Existence and Nonexistence of Weak Positive Solution for Classes of 3 × 3 P-Laplacian Elliptic Systems
1LAMIS Laboratory, Tebessa University, Tebessa, Algeria
2LANOS Laboratory, Badji Mokhtar University, Annaba, Algeria
In this work, we are interested to obtain some result of existence and nonexistence of large positive weak solution for the following p-laplacian system where λ, µ and ν are a positive parameter, and Ω is a bounded domain in with smooth boundary ∂Ω. The proof of the main results is based to the sub-supersolutions method.
Keywords: Positive solutions, Sub-supersolutions, p-Laplacian systems
International Journal of Partial Differential Equations and Applications, 2013 1 (1),
Received November 12, 2013; Revised November 29, 2013; Accepted December 23, 2013Copyright: © 2013 Science and Education Publishing. All Rights Reserved.
Cite this article:
- Rafik, Guefaifia, Akrout Kamel, and Saifia Warda. "Existence and Nonexistence of Weak Positive Solution for Classes of 3 × 3 P-Laplacian Elliptic Systems." International Journal of Partial Differential Equations and Applications 1.1 (2013): 13-17.
- Rafik, G. , Kamel, A. , & Warda, S. (2013). Existence and Nonexistence of Weak Positive Solution for Classes of 3 × 3 P-Laplacian Elliptic Systems. International Journal of Partial Differential Equations and Applications, 1(1), 13-17.
- Rafik, Guefaifia, Akrout Kamel, and Saifia Warda. "Existence and Nonexistence of Weak Positive Solution for Classes of 3 × 3 P-Laplacian Elliptic Systems." International Journal of Partial Differential Equations and Applications 1, no. 1 (2013): 13-17.
|Import into BibTeX||Import into EndNote||Import into RefMan||Import into RefWorks|
Problems involving the p-Laplacian arise from many branches of pure mathe-matics as in the theory of quasiregular and quasiconformal mapping as well as from various problems in mathematical physics notably the flow of non-Newtonian fluids.
Hai, Shivaji  studied the existence of positive solution for the p-Laplacian system
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 , the author studied the existence and nonexistence of positive weak solution to the following quasilinear elliptic system
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
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 (ψ1, ψ2, ψ3) and supersolution (z1, z2, z3) of (1.3) such that, satisfies
for all with
We suppose that α, β, γ, f, g and h verify the following assumptions;
(H1) are monotone functions such that
Let λ1, µ1 and ν1 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
i.e. (ψ1, ψ2, ψ3) is a subsolution of (1.3). Next, let, and are the solution of
Where is a large number to be chosen later: We shall verify that (z1, z2, z3) 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
which imply that
Then we have
in another hand
i.e (z1, z2 , z3) 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
Proof. Multiplying the first equation by u; we have from Young inequality that
then, we have
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.
Theorem 3. consider the following system in
1) the system has a large positive solutions if
2) In the case where
the system has not non trivial positive solutions if
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
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
this inequality hold if (5.4) hold.
Theorem 4. The following problem has a large positive solution if λ large
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 t1 and t3
is decreasing with respect to t2
Proof. The problem (5.7) can be written under the following system form
In this case, the assumptions of theorem (3.1) holds.
|||S. Ala, G. A. Afrouzi, Q. Zhang & A. Niknam, Existence of positive solutions for variable exponent elliptic systems, Boundary Value Problems 2012, 2012:37.|
|||G. A. Afrouzi & J. Vahidi, On critical exponent for the existence and stability properties of positive weak solutions for some nonlinear elliptic systems involving the (p, q)-Laplacian and indefinite weight function, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 121, No. 1, February 2011, pp. 83-91.|
|||G. A. Afrouzi & Z. Valinejad, Nonexistence of result for some p-Laplacian Systems, The Journal of Mathematics and Computer Science Vol. 3 No. 2 (2011) 112-116.|
|||J. Ali, R. Shivaji, Existence results for classes of Laplacian systems with sign-changing weight, Applied Mathematics Letters 20 (2007) 558-562.|
|||J. Ali, R. Shivaji, Positive solutions for a class of p-Laplacian systems with multiple parameters, J. Math. Anal. Appl. 335 (2007) 1013-1019.s.|
|||C. Chen, On positive weak solutions for a class of quasilinear elliptic systems, Nonlinear Analysis 62 (2005) 751-756.|
|||R. Dalmasso, Existence and uniqueness of positive solutions of semilinear elliptic systems, Nonlinear Analysis 39 (2000) 559-568.|
|||H. Dang, S. Oruganti & R. Shivaji, nonexistence of non positive solutions for a class of semilinear elliptic systems, Rocky Mountain Journal of mathematics, Volume 36, Number 6, 2006.|
|||D. D. Hai, R. Shivaji, An existence result on positive solutions for a class of p-Laplacian systems, Nonlinear Analysis 56 (2004) 1007-1010.|
|||P. Drˆabek & J. Hern´andez, Existence and uniqueness of positive solutions for some quasilinear elliptic problems, Nonlinear Analysis 44 (2001).|
|||S. Haghaieghi & G. A. Afrouzi, Sub-super solutions for (p-q) Laplacian systems, Boundary Value Problems 2011, 2011:52.|
|||R. Shivaji a & J. Ye, Nonexistence results for classes of 3 × 3 elliptic systems, Nonlinear Analysis 74 (2011) 1485-1494. 189-204.|