This work is concerned with the entire positive solutions for a -Laplacian elliptic system of equations with a gradient term. We find the sufficient condition for nonexistence of entire large positive solutions and existence of infinitely many entire solutions, which are large or bounded.
In this paper, we consider a class of -Laplacian elliptic system of equations with a gradient term
where , , , the nonlinearities are positive, continuous and nondecreasing functions for each variable, and are continuous functions, and the potentials are c-positive functions (or circumferentially positive) in a domain which are nonnegative in Ω and satisfy the following:
• If and , then there exists a domain such that and for all .
Problem (1.1) arises in the theory of quasiregular and quasiconformal mappings, stochastic control and non-Newtonian fluids, etc. In the non-Newtonian theory, the quantity is a characteristic of the medium. Media with are called dilatant fluids, while are called pseudoplastics. If , they are Newtonian fluids.
We are concerned only with the entire positive solutions of problem (1.1). An entire large (or explosive) solution of problem (1.1) means a pair of functions for solving problem (1.1) in the weak sense and , as .
In recent years, existence and nonexistence of entire solutions for the semilinear elliptic system
have been studied by many authors; see [1-3] and the references therein. For example, Ghergu and Radulescu , Lair and Wood , Kawano and Kusano  discussed the entire solutions under proper conditions. For other works for a single equation, we refer to [4,5] and the references therein. Moreover, a comprehensive discussion on entire solutions for a large class of semilinear systems
can be found in Ghergu and Radulescu . Later, Yang  extended their results to a class of quasilinear elliptic systems. To our best knowledge, problem (1.1) of equations with a gradient term has not been sufficiently investigated. Only a few papers have dealt with this problem (1.1). In , Ghergu and Radulescu studied the existence of blow-up solutions for the system
They proved that boundary blow-up solutions fail to exist if f and g are sublinear, whereas this result holds if is bounded and a, b are slow decay at infinity. They also showed the existence of infinitely blow-up solutions in if a, b are of fast decay and f, g satisfy a sublinear-type growth condition at infinity. In , Cirstea and Radulescu studied a related problem. Recently, Zhang and Liu  studied the semilinear elliptic systems with a gradient term
and obtained the sufficient condition of nonexistence and existence of positive entire solutions. Furthermore, for the single equation with a gradient term, we read [10-12] and the references therein.
Motivated by the results of the above cited papers, we study the nonexistence and existence of positive entire solutions for system (1.1) deeply, and the results of the semilinear systems are extended to the quasilinear ones. In , the authors studied the existence and nonexistence of entire large positive solutions of -Laplacian system (1.1) with , . However, they obtained different results under the suitable conditions. In this paper, our main purpose is to establish new results under new conditions for system (1.1). Roughly speaking, we find that the entire large positive solutions fail to exist if f, g are sublinear and a, b have fast decay at infinity, while f, g satisfy some growth conditions at infinity, and a, b are of slow decay or fast decay at infinity, then the system has many infinitely entire solutions, which are large or bounded. Unfortunately, it remains unknown whether an analogous result holds for system (1.1) with different gradient power , for .
2 Main results and proof
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
Firstly, we give a nonexistence result of a positive entire radial large solution of system (1.1).
Theorem 1Suppose thatfandgsatisfy
anda, bsatisfy the decay conditions
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
Combining (2.3) and (2.4), we can get
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
It follows that , and we can find such that
Thus, we have
From (2.6), we can get
where , . Similarly,
then we can get
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 2Suppose that
Then system (1.1) has infinitely many positive entire solutions . Moreover, the following hold:
(i) Ifaandbsatisfy , then all entire positive solutions of (1.1) are large.
(ii) Ifaandbsatisfy , , 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
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
it follows that
Then we can get
It follows from is increasing on and (2.10) that
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 3If , , , and there exist , such that
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. □
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.
(i) It follows from the proof of Theorem 2 that
Choosing an arbitrary , from (2.14) and (2.15), we can get
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.
holds, then by (2.16), we have
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. □
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
The first and second authors were supported by the National Science Foundation of Shandong Province of China (ZR2012AM018) and Changwon National University in 2013, respectively. The authors would like to express their sincere gratitude to the anonymous reviewers for their insightful and constructive comments.
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