is a homeomorphism from to for any fixed . For any , denote by the inverse operator of , then
It is clear that is continuous and sends bounded sets into bounded sets.
Let us now consider the following problem:
with the boundary value condition
where . If u is a solution of (1) with (2), by integrating (1) from 0 to t, we find that
Denote . It is easy to see that is dependent on . Define operator as
From (3), we have
By integrating (4) from 0 to t, we find that
From (2), we have
and
It is easy to obtain the following lemma.
Lemma 2.2is continuous and sends bounded sets ofto bounded sets of. Moreover,
It is clear that is a compact continuous mapping.
Let us now consider another problem
with the boundary value condition
where . Similar to the discussion of the solutions of (1) with (2), we have
and
From (9) and (10), we have
Lemma 2.3The functionhas the following properties:
(i) For any fixed, the equation
(ii) The function, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover,
Proof (i) It is easy to see that
From Lemma 2.1, it is immediate that
and hence, if (11) has a solution, then it is unique.
Let . Suppose . Since , it is easy to see that there exists an such that the ith component of satisfies
Thus the ith component of is nonzero and keeps sign, and then we have
Let us consider the equation
It is easy to see that all the solutions of (12) belong to . So, we have
which implies the existence of solutions of .
In this way, we define a function , which satisfies
(ii) By the proof of (i), we also obtain that sends bounded sets to bounded sets, and
It only remains to prove the continuity of . Let be a convergent sequence in C and as . Since is a bounded sequence, then it contains a convergent subsequence . Let as . Since , letting , we have . From (i), we get , it means that is continuous. The proof is completed. □
Now, we define the operator as
It is clear that is continuous and sends bounded sets of into bounded sets of , and hence it is a compact continuous mapping.
If u is a solution of (1) with (2), we have
and
If u is a solution of (7) with (8), we have
and
We denote
Lemma 2.4The operators () are continuous and send equiintegrable sets into relatively compact sets in.
Proof We only prove that the operator is continuous and sends equiintegrable sets in to relatively compact sets in , the rest is similar.
It is easy to check that for all . Since
it is easy to check that is a continuous operator from to .
Let now U be an equiintegrable set in , then there exists such that
We want to show that is a compact set.
Let be a sequence in , then there exists a sequence such that . For any , we have
Hence the sequence is uniformly bounded and equicontinuous. By the AscoliArzela theorem, there exists a subsequence of (which we still denote by ) convergent in C. According to the bounded continuous of the operator , we can choose a subsequence of (which we still denote by ) which is convergent in C, then is convergent in C.
From the definition of and the continuity of , we can see that is convergent in C. Thus, is convergent in . This completes the proof. □
It is easy to see that and are both compact continuous.
We denote () the Nemytskii operator associated to defined by
Lemma 2.5is a solution of (P) if and only ifis a solution of the following abstract equation:
Proof If is a solution to (P), according to the proof before Lemma 2.5, it is easy to obtain that is a solution to (S).
Conversely, if is a solution to (S), then
which implies
It follows from (S) that
then
By the condition of the mapping , we have
and then
From (S), we have
and
then
Thus
From (S), we have
By the condition of the mapping , we have
which implies that
Moreover, from (S), it is easy to obtain
and
This completes the proof. □
3 Existence of solutions
In this section, we apply LeraySchauder’s degree to deal with the existence of solutions for (P), when satisfies a sub growth condition or a general growth condition ().
We denote (S) as
where
Theorem 3.1Ifsatisfies a subgrowth condition, then the problem (P) has at least one solution for any fixed parameter ().
Proof Denote
where
According to Lemma 2.5, we know that (P) has the same solution of
It is easy to see that the operators and are compact continuous. According to Lemma 2.2, Lemma 2.3 and Lemma 2.4, we can see that and are compact continuous from to , thus is compact continuous from to W.
We claim that all the solutions of (14) are uniformly bounded for . In fact, if it is false, we can find a sequence of solutions for (14) such that as n.
From Lemma 2.2, we have
which together with the sub growth condition of implies that
From (14), we have
then
Denote . From the above inequality we have
It follows from (14) and (15) that
which implies that
Thus
It follows from (16) and (17) that .
Thus, we can choose a large enough such that all the solutions of (14) belong to . Therefore, the LeraySchauder degree is well defined for each , and
Denote
where and are defined in (5) and (13), then is the unique solution of .
It is easy to see that is a solution of if and only if is a solution of the following system:
Obviously, (19) possesses a unique solution . Note that , we have
Therefore (P) has at least one solution. This completes the proof. □
In the following, we investigate the existence of solutions for (P) when satisfies a general growth condition.
Denote
Assume the following.
(A_{1}) Let a positive constant ε be such that , , and , , where is defined in (18), and are defined in (5) and (13), respectively.
It is easy to see that is an open bounded domain in W. We have the following theorem.
Theorem 3.2Assume that (A_{1}) is satisfied. If positive parametersandare small enough, then the problem (P) has at least one solution on.
Proof Similarly, we denote . By Lemma 2.5, is a solution of
with (2) and (8) if and only if is a solution of the following abstract equation:
From the proof of Theorem 3.1, we can see that is compact continuous from to W. According to LeraySchauder’s degree theory, we only need to prove that
(1^{∘}) has no solution on for any ,
then we can conclude that the system (P) has a solution on .
(1^{∘}) If there exists a and is a solution of (20), then and λ satisfy
and
Since , there exists an i such that or .
() Suppose that , then . On the other hand, for any , we have
This implies that for each .
Note that , then , holding . Since is continuous, when is small enough, from (A_{1}), we have
It is a contradiction to for each .
() Suppose that , then . This implies that for some , and we can find
Since and is Carathéodory, it is easy to see that
thus
From Lemma 2.2, is continuous, then we have
When is small enough, from (A_{1}) and (21), we can conclude that
(ii) If . Similar to the proof of (i), we get a contradiction. Thus .
Summarizing this argument, for each , has no solution on when positive parameters and are small enough.
(2^{∘}) Since (where is defined in (18)) is the unique solution of , and (A_{1}) holds , we can see that the LeraySchauder degree
This completes the proof. □
As applications of Theorem 3.2, we have the following.
Corollary 3.3Assume that, where; satisfy, . Ifandare small enough, then the problem (P) possesses at least one solution.
Proof It is easy to have
From and the definition of , we have
Since , then there exists a small enough ε such that
From Lemma 2.2 and the small enough , we have
Obviously, it follows from , and the small enough that , , and .
Thus, the conditions of (A_{1}) are satisfied, then the problem (P) possesses at least one solution. □
Corollary 3.4Assume that, where; satisfy. If, andare small enough, then the problem (P) possesses at least one solution.
Proof From Lemma 2.2, we have
Since is dependent on the small enough and , then it follows from the continuity of that is small enough, which implies that
From , and the small enough and , it is easy to have that , , and .
Thus, the conditions of (A_{1}) are satisfied, then the problem (P) possesses at least one solution. □
We denote
Assume the following.
(A_{2}) Let positive constants and be such that , , and , , where is defined in (18), and are defined in (5) and (13), respectively.
It is easy to see that is an open bounded domain in W. We have the following.
Corollary 3.5Assume that
whereϵ, are positive constants; satisfy, and, . Then the problem (P) possesses at least one solution.
Proof Similar to the proof of Theorem 3.2, we only need to prove that (A_{2}) is satisfied, then we can conclude that the problem (P) possesses at least one solution.
From and the definition of , it is easy to have that
where we suppose . Since , then there exists a big enough such that
From Lemma 2.2, we have
From and the definition of , we have
Since , then there exists a such that (where ), which implies that
From Lemma 2.3, we have
Obviously, it follows from and that , , and .
Thus, the conditions of (A_{2}) are satisfied, then the problem (P) possesses at least one solution. □
Corollary 3.6Assume that
whereϵ, are positive constants; satisfy, , and, . Ifandare small enough, then the problem (P) possesses at least one solution.
Proof Similar to the proof of Corollary 3.5, we conclude that (A_{2}) is satisfied. Then the problem (P) possesses at least one solution. □
4 Existence of nonnegative solutions
In the following, we deal with the existence of nonnegative solutions of (P). For any , the notation () means () for any . For any , the notation means , the notation means .
Theorem 4.1We assume that
Then every solution of (P) is nonnegative.
Proof (i) We shall show that is nonnegative.
If is a solution of (P), from Lemma 2.5, we have
which together with (5), (1^{0}) and (4^{0}) implies that
Thus for any . Holding is decreasing, namely for any with .
According to the boundary value condition (2) and condition (3^{0}), we have
then
(ii) We shall show that is nonnegative.
If is a solution of (P), From Lemma 2.5, we have
We claim that . If it is false, then there exists some such that , which together with condition (2^{0}) implies that
Similar to the proof of Lemma 2.3, the boundary value condition (8) implies
From (22) and , we get a contradiction to (23).
We claim that
If it is false, then there exists some such that
which together with condition (2^{0}) implies
From (25) and , we get a contradiction to (23). Thus (24) is valid.
Denote
Obviously, , , and is decreasing, i.e., for any with . For any , there exist such that
We can conclude that is increasing on , and is decreasing on . Thus
which together with (8) implies that
It follows from (26) and (27) that
If
from (8) and (28), we have
Combining (29) and (30), we have
Combining (i) and (ii), we find that every solution of (P) is nonnegative. □
Corollary 4.2We assume that
Then we have
(a) Under the conditions of Theorem 3.1, (P) has at least one nonnegative solution;
(b) Under the conditions of Theorem 3.2, (P) has at least one nonnegative solution.
Proof (a) Define
where
Denote
where , then satisfies the Carathéodory condition, and .
We assume the following.
(A_{2}) for uniformly, where and .
Obviously, satisfies a sub growth condition.
Let us consider the existence of solutions of the following system:
According to Theorem 3.1, (31) has at least a solution . From Theorem 4.1, we can see that is nonnegative. Thus, is a nonnegative solution of (P).
(b) It is similar to the proof of (a).
This completes the proof. □
5 Examples
Example 5.1 Consider the following problem:
Obviously, and are Caratheodory, , , , then the conditions of Theorem 3.1 are satisfied, then () has a solution.
Example 5.2 Consider the following problem
Obviously, and are Caratheodory, , , , the conditions of Corollary 4.2 are satisfied, then () has a nonnegative solution.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors typed, read and approved the final manuscript.
Acknowledgements
Partly supported by the National Science Foundation of China (10701066 & 10971087).
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