Keywords: Positive solutions, Subsupersolutions, pLaplacian systems
International Journal of Partial Differential Equations and Applications, 2013 1 (1),
pp 1317.
DOI: 10.12691/ijpdea113
Received November 12, 2013; Revised November 29, 2013; Accepted December 23, 2013
Copyright © 2014 Science and Education Publishing. All Rights Reserved.
1. Introduction
Problems involving the pLaplacian arise from many branches of pure mathematics as in the theory of quasiregular and quasiconformal mapping as well as from various problems in mathematical physics notably the flow of nonNewtonian fluids.
Hai, Shivaji ^{[9]} studied the existence of positive solution for the pLaplacian system
 (1.1) 
which f(s); g(s) are the increasing functions in and satisfy
the authors showed that the problem (1.2) has at least one positive solution provided that is large enough.
In ^{[6]}, the author studied the existence and nonexistence of positive weak solution to the following quasilinear elliptic system
 (1.2) 
The first eigenfunction is used to construct the subsolution of problem (1.3), the main results are as follows:
(i) If then problem (1.3) has a positive weak solution for each
(ii) If then there exists such that for then problem (1.3) has no nontrivial nonnegative weak solution.
In this paper, we are concerned with the existence and nonexistence of positive weak solution to the quasilinear elliptic system
 (1.3) 
where λ, µ and ν are a positive parameter, and Ω is a bounded domain in with smooth boundary . We prove the existence of a large positive weak solution for λ, µ and ν large when
2. Definitions and Notations
Definition 1. We called positive weak solution (u; v; w) of (1.3) such that satisfies
for all with
Definition 2. We called positive weak subsolution (ψ_{1}, ψ_{2}, ψ_{3}) and supersolution (z_{1}, z_{2}, z_{3}) of (1.3) such that, satisfies
and
for all with
We suppose that α, β, γ, f, g and h verify the following assumptions;
(H1) are monotone functions such that
for all
(H2)
(H3)
Let λ_{1}, µ_{1} and ν_{1} be the first eigenvalue of −∆_{p}, −∆_{q} and −∆_{r} with Dirichlet boundary conditions and , and the corresponding positive eigenfunction with , and m, δ > 0, such that
We denote by
3. Existence Results
Theorem 1. Let (H1) and (H2) hold. Then for λ large, the system (1.3) has a large positive solution (u, v, w):
Proof. We shall verify that
is a subsolution of (1.3) for λ large: Let with .A calculation shows that
Now, on we have
Next, on we have for some , and therefore for λ, µ and ν large
Hence
i.e. (ψ_{1}, ψ_{2}, ψ_{3}) is a subsolution of (1.3). Next, let, and are the solution of
Let
Where is a large number to be chosen later: We shall verify that (z_{1}, z_{2}, z_{3}) is a supersolution of (1.3) for λ, µ and ν large.
To this end, let with
By (H1) and (H2), we can choose large enough so that
then
and
then
which imply that
Then we have
in another hand
similar
i.e (z_{1}, z_{2} , z_{3}) is a supersolution of (1.3) with for large Thus, there exists a solution (u, v, w) of (1.3) with
4. Nonexistence Results
Theorem 2. If f, g and h verify (H 3) ,the system (1.3) has not nontrivial positive solutions for
 (4.1) 
Proof. Multiplying the first equation by u; we have from Young inequality that
then, we have
 (4.2) 
Note that
Combining (4.2) and (4.3), we obtain
which is a contradiction if (4.2) hold.
Thus (1.3) has no nontrivial nonnegative weak solution.
5. Applications
Theorem 3. consider the following system in
 (5.1) 
1) the system has a large positive solutions if
 (5.2) 
2) In the case where
 (5.3) 
the system has not non trivial positive solutions if
 (5.4) 
Proof. 1) (5.2) imply that (H2). By using theorem 1, the system has a large positive solutions.
2) The first equation in (5.3) imply that
 (5.5) 
then, the generalized Young inequality gives
By the same manner, we conclude that, the assumption (H3) holds. Then he system (5.1) has not nontrivial positive solutions if
which imply that
 (5.6) 
this inequality hold if (5.4) hold.
Theorem 4. The following problem has a large positive solution if λ large
 (5.7) 
where Ω is a bounded domain in with smooth boundary , λ is a positive parameter and γ is a function of class and
is of class ,
is increasing with respect to t_{1} and t_{3}
is decreasing with respect to t_{2}
Proof. The problem (5.7) can be written under the following system form
In this case, the assumptions of theorem (3.1) holds.
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