We now state and prove the main results of this paper.
In order to describe our results conveniently, let us define , and let , denote inverse functions of and , where , . Moreover, we define
where .
Firstly, we give a nonexistence result of a positive entire radial large solution of system (1.1).
Theorem 1
Suppose that
f
and
g
satisfy
(2.1)
and a, b satisfy the decay conditions
(2.2)
where , then problem (1.1) has no positive entire radial large solution.
Proof Our proof is by the method of contradiction. That is, we assume that system (1.1) has the positive entire radial large solution . From (1.1), we know that
Now, we set
It is easy to see that are positive and nondecreasing functions. Moreover, we have , and as . It follows from (2.1) that there exists such that
(2.3)
and
(2.4)
Combining (2.3) and (2.4), we can get
(2.5)
Then we have
Thus, for all , we obtain
where C is a positive constant. Because of , the last inequality above is valid for . Noticing that (2.2), we choose such that
(2.6)
It follows that , and we can find such that
(2.7)
Thus, we have
From (2.6), we can get
that is,
where , . Similarly,
then we can get
(2.8)
which means that U and V are bounded and so u and v are bounded, which is a contradiction. It follows that (1.1) has no positive entire radial large solutions. □
Remark 1 In fact, through a slight change of the proofs of Theorem 1, we can obtain the same result as that of problem (1.1). That is, if and f, g satisfy
and a, b satisfy the decay conditions (2.2), then problem (1.1) still has no positive entire radial large solution.
Secondly, we give existence results of positive entire solutions of system (1.1).
Theorem 2
Suppose that
Then system (1.1) has infinitely many positive entire solutions . Moreover, the following hold:
-
(i)
If a and b satisfy , then all entire positive solutions of (1.1) are large.
-
(ii)
If a and b satisfy , , then all entire positive solutions of (1.1) are bounded.
Proof We start by showing that (1.1) has positive radial solutions. To this end, we fix and show that the system
(2.9)
has solutions . Thus, , are positive solutions of system (1.1). Integrating (2.9), we have
Let and be sequences of positive continuous functions defined on by
Obviously, , , , for all . And the monotonicity of f and g yields , for .
Repeating such arguments, we can deduce that
and , are nondecreasing sequences on . Noticing that
and
it follows that
Then we can get
that is,
(2.10)
It follows from is increasing on and (2.10) that
(2.11)
And from , we know that . By (2.11), the sequences and are bounded and increasing on for any . Thus, and have subsequences converging uniformly to u and v on . Consequently, is a positive solution of (2.9); therefore, is an entire positive solution of (1.1). Noticing that , and are chosen arbitrarily, we can obtain that system (1.1) has infinitely many positive entire solutions.
-
(i)
If , since , for , we have
which yields is the positive entire large solution of (1.1).
-
(ii)
If , , then
which implies that is the positive entire bounded solution of system (1.1). Thus, the proof of Theorem 2 is finished.
□
Theorem 3 If , , , and there exist , such that
(2.12)
then system (1.1) has an entire positive radial bounded solution (for ) satisfying
Proof If the condition (2.12) holds, then we have
Since is strictly increasing on , we have
The rest of the proof obviously holds from the proof of Theorem 2. The proof of Theorem 3 is now finished. □
Theorem 4
-
(i)
If
and
(2.13)
then system (1.1) has infinitely many positive entire large solutions.
-
(ii)
If , , and
then system (1.1) has infinitely many positive entire bounded solutions.
Proof
-
(i)
It follows from the proof of Theorem 2 that
(2.14)
and
(2.15)
Choosing an arbitrary , from (2.14) and (2.15), we can get
which implies
(2.16)
Taking account of the monotonicity of , there exists
We claim that is finite. Indeed, if not, we let in (2.16) and the assumption (2.13) leads to a contradiction. Thus, is finite. Since , are increasing functions, it follows that the map is nondecreasing and
Thus, the sequences and are bounded from above on bounded sets. Let
then is a positive solution of (2.9).
In order to conclude the proof, we need to show that is a large solution of (2.9). By the proof of Theorem 2, we have
And because f and g are positive functions and
we can conclude that is a large solution of (2.9) and so is a positive entire large solution of (1.1). Thus, any large solution of (2.9) provides a positive entire large solution of (1.1) with and . Since was chosen arbitrarily, it follows that (1.1) has infinitely many positive entire large solutions.
-
(ii)
If
holds, then by (2.16), we have
Thus,
Thus, the sequences and are bounded from above on bounded sets. Let
then is a positive solution of (2.9).
It follows from (2.14) and (2.15) that is bounded, which implies that (1.1) has infinitely many positive entire bounded solutions. □