Existence and Uniqueness of Positive Radial Solutions for a Class of Quasilinear Elliptic Systems

LI Qinand YANG Zuodong?,,2

1Institute of Mathematics,School of Mathematical Sciences,Nanjing Normal University,Nanjing 210023,China.

2School of Teacher Education,Nanjing Normal University,Nanjing 210097,China.

Existence and Uniqueness of Positive Radial Solutions for a Class of Quasilinear Elliptic Systems

LI Qin1and YANG Zuodong?,1,2

1Institute of Mathematics,School of Mathematical Sciences,Nanjing Normal University,Nanjing 210023,China.

2School of Teacher Education,Nanjing Normal University,Nanjing 210097,China.

.This articleisconcerned with the existenceanduniqueness ofpositive radial solutions for a class of quasilinear elliptic system.With some reasonable assumptions on the nonlinear source functions and their coefflcients,the existence and the upper and lower bounds of the positive solutions will be provided by using the flxed point theorem and the maximum principle for the quasilinear elliptic system.

Quasilinear elliptic equation(system);existence;uniqueness;flxed point theorem;the maximum principle.

1 Introduction

In this paper,we study the existence and uniqueness of positive solutions for the following quasilinear elliptic system

where ? is the open unit ball in RNwith N≥1,p,q,m＞1,a,b,c:[0,∞)→(0,∞)are continuous functions and f,g,h:[0,∞)→[0,∞)are continuous and nondecreasing.

In recent years,much attention has been paid to the existence and uniqueness of solutionsforthequasilinear elliptic systemswithtwo equations,in particular,fortheproblem

See,for example,([1–10])and the reference therein.

When f(v)=|v|δ?1v,g(u)=|u|μ?1u,Guo[4]has proved that the problem(1.2)has at least one positive radial solutions.

More recently,Cui,Yang and Zhang[6]studied(1.2)when a(x)≡λ,b(x)≡μ,f,g are smooth functions that are negative at the origin and f(x)∽xm,g(x)～xnfor x large with m,n≥0,mn＜(p?1)(q?1).By using the flxed point theorem in a cone,the authors obtained the existence and uniqueness of positive solutions for(1.2).

For systems with three equations,Yang[2]studied the following problem

By the blowing up argument and degreetheory,the authorhas proved an existenceresult ofpositivesolutionsand obtaineda prioriboundsforthepositive radial solutionsof(1.3).

Compared to the case of systems with two equations,there are some extra difflculties in the study of systems with three or more equations.For example,some systems with two equations could have a variational structure,but not for most systems with three or more equations.The readers can flnd this difflculty in our result of uniqueness.

Motivated by the above results,we aim to investigate the existence and uniquenessof positive solutions for(1.1)by using the flxed point theorem and the maximum principle for the quasilinear elliptic system.And the readers can flnd the related results for p=q= m=2 in[11].

Throughout this paper,we suppose a,b,c,f,g,h satisfy the following conditions:

(H2)There exist nonnegative numbers α,β,γ,A,B,C with A,B,C＞0,αβγ＜1 such that

and

Now,we state our main theorems.

Theorem 1.2.Let(u,v,w)be a positive solution of system(1.1),then there exist positive constants Mi(i=1,2,...6)and σ＞0(independent of u,v,w)such that

2 Proof of Theorems 1.2

By[2],positive solutions of system(1.1)are radially symmetric and decreasing in radial direction.Then positive solutions of(1.1)satisfy

where Φp(u)=|u|p?2u,Φq(v)=|v|q?2v,Φm(w)=|w|m?2w.

Let(u,v,w)be a positive solution of system(2.1).Then we have

For the readers’convenience,we denote Ci(i=1,2,...)positive constants independent of u,v,w,a0,b0,c0.Since w is decreasing and g is nondecreasing,we have

Similarly,we get

By(H2),there exist positive constants K1,K2,K3such that

By(2.2)-(2.4),we have

Similarly,we obtain

and after integrating from r to 1,we have

Similarly,we can show that

Since u,v,w are decreasing in(0,1),there exist positive constants M1,M3,M5(independent of u,v,w)such that the left-side inequalities for u,v,w in Theorem 1.2 hold.

Next,we will show that the right-side inequalities hold.By the equations of u,v,w, we get

By(H2)and(2.9),we obtain

Consequently,we have

Similarly,we have

Then by some direct computation,we get

This implies that

By using the same method,we can easily get the upperestimates for v(r)and w(r).Thus, we complete the proof.

3 Existence and uniqueness

For X=C([0,1])×C([0,1])×C([0,1]),we denote the norm on X by

Let K={(u,v,w)∈X:u≥0,v≥0,w≥0}be a cone.Then for each(u,v,w)∈K,we deflne

It is easy to check that T:K→K is completely continuous and flxed points of T are nonnegative solutions of(2.1).

First,we give the following flxed point theorem in a cone:

Theorem 3.1.(Gustafson and Schmitt[12])Let K be a cone in a Banach space and T:K→K be a completely continuous mapping satisfying

(a)There exists k∈K,||k||=1,and a number r＞0 such that all solutions y∈K of y=Ty+θk, 0＜θ＜∞satisfy||y||/=r.

(b)There exists R＞r such that all solutions z∈K of z=θTz,0＜θ＜1 satisfy||z||/=R. Then T has a flxed point x∈K,r≤||x||≤R.

For the readers’convenience,we give the following lemma.

Lemma 3.1.([13,Lemma 3])Let H(x)be continuous on[0,∞)and C1on(0,∞)such that

Let M,?,μbe positive numbers with ?＜1.Then there exists a positive number M′such that

for ?≤ν≤1 and 0≤x≤M.

Now,we prove Theorem 1.1.

Proof.Existence:We shall verify the conditions of Theorem 3.1.Let(u,v,w)∈K satisfy (u,v,w)=T(u,v,w)+θ(1,1,1)forsome θ＞0.Then(u,v,w)are positive and non-increasing on(0,1).By a simple argument similar to that of Theorem 1.2,we have

Let(u,v,w)∈K satisfy(u,v,w)=θT(u,v,w)for some 0＜θ＜1.Then it follows from the proof of Theorem 1.2 that

Clearly,we can flnd a number R＞r such that|(u,v,w)|0/=R.

Then,Theorem 3.1 implies the existence of a nonnegative solution(u,v,w)of(2.1) with r≤|(u,v,w)|0≤R.By the maximum principle,we deduce u≥0,v≥0,w≥0 in(0,1). Thus,the existence result is proven.

We claim that λ≥1,μ≥1 and θ≥1.Without loss of generality,we may assume that λ≤μ≤θ,then we only need to prove that λ≥1.Assume that λ＜1 by contradiction.For the convenience,we deflne

Since

Then,we have

Let α1＞α2＞α,β1＞β2＞β,γ1＞γ2＞γ and α1β1γ1＜1,we claim that

This implies that

and therefore,for ξ≤T,one has

On the other hand,for ξ＞T,we can easily get

here we have used Lemma 3.1 with H(x)=h(x).By(2.4),(2.5)and(2.6),we obtain

Since there exists a positive number k1＞0 such that

we have

if T is sufflciently close to 1.Then,(3.2)holds.Similarly,we have

Substituting(3.3)into(3.1)and integrating from 0 to z imply that

For r≤T,by(2.4)and Theorem 1.2,we have

that is,

Note that there exists a positive number k2＞0 such that

then for r≤T,we obtain

where c2(T)=K1(c1(T))αk2.This implies that

For z＞T,by Lemma 3.1 and(3.4),we get

that is,

4 Conclusion

In this paper,by using the flxed point theorem and the maximum principle,we studied the existence,uniqueness and the upper and lower bounds of the positive solutions for (1.1).Just as we know,the systems with three or more equations are much more complicated than those with two equations,thus,our results are new and useful.But,this paper only discussed the radial case,that is,? is a ball.The case when ? is a general bounded domain in RNis still a problem.

Acknowledgement

This work was supportedby the National Natural Science Foundation of China(No.1117 -1092 and 11471164);theGraduateStudentsEducationandInnovationofJiangsuProvince (No.KYZZ?0209)and the Natural Science Foundation of Educational Department of Jiangsu Province(No.08KJB110005).

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?Corresponding author.Email addresses:zdyang jin@263.net(Z.D.Yang)

Received 29 August,2015;Accepted 28 October,2015

AMS Subject Classiflcations:35B30,35J65,35J92

Chinese Library Classiflcations:O175.2