Existence and Nonexistence of Entire Positive Solutions for a Class of Singular p-Laplacian Elliptic System *

In this paper, we show the existence and nonexistence of entire positive solutions for a class of singular elliptic system We have that entire large positive solutions fail to exist if f and g are sublinear and b and d have fast decay at infinity. However, if f and g satisfy some growth conditions at infinity, and b, d are of slow decay or fast decay at infinity, then the system has infinitely many entire solutions, which are large or bounded.

We have that entire large positive solutions fail to exist if f and g are sublinear and b and d have fast decay at infinity.However, if f and g satisfy some growth conditions at infinity, and b, d are of slow decay or fast decay at infinity, then the system has infinitely many entire solutions, which are large or bounded.

Introduction
In this paper, we mainly consider the existence and nonexistence of positive solutions for the following singular p-laplacian elliptic system: where , and are continuous, positive and nondecreasing functions in ) When , the following semi-linear elliptic system: = 0, = = 2 a p q     , , = 0, in , , , = 0, in , has been studied extensively over the years, for example see [1][2][3][4].If for which existence results for boundary blow-up positive solution can be found in a recent paper by Lair and Wood [5].The authors established that all positive entire radial solutions of system above are boundary blow-up provided that On the other hand, if then all positive entire radial solutions of this system are bounded.F. Cìrstea and V. Rădulescu [2] extended the above results to a larger class of systems Yang [6] extended the above results to a class of quasi-linear elliptic system: are positive and non-decreasing.When and g satisfy then there exists an entire positive radial solution, and in addition, the function , then all entire positive radial solutions are large.
On the other hand, if and satisfy b d , then all entire positive radial solutions are bounded.
While in [7], the author got the relevant results of the same system only on the following conditions (H7) are continuous; , , , : [0, ) [0, ) b d g f    (H8) g and f are non-decreasing functions on [0, ). However, when , there are few results about the existence and nonexistence of singular p-Laplacian elliptic system (1).And as to the single equation, we can refer to [8].The present results are complements and extensions of some results in [7,9], which to be more precise, if , you can get the relevant existence and nonexistence results for a class of semilinear elliptic system with gradient term in [9]; Meanwhile, if , you can get the relevant existence results for a class of quasi-linear elliptic system in [7].
For convenience, we need the following definition: is called an entire large solution(or explosive solution, or blow-up solution), if it is classical solution of (2) on and N R   u x   and   v x   as x   .Now we give our main theorem: Theorem 1. Suppose f and g satisfy and satisfy the decay conditions , b d where , then problem (1) has no positive entire radial large solution.

= min{ , } m pq
In order to state our results conveniently, let us write where satisfies   , and F   Then the system (1) has infinitely many positive entire  .Moreover, the following hold: 1) If

  <
B   and , then u and are bounded; , that is all positive entire solutions of (1) are large.

Proof of Theorem
and there exist > In this section, we consider the proof of Theorem 1 by contradictions.Assume that the system (1) has the positive entire radial large solution .From (1), we know that ( , ) u v the system (1) has a positive radial bounded solution satisfying then the system (1) has infinitely many positive entire large solutions; it is easy to see that are positive and nondecreasing functions.Moreover, we have and

as
. It follows from (3) that there exists such that and Then by ( 8) and ( 9), we have Then we can get     then the system (1) has infinitely many positive entire bounded solutions.
So, for all , we obtain where is a positive constant.As , we have , so the last inequality above is valid.Notice that (4), we choose such that Thus, we have By (11), we get where (13) means that U and V are bounded and so and are bounded which is a contradiction.It follows that (1) has no positive entire radial large solutions and the proof is now completed.

u v
Remark.In Theorem 1, if , and , > 2 p q f g satisfy and satisfy the same decay conditions (4), we can also get the same result that problem (1) has no positive entire radial large solution.

, b d
In the following,we will give the detailed proof.Proof.We also consider the proof by contradiction.If using the same process in Theorem 1, we will omit that items here.Assume that the system (1) has the positive entire radial large solution , we can get from the given condition above that there exists such that Thus, we can get , for 0, here     , U r V r are the same functions defined in Theorem 1.
As the proof of Theorem 1, we omit the same process here, for all , we obtain where is a positive constant.Notice the condition (4), we choose such that together with (12), we get We claim the above inequality is invalid.In fact, set a function has no positive entire radial large solution and the proof of the remark is completed.

Proofs of Theorem 2 and Theorem 3
Proof of Theorem 2. We start by showing that (1) has positive radial solutions.On this purpose we fix >   and >   and we show that the system  is an increasing function on ,and it can not be always controlled by a fixed constant, which is a contraction.It follows that system (1) [0, )  has positive solution ( , (where V x are positive solutions of (1).Integrating ( 14) we have Let and be the sequences of posi-tive continuous functions defined on [0 by Obviously, for all , we have The monotonicity of f and g yield Repeating such arguments we deduce that and we obtain that sequences and are nondecreasing on where has been defined before.And then integrat-ing on we obtain It follows from By the arbitrariness of 0 c , we see that is a positive solution of (15), that is, is an entire positive solution of (1).Notice , ) (0, ) (0, )       was chosen arbitrarily, it follows that (1) has infinitely many positive entire solutions.1) If and , then which implies that are the positive entire bounded solutions of (1).
The last part of the proof is clear from the proof of Theorem 2. The proof of Theorem 3 is now finished.

Proof of Theorem 4
1) It follows from the proof of Theorem 3, we have and .
Let be arbitrary.From ( 18) and (19 Taking into account the monotonicity of , there exists We claim that is finite.Indeed, if not, we let and the assumption (6) leads us to a contradiction,thus are increasing functions, it follows that the map is nondecreasing and Thus the sequences are bounded from above on bounded sets.Let Then is a positive solution of ( 14). ( , ) u v In order to conclude the proof, it is enough to show that is a large solution of (14).We see , , , , 0 Since f and g are positive functions and we can conclude that is a large solution of (14) and so is a positive entire large solution of (1).Thus any large solution of (14) provide a positive entire large solution of (1) with Since ( , ) (0, ) (0, )       was chosen arbitrarily, it follows that (1) has infinitely many positive entire large solutions. 2

The Existence and Nonexistence of Entire Positive Solutions of the Corresponding Singular Elliptic Systems with Gradient Term
In this section, we consider the following singular elliptic systems with gradient term:  We can get the same four theorems under the same conditions in the foregoing items.In the detailed proofs, only a few modifications should be noticed.Such as, we note where 0 is defined as before and other changes are similar, so we omit here.

F and 1 F
 are both increasing functions on [ , )

4
are the positive entire large solutions of (1).The proof of theorem is now completed.)U V Proof of Theorem 3. If condition (5) holds, then we have It follows from (18) and (19) that is bounded, which implies that (1) has infinitely many positive entire bounded solutions.The proof is completed.( , ) u v m