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Existence and Nonexistence of Weak Positive Solution for Classes of 3 × 3 P-Laplacian Elliptic Systems

Guefaifia Rafik, Akrout Kamel*, Saifia Warda

LAMIS Laboratory, Tebessa University, Tebessa, Algeria

Abstract: 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

Received November 12, 2013; Revised November 29, 2013; Accepted December 23, 2013

Citation: Guefaifia Rafik, Akrout Kamel, 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). 13-17. January 2013. doi:10.12691/ijpdea-1-1-3

1. Introduction

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 [9] studied the existence of positive solution for the p-Laplacian system

[picture] | (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

[picture] | (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

[picture] | (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 (ψ₁, ψ₂, ψ₃) and supersolution (z₁, z₂, z₃) 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 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

Hence

i.e. (ψ₁, ψ₂, ψ₃) 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 (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

then

and

then

which imply that

Then we have

in another hand

similar

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

[picture] | (4.1)

Proof. Multiplying the first equation by u; we have from Young inequality that

then, we have

[picture] | (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

[picture] | (5.1)

1) the system has a large positive solutions if

[picture] | (5.2)

2) In the case where

[picture] | (5.3)

the system has not non trivial positive solutions if

[picture] | (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

[picture] | (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

[picture] | (5.6)

this inequality hold if (5.4) hold.

Theorem 4. The following problem has a large positive solution if λ large

[picture] | (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 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.

References

[1]   | S. Ala, G. A. Afrouzi, Q. Zhang & A. Niknam, Existence of positive solutions for variable exponent elliptic systems, Boundary Value Problems 2012, 2012:37.
[2]   | 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.
[3]   | 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.
[4]   | J. Ali, R. Shivaji, Existence results for classes of Laplacian systems with sign-changing weight, Applied Mathematics Letters 20 (2007) 558-562.
[5]   | 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.
[6]   | C. Chen, On positive weak solutions for a class of quasilinear elliptic systems, Nonlinear Analysis 62 (2005) 751-756.
[7]   | R. Dalmasso, Existence and uniqueness of positive solutions of semilinear elliptic systems, Nonlinear Analysis 39 (2000) 559-568.
[8]   | 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.
[9]   | D. D. Hai, R. Shivaji, An existence result on positive solutions for a class of p-Laplacian systems, Nonlinear Analysis 56 (2004) 1007-1010.
[10]   | P. Drˆabek & J. Hern´andez, Existence and uniqueness of positive solutions for some quasilinear elliptic problems, Nonlinear Analysis 44 (2001).
[11]   | S. Haghaieghi & G. A. Afrouzi, Sub-super solutions for (p-q) Laplacian systems, Boundary Value Problems 2011, 2011:52.
[12]   | R. Shivaji a & J. Ye, Nonexistence results for classes of 3 × 3 elliptic systems, Nonlinear Analysis 74 (2011) 1485-1494. 189-204.