Existence and Nonexistence of Weak Positive Solution for a Class of p-Laplacian Systems

AKROUT Kameland GUEFAIFIA Rafik

LAMIS Laboratory,Tebessa University,Tebessa,Algeria.

LANOS Laboratory,Badji Mokhtar University,Annaba,Algeria.

Existence and Nonexistence of Weak Positive Solution for a Class of p-Laplacian Systems

AKROUT Kamel?and GUEFAIFIA Rafik

LAMIS Laboratory,Tebessa University,Tebessa,Algeria.

LANOS Laboratory,Badji Mokhtar University,Annaba,Algeria.

Received 22 October 2013;Accepted 24 March 2014

.In this work,we are interested to obtain some result of existence and nonexistence of positive weak solution for the following p-Laplacian system

AMS Subject Classifications:35J25,35J60

Chinese Library Classifications:O175.8,O175.25,O175.29

Positive solutions;sub-supersolutions;elliptic systems.

1 Introduction

In this paper,we are concerned with the existence and nonexistence of positive weak solution to the quasilinear elliptic system

where are a positive parameter,and ? is a bounded domain in RNwith smooth boundary??.We prove the existence of a positive weak solution for λi＞,1≤i≤m when

Problems involving the p-Laplacian arise from many branches of pure mathematics as in thetheoryofquasiregularandquasiconformalmappingas wellasfromvarious problems in mathematical physics notably the flow of non-Newtonian fluids.

Hai,Shivaji[1]studied the existence of positive solution for the p-Laplacian system

where f(s),g(s)are the increasing functions in[0,∞)and satisfy

the authors showed that the problem(1.2)has at least one positive solution provided that λ＞0 is large enough.

In[2],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 α,β≥0,γ,δ＞0,θ=(p-1-α)(q-1-β)-γδ＞0,then problem(1.3)has a positive weak solution for each λ＞0;

(ii)If θ=0 and pγ=q(p-1-α),then there exists λ0＞0 such that for 0＜λ＜λ0,then problem(1.3)has no nontrivial nonnegative weak solution.

2 Definitions and notations

Definition 2.1.We called positive weak solution u=(u1,…,um)∈X of(1.1)such that satisfies

for all φ=(φ1,…,φm)∈X with φi≥0,1≤i≤m.

Definition 2.2.We called positive weak subsolution ψ=(ψ1,…,ψm)∈X and supersolution z=(z1,…,zm)∈X of(1.1)such that ψi≤zi,?i,1≤i≤m,satisfies Z

and

for all φ=(φ1,…,φm)∈X with φi≥0,1≤i≤m.

the following assumptions;

and

An example:

Let λpibe the first eigenvalue of-Δpiwith Dirichlet boundary conditions and ?ithe corresponding positive eigenfunction withk?ik∞=1,and Mi,σi,δ＞0,1≤i≤m such that

The assumption(H1)assume that

3 Main results

Proof.We shall verify that ψi,1≤i≤m,where

then

Hence

i.e.,ψ=(ψ1,…,ψm)∈X is a subsolution of(1.1).

Next,let ωibe the solution of

Let

By(H1)and(H2),we can choose C large enough so that

whereμi=kωik∞.Then

which imply that

Then we have

i.e.,z=(z1,…,zm)∈X is a supersolution of(1.1)with zi≥ψi,1≤i≤m for C large.Thus, there exists a solution u=(u1,…,um)∈X of(1.1)with ψi≤ui≤zi,1≤i≤m.

Proof.Multiplying Eq.(1.1)by uiand integrating over ?,we obtain

in an other hand

Then,we have

Corollary 3.1.Consider the following system in X

1)The system(3.1)has a positive weak solution if

2)The system(3.1)has not positive weak solution ifˉλi＜λpi

where

Proof.1)Using Theorem 3.1,the assumption(3.2)imply the desired result. 2)(3.3)and the following generalized Young inequality

imply that

Multiplying the equation(i)in(3.1)by uiand integrating over ?,we obtain by using (3.5)

then

which is a contradiction ifˉλi＜λpi.

Corollary 3.2.The following problem has a positive solution if λ large

where ? is a bounded domain in RNwith smooth boundary??,λ is a positive parameter and γ is a function of class L∞(?)and H is of class C1(Rm)verify

The problem(3.6)can be written under the following system form

In this case,we have

Then the assumptions of theorem(3.1)holds.

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10.4208/jpde.v27.n2.6 June 2014

?Corresponding author.Email addresses:akroutkamel@gmail.com(K.Akrout),nabilrad12@yahoo.fr(R. Guefaifia)