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A coupled p-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior

  • Received: 01 April 2020 Revised: 01 June 2020 Published: 31 July 2020
  • 35J60, 35J66, 35J92

  • We prove uniqueness, existence and asymptotic behavior of positive solutions to the system coupled by p-Laplacian elliptic equations

    {Δpz1=λ1g1(z2)  in Ω,Δpz2=λ2g2(z1)  in Ω,z1=z2=0  on  Ω,

    where Δpu=div(|u|p2u), 1<p<, λ1 and λ2 are positive parameters, Ω is the open unit ball in RN, N2.

    Citation: Yichen Zhang, Meiqiang Feng. A coupled p-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior[J]. Electronic Research Archive, 2020, 28(4): 1419-1438. doi: 10.3934/era.2020075

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  • We prove uniqueness, existence and asymptotic behavior of positive solutions to the system coupled by p-Laplacian elliptic equations

    {Δpz1=λ1g1(z2)  in Ω,Δpz2=λ2g2(z1)  in Ω,z1=z2=0  on  Ω,

    where Δpu=div(|u|p2u), 1<p<, λ1 and λ2 are positive parameters, Ω is the open unit ball in RN, N2.



    It is well known that p-Laplace equations are quasilinear equations when p2(see [22], [12]), and there are many important applications in physics, game theory and image processing (see [14], [5]). In the past few decades, a good many of results have been developed for single p-Laplace equations by different methods, for instance, see [21,23,19,18] and the references cited therein. Specially, in [24], Zhang and Li considered the following p-Laplacian equation

    {Δpu=g(u)  in Ω,u=0  on Ω, (1)

    where pu=div(|u|p2u) is the p-Laplacian operator, N<p<, Ω is a smooth bounded domain in RN,N1. The authors applied differential equations theory in Banach spaces and dynamics theory to study problem (1), and obtained excellent multiple solutions and sign-changing solutions theorems of p-Laplacian.

    At the same time, we notice that many authors have paid more attention to existence and uniqueness problems, for example, see Castro, Sankar and Shivaji [2], Lin [13], Hai [10], and Guo [6]. Specially, Guo and Webb [7] considered the following p-Laplacian equation

    {Δpu=λf(u) in Ω,u=0  on Ω. (2)

    They obtained existence and uniqueness results to problem (2) for large λ if f0,(f(x)/xp1)<0 for x>0 and f satisfies some p-sublinearity conditions at and 0. In [11], using sub-supersolution method together with sharp estimates near the boundary, Hai and Shivaji improved the results of (2) in a unit ball under much weaker assumptions than in [7]. Recently, Shivaji, Sim and Son [20], and Chu, Hai and Shivaji [3] generalize the study in [7] from a bounded domain to the exterior domains and obtained some excellent results.

    Moreover, we notice that various of system problems have become an important area of investigation in recent years. To identify a few, we refer the reader to [4,15,16,17]. In [11], Hai considered the existence and uniqueness of positive solutions to the following elliptic system

    {Δu=λf(v)  in Ω,Δv=μg(u)  in Ω,u=v=0  on Ω,

    where Ω is the open ball in RN, f, g:R+R+, λ and μ are positive parameters.

    Inspired by the above works, we are interested in the existence and uniqueness of positive radial solutions to the following system

    {Δpz1=λ1g1(z2)  in Ω,Δpz2=λ2g2(z1)  in Ω,z1=z2=0  on  Ω. (3)

    Here p denotes the singulardegenerate p-Laplace operator pu=div(|u|p2u), 1<p<, λ1>0 and λ2>0 are parameters, g1 and g2 are continuous nonlinearities, and Ω={xRN:|x|<1}, N2.

    We also give new existence results for system (3). Our main tool is the eigenvalue theory in cones. However, based on the idea of decoupling method we will investigate composite operators. Besides, the exactly determined intervals of positive parameters λ1×λ2 are established.

    The rest of the paper is organized as follows. In Section 2, we present some necessary definitions, Lemmas and theorems that will be used to prove our main results, Theorem 2.4. Section 3 is devoted to proving the existence and uniqueness of positive solution to system (3). In Section 4, we establish the exactly determined intervals of positive parameter λ1×λ2 in which system (3) admits at least one positive solution. Section 5 verifies the existence and asymptotic behavior of positive radial solutions to system (3). Finally, in Section 6, we will give some remarks on our main results.

    In order to study the existence of the positive radial solutions for system (3), let us firstly introduce the radial coordinates form of the p-Laplacian operator. Letting r=|x|, and u(r)=z1(x),v(r)=z2(x), then

    Δpz1(x)=r1N(rN1|u(r)|p2u(r)),
    Δpz2(x)=r1N(rN1|v(r)|p2v(r)).

    Therefore, the study of positive radial solutions of system (3) is reduced to the study of positive solutions to the following system:

    {(rN1φp(u))=λ1rN1g1(v)  in  0<r<1,(rN1φp(v))=λ2rN1g2(u)  in  0<r<1,u(0)=v(0)=u(1)=v(1)=0, (4)

    where φp(s)=|s|p2s, (φp)1=φq, 1p+1q=1.

    Lemma 2.1. Let p>1,q>1 satisfy 1p+1q=1. Then φp(s)=|s|p2s is odd, and

    sφp(s)>0 if s0, φp(st)=φp(s)φp(t),φp(0)=0, φp(1)=1, φp(1)=1,
    φp(s+t)={2p1(φp(s)+φp(t)),  if  p2, s,t>0,φp(s)+φp(t),  if  1<p<2, s,t>0.

    On the other hand, φp(s) is increasing on [0,), and for a0, φp(sa)=φap(s) on [0,).

    Next we mainly analyze the existence of positive solutions for system (4). In order to get our theorems, we let R+=[0,+) and gi satisfy

    (C0) g1 and g2:R+R+ are continuous.

    (C1) g1 and g2:R+R+ are nondecreasing, C1 on (0,) and

    limx0+supxg1(x)<,  limx0+supxg2(x)<.

    (C2) There exist nonnegative numbers a,b,A,D, where ab<1 and A,D>0 such that

    limx0+infg1(x)φp(xa)>0,   limx0+infg2(x)φp(xb)>0,
    limxg1(x)φp(xa)=A,   limxg2(x)φp(xb)=D

    and for a1>a and b1>b,

    g1(x)φp(xa1) and g2(x)φp(xb1)

    are nonincreasing for x large.

    A pair of functions u,vC[0,1]C1(0,1) with ϕp(u),ϕp(v)C1(0,1) is called to be a positive solution of (4) if u(t),v(t)>0 for all t(0,1), and u and v satisfy (4).

    Let

    E=C([0,1],R)×C([0,1],R).

    Then E is a Banach space with the norm (u,v)=max{u,v}, where u=maxt[0,1]|u(t)|.

    Define a cone P by

    P={(u,v)E:u0,v0}.

    Define an operator F: EE by

    F(u,v)(t)=(A(u,v)(t),B(u,v)(t)),  t[0,1],

    where

    A(u,v)(t)=1tφq(1sN1s0λ1τN1g1(v(τ))dτ)ds,

    and

    B(u,v)(t)=1tφq(1sN1s0λ2τN1g2(u(τ))dτ)ds.

    It is easy to check that F:PP is completely continuous and the solution of system (4) is equivalent to the fixed point equation

    F(u,v)=(u,v).

    Therefore, the task of the present paper is to search nonzero fixed points of F.

    The following well-known results are crucial in the proofs of our results.

    Lemma 2.2. (See Lemma 2.4 of [9], on page 131) Let P be a cone in a Banach space E and T: PP be a completely continuous mapping satisfying

    (a) There exist kK, k=1, and a number r>0 such that all solutions yP of

    y=Ty+θk,  0<θ<

    satisfy yr.

    (b) There exists R>r such that all solutions zP of

    z=θTz,  0<θ<1.

    Then T has a fixed point xP, rxR.

    In this section, we analyze the uniqueness of fixed point of F for λ1λa2 and λb1λ2 sufficiently large.

    Lemma 3.1. Let h be continuous on R+ and C1 on (0,) such that

    limx0+supxh(x)<.

    Let M, ε, r be positive numbers with ε<1. Then there is a positive constant C such that

    |h(γx)φp(γr)h(x)|C(1γ)

    for εγ<1 and 0xM.

    Proof. Let 0xM. Define H(γ)=h(γx)φp(γr)h(x), εγ<1. Using the mean value theorem, there is a c(γ,1) such that

    |H(γ)|=|H(γ)H(1)|=(1γ)|xh(cx)r(p1)cr(p1)1h(x)|C(1γ),

    where

    C=sup{|yh(y)|:0<yM}ε+r(p1)max(εr(p1)1,1)sup{|h(y)|:yM}.

    Next we will check the upper and lower estimates for possible positive solutions of system (4).

    Lemma 3.2. Let (u,v) be a positive solution of (4). Then there exist positive constants Mi, i{1,2,3,4} and M independent of u,v such that

    M1(φq(λ1λa2))11ab(1t)u(t)M2(φq(λ1λa2))11ab(1t), 0<t<1,
    M3(φq(λ2λb1))11ab(1t)v(t)M4(φq(λ2λb1))11ab(1t), 0<t<1

    for min{λ1λa2,λ2λb1}M.

    Proof. Suppose that u and v are a pair of positive solutions for system (4). By integrating, we get

    u(t)=1tφq(1sN1s0λ1τN1g1(v(τ))dτ)ds,v(t)=1tφq(1sN1s0λ2τN1g2(u(τ))dτ)ds.

    Next, we can denote by ci, i=1,2,, positive constants independent of u,v,λ1,λ2. Since v is decreasing, we have

    u(12)112φq(1sN1120λ1τN1g1(v(τ))dτ)ds=φq(λ1)112φq(1sN1120τN1g1(v(τ))dτ)dsφq(λ1)112φq(1sN1)dsφq(120τN1dτ)φq(g1(v(12)))12φq(λ1N2N)φq(g1(v(12))). (5)

    Similarly we can get

    v(12)12φq(λ2N2N)φq(g2(u(12))). (6)

    By (C2), there are two positive constants K1 and K2 such that

    g1(x)φp(K1xa),  g2(x)φp(K2xb) for x0. (7)

    This together with (5) and (6), shows that

    u(12)12φq(λ1N2N)K1[12φq(λ2N2N)]aKa2(u(12))ab=c1φq(λ1λa2)(u(12))ab.

    Thus,

    u(12)(c1)11abφq(λ1λa2)11ab=c2(φq(λ1λa2))11ab. (8)

    Similarly, we can get

    v(12)(c1)11abφq(λ2λb1)11ab=c2(φq(λ2λb1))11ab. (9)

    It follows from (7), (8) and (9) that for t12

    u(t)=φq(λtN1t0sN1g1(v)ds)φq(λ1120sN1g1(v)ds)φq(λ1N2Ng1(v(12)))=φq(λ1N2N)φq(g1(v(12)))φq(λ1N2N)K1ca2(φq(λ1μa))11ab=c3(φq(λ1λa2))11ab.

    Then by integrating, for t12, we get

    u(t)c3(φq(λ1λa2))11ab(1t). (10)

    Similarly,

    v(t)c4(φq(λ2λb1))11ab(1t). (11)

    Because of u,v being decreasing, this shows the left-side inequalities for u,v in Lemma 3.2.

    According to the formulas for u,v, we can see that

    u(t)|u|10φq(1sN1s0λ1τN1g1(|v|)dτ)ds10φq(1sN110λ1τN1g1(|v|)dτ)dsφq(λ1g1(|v|)) (12)

    and

    |v|φq(λ2g2(|u|)). (13)

    By (7) and (11), for large λ1λa2 and λ2λb1, we get

    |v(t)|c4(φq(λ1λb2))11ab(1t)1  (i.e. |v| is large) (14)

    and

    λ1g1(|v|)λ1φp(K1|v|a)λ1φp(K1[c4(φq(λ2λb1))11ab]a)λ1φp(K1[ca4(φq(λ1λa2))11ab])1.

    Note that from (C2) it follows that

    |v|φq(λ2g2(|u|))φq(λ2g2(φq(λ1g1(|v|))))φq(λ2g2(φq(λ1K1)|v|a))φq(λ2K2φp((φq(λ1K1))b(|v|)ab))=φq(Kb1K2)φq(λ2λb1)(|v|)ab=c5φq(λ2λb1)(|v|)ab,

    or

    |v|c11ab5(φq(λ2λb1))11ab. (15)

    By combining the equation of u and (15), we can get

    u(t)φq(λ1tN1t0sN1g1(|v|)ds)φq(λ1g1(|v|))φq(λ1f(c11ab5(φq(λ2λb1))11ab))φq(λ1)K1c11ab5(φq(λ1λa2)11ab)=c6(φq(λ1λa2)11ab), (16)

    and then it follows from integrating that

    u(t)c6(φq(λ1λa2)11ab)(1t),  0<t<1.

    Similarly, we can get the upper estimate for v(t). This completes the proof.

    Theorem 3.3. Assume (C0)(C2) hold. Then there is a constant β>0 such that the system (4) admits a unique positive solution for min(λ1λa2,λ2λb1)β.

    Proof. We shall check the conditions of Lemma 2.2 to prove the existence of solution. Let (u,v)P satisfy

    (u,v)=F(u,v)+θ(1,1)

    for some θ>0. Because of u,v are nonincreasing and u,v>0 on (0,1), the proof is analogous to that of Lemma 3.2 that

    u(12)c2(φq(λ1λa2))11ab.

    So, (u,v)r, where 0<r<c2(φq(λ1λa2))11ab.

    Next, set (u,v)P with

    (u,v)=θF(u,v)

    for some θ(0,1). Then, by (12) and (13) we get

    |v|φq(λ2g2(φq(λ1g1(|v|)))),  |u|φq(λ1g1(φq(λ2g2(|u|)))).

    In addition, if |v|, by (C2) and ab<1 we can see that

    1φq(λ2g2(φq(λ1g1(|v|))))|v|c5φq(λ2λb1)(|v|)ab|v|=0,

    which is impossible. So, there is a number R>r such that (u,v)R. Therefore, Lemma 2.2 shows that F has a fixed point (u,v) with r(u,v)R. Thus, it follows that (4) admits one positive solution, and the existence is proved.

    Next, we shall show that the solution is unique. Suppose that (u,v) and (u1,u2) are positive solutions of system (4) and let min{λ1λa2, λ2λb1,} be large enough such that Lemma 3.2 holds. It follows from Lemma 3.2 that

    M1M2u1uM2M1u1  on  (0,1).

    Let α=sup{d>0:udu1 in (0,1)}. Then obviously d0α and uαu1 in (0,1), where α0=M1M2. We assert that α1. In fact, we can assume by contradiction that α<1. Since u,v are decreasing and

    (tN1φp(u))=λ1tN1g1(1tφq(1sN1s0λ2τN1g2(u(τ))dτ)ds),
    (tN1φp(αu1))=λ1tN1φp(α)g1(1tφq(1sN1s0λ2τN1g2(u1(τ))dτ)ds),

    we can get

    [tN1(φp(u)φp(αu1))]λ1tN1[g1(1tφq(λ2sN1s0τN1g2(αu1(τ))dτ)ds)φp(α)g1(1tφq(λ2sN1s0τN1g2(u1(τ))dτ)ds)]. (17)

    Let b1>b2>b, a1>a and a1b1<1. Then we assert that

    s0τN1g2(αu1(τ))dτφp(αb1)s0τN1g2(u1(τ))dτ. (18)

    According to g2(x)φp(xb2) is nonincreasing for x1 and αα0, we can get

    g2(αx)φp((αx)b2)g2(x)φp(xb2),

    or

    g2(αx)φp(αb2)g2(x).

    By Lemma 3.2, we can get

    u(r)M1(φq(λ1λa2))11ab(1T)1,

    where T(12,1).

    Since α<1, then for sT, we have

    s0τN1(g2(αu1)φp(αb1)g2(u1))dτ(φp(αb2)φp(αb1))s0τN1g2(u1)dτ0.

    For s>T, combined with Lemma 3.1, we have

    s0τN1(g2(αu)φp(αb1)g2(u))dτ=T0τN1(g2(αu)φp(αb1)g2(u))dτ+sTτN1(g2(αu)φp(αb1)g2(u))dτ(φp(αb2)φp(αb1))s0τN1g2(u)dτC(1T)(1α).

    Because

    T0τN1g2(u)dτ120τN1g2(u)dτg2(u(12))N2NK2N2N

    and since there is a positive number l>0 such that

    (φp(αb2)φp(αb1))l(1αp1)  for α0α1.

    This follows that

    s0τN1(g2(αu)φp(αb1)g2(u))dτ>0,s>T

    when T is sufficiently close to 1. So, we can see that

    s0τN1(g1(αu)φp(αb1)g1(u))dτ>0,  for s>T

    when T is sufficiently close to 1. So, the proof of (18) is finished.

    Substituting (18) into (17) and integrating gets

    zN1(uαu1)(z)λ1z0G(α,t)dt,

    where

    G(α,t)=tN1[g1(αb11tφq(λ2sN1s0τN1g2(u1(τ))dτ)ds)             φp(α)g1(1tφq(λ2sN1s0τN1g2(u1(τ))dτ)ds)].

    Applying (7) and Lemma 3.2, for tT, we get

    1rφq(1sN1s0λ2τN1g2(u1(τ))dτ)ds=φq(λ2)1rφq(1sN1s0τN1g2(u1(τ))dτ)dsφq(λ2)1Tφq(1sN1T0τN1g2(u1(τ))dτ)ds
    φq(g2(u1(T)))φq(λ2)1Tφq(1sN1T0τN1dτ)ds(1T)φq(λ2TNN)φq(g2(u1(T)))(1T)φq(λ2TNN)K2ub1(T)(1T)φq(λ2TNN)K2[M1(φq(λ1λa2))11ab(1t)]bφq(λ2TNN)K2Mb1(φq(λ1λa2))b1ab(1T)b+1=h(T)(φq(λ2λb1))11ab1,

    where h(T)=φq(TNN)K2Mb1(1T)b+1.

    Because g1(x)φp(xa1) is nonincreasing for x1, we get

    g1(αb1x)φp(αa1b1)g1(x).

    Thus

    G(α,t)=tN1(φp(αa1b1)φp(α))g1(1tφq(1sN1s0μτN1g2(u1(τ))dτ)ds)tN1l0(1αp1)g1(h(T)(φq(λ2λb1))11ab)H(T)tN1φq(λ2λb1))a1ab(1αp1)>0,  tT, (19)

    where H(T)=l0kK2ha(T)) and l0 is a positive constant so that

    φp(αa1b1)φp(α)l0(1αp1) for α0α1.

    This proves that

    zN1(uαu1)(z)<0,   0<zT.

    On the other hand, if z>T, then for large λ1λa2 and λ2λb1, and T sufficiently close to 1, it follows from Lemma 3.1 and (19) that

    z0G(α,r)dr120G(α,t)dt+zTG(α,t)dt                 H(12)N2Nφq(λ2λb1))a1ab(1αp1)C(1T)(1α1q1)                 >0.

    Therefore, we have

    (uαu1)(z)<0  for  0<z1.

    This shows that there is a constant ˜α>α in (0,1) such that u˜αu1, which is a contradiction. Thus α1 and hence u=u1 in (0, 1). Similarly, we can verify v=v1 in (0,1) and so we finish the proof of Theorem 3.3.

    In this section, we will establish some new existence results of positive solutions for system (4). To achieve this goal, we will define a new cone P and a composite operator T.

    Lemma 4.1. (See Theorem 2.3.6 of [8], on page 99) Suppose that D is an open subset of the an infinite-dimensional real Banach space E, θD, and P is a cone of E. If the operator Γ:PDP is completely continuous with Γθ=θ and satisfies

    infxPDΓx>0,

    then Γ has a proper element on PD associated with a positive eigenvalue. That is, there exist x0PD and μ0>0 such that Γx0=μ0x0.

    Let E=C[0,1]. Then E is a real Banach space with the norm defined by

    x=maxtJ|x(t)|.

    Let J=[0,1] and P be the cone

    P:={vE:v(t)0, tJ, v(t)14v, t[14,34]}. (20)

    It is easy to see that P is a normal cone of E.

    For vP, define Ti:PE(i=1,2) as

    (T1v)(t)=1tφq(1τN1τ0sN1g1(v(s))ds)dτ, (21)
    (T2v)(t)=φq(λ2)1tφq(1τN1τ0sN1g2(v(s))ds)dτ. (22)

    It follows from Lemma 3 in [1] that Ti(i=1,2) maps P into itself. Moreover, T1 and T2 are completely continuous by standard arguments.

    Define a composite operator T=T1T2, which is also completely continuous from P to itself. So the operator T also maps P into P. Therefore the next task of this paper is to search nonzero fixed points of operator T.

    Let

    g1:=limvg1(v)φp(v),  g01:=limv0g1(v)φp(v);
    g2:=limvg2(v)φp(v),  g02:=limv0g2(v)φp(v),

    and

    A=3414sN1ds=3N1N4N, B=134φq(1τN1)dτ, B=10φq(1τN1)dτ. (23)

    Theorem 4.2. Suppose that (C0) holds. If 0<gi<+(i=1,2), then there exists β0>0 such that, for every R>β0, system (4) admits a pair of positive solutions uR,vR satisfying uR=R for any

    λ1Rλ2[λR,ˉλR],

    where λR and ˉλR are positive finite numbers.

    Proof. Since 0<gi<+, there exist 0<l1<l2,  μ>0 so that

    l1φp(v)<g1(v)<l2φp(v), vμ;
    l1φp(u)<g2(u)<l2φp(u), uμ.

    Next, we verify that β0=4μ is required. Letting

    ΩR={xE:x<R},

    then 0ΩR and ΩR is a bounded open subset of Banach space E.

    Since R>β0, for any u,vPΩR, we get

    u(t)14u=14R, v(t)14v=14R,  t[14,34],

    and

    u(t)14u>14β0=μ, v(t)14v>14β0=μ,  t[14,34].

    So, for any vPΩR, we have

    (T1v)(t)134φq(1τN1340sN1g1(v(s))ds)dτ134φq(1τN13414sN1g1(v(s))ds)dτ134φq(1τN13414sN1l1φp(v(s))ds)dτ134φq(1τN13414sN1l1φp(14v)ds)dτ=14vφq(l1A)B, tJ.

    Analogously, for uPΩR, we obtain

    (T2u)(t)134φq(1τN1340sN1g2(u(s))ds)dτ134φq(1τN13414sN1g2(u(s))ds)dτ134φq(1τN13414sN1l1φp(u(s))ds)dτ134φq(1τN13414sN1l1φp(14u)ds)dτ=14uφq(l1A)B, tJ.

    Therefore, we get

    (Tu)(t)=(T1T2u)(t)14T2uφq(l1A)B116u(φq(l1A)B)2.

    This gives that

    infuPΩRTu116u(φq(l1A)B)2>0.

    For any R>β0, Lemma 4.1 yields that operator T admits a proper element uRP associated with the eigenvalue μ1R>0, and uR satisfies uR=R.

    For operator T, we can denote vR=T2uR, then uR and vR are the solutions of system (4).

    Let λ1R=1φp(μ1R). Then we get

    TuR=μ1RuR=1φq(λ1R)uR. (24)

    It follows from the proof above that, for any R>β0, system (4) has a pair of positive solutions uR and vR with uRPΩR associated with λ1=λ1R>0. Thus, by (24) we get

    uR(t)=φq(λ1R)TuR,

    and so

    uR(t)=φq(λ1R)1tφq(1τN1τ0sN1g1(vR(s))ds)dτ,
    vR(t)=φq(λ2R)1tφq(1τN1τ0sN1g2(uR(s))ds)dτ

    with uR=R.

    On the one hand,

    uR(t)=φq(λ1R)1tφq(1τN1τ0sN1g1(vR(s))ds)dτφq(λ1R)10φq(1τN110sN1g1(vR(s))ds)dτφq(l2λ1R)vR10φq(1τN1)dτ=φq(l2λ1R)BvR, tJ.

    Analogously,

    vR(t)φq(l2λ2)BuR, tJ.

    This verifies that

    uR=Rφq(l22(B)2λ1Rλ2)uR,

    and so,

    λ1Rλ21l22φp((B)2)=λR.

    On the other hand,

    (uR)(t)φq(λ1R)134φq(1τN1340sN1g1(vR(s))ds)dτφq(λ1R)134φq(1τN13414sN1g1(vR(s))ds)dτφq(λ1R)134φq(1τN13414sN1l1φp(vR(s))ds)dτφq(λ1R)134φq(1τN13414sN1l1φp(14vR)ds)dτ=14φq(λ1Rl1A)BvR, tJ.

    Analogously, we can show that

    (vR)(t)14φq(λ2l1A)BuR, tJ.

    Therefore, we get

    uR116φq(λ1Rλ2l21A2)B2uR,

    and so,

    λ1Rλ2φp(16)l21A2φp(B2)=ˉλR. (25)

    We hence get λ1Rλ2[λR,ˉλR]. This gives the proof.

    If we define another composite operator T=T2T1, where

    (T1v)(t)=φq(λ1)1tφq(1τN1τ0sN1g1(v(s))ds)dτ, (26)
    (T2v)(t)=1tφq(1τN1τ0sN1g2(v(s))ds)dτ. (27)

    Corollary 1. Let T=T2T1. Suppose that (C0) holds. If 0<gi<+(i=1,2), then there exists β0>0 such that, for every R>β0, system (4) admits a pair of positive solutions uR,vR satisfying vR=R for any

    λ1λ2R[λR,ˉλR], (28)

    where λR and ˉλR are positive finite numbers.

    Proof. Similar to the proof of Theorem 4.2, we can prove Corollary 1.

    Theorem 4.3. Suppose that (C0) holds. If 0<g0i<+(i=1,2), then there exists β0>0 such that, for every 0<r<β0, system (4) admits a pair of positive solutions ur,vr satisfying ur=r for any

    λ1rλ2[λr,ˉλr],

    where λr and ˉλr are positive finite numbers.

    Proof. Similar to the proof of Theorem 4.2, we can prove Theorem 4.3.

    Theorem 4.4. Suppose that (C0) holds. If gi=+(i=1,2), then there exists ˉβ0>0 such that, for every R>ˉβ0, system (4) admits a pair of positive solutions uR,vR satisfying uR=R for any

    λ1Rλ2(0,λR], (29)

    where λR is a positive finite number.

    Proof. Since gi=+, there exist l>0,  μ>0 so that

    g1(v)>lφp(v), vμ;
    g2(u)>lφp(u), uμ.

    Now, we show that ˉβ0=4μ is required. Set

    ΩR={xE:x<R}.

    Since R>ˉβ0, for any u,vPΩR, we get

    u(t)14u=14R, v(t)14v=14R,  t[14,34],

    and

    u(t)14u>14ˉβ0=μ, v(t)14v>14ˉβ0=μ,  t[14,34].

    So, for any vPΩR, we have

    (T1v)(t)134φq(1τN1340sN1g1(v(s))ds)dτ134φq(1τN13414sN1g1(v(s))ds)dτ134φq(1τN13414sN1lφp(v(s))ds)dτ134φq(1τN13414sN1lφp(14v)ds)dτ=14vφq(lA)B, tJ.

    Analogously, for uPΩR, we obtain

    (T2u)(t)134φq(1τN1340sN1g2(u(s))ds)dτ134φq(1τN13414sN1g2(u(s))ds)dτ134φq(1τN13414sN1lφp(u(s))ds)dτ134φq(1τN13414sN1lφp(14u)ds)dτ=14uφq(lA)B, tJ.

    Therefore, we get

    (Tu)(t)=(T1T2u)(t)14T2uφq(lA)B116u(φq(lA)B)2.

    This gives that

    infuPΩRTu116u(φq(lA)B)2>0.

    For any R>ˉβ0, Lemma 4.1 yields that operator T admits a proper element uRP associated with the eigenvalue μ1R>0, and uR satisfies uR=R.

    For operator T, we denote vR=T2uR, then uR and vR are the solutions of system (4).

    Let λ1R=1φp(μ1R). Next, similar to the proof of (25), we can verify that (29) holds. This finishes the proof of Theorem 4.4.

    Theorem 4.5. Suppose that (f) holds. If g0i=+(i=1,2), then there exists β1>0 such that, for every 0<r<β1, system (4) admits a nontrivial radial solution ur=(u1r,u2r) satisfying u1r=r for any

    λ1rλ2(0,λ],

    where λ is a positive finite number.

    Proof. Similar to the proof of Theorem 4.4, we can prove Theorem 4.5.

    In this section, we study the asymptotic behavior of positive solutions for system (4).

    Let P be defined as (20), and T1 and T2 be respectively defined in (26) and (22). Define a composite operator ~T1=T1T2, which is completely continuous from P to itself. So the operator ~T1 also maps P into P. We also define another composite operator

    ~T2=T2T1,

    which has the same meaning as ~T1.

    Theorem 5.1. Suppose that (C0) holds. For i{1,2}, then we have the following two conclusions.

    (C3) If g0i=0 and gi=, then for every λi>0 system (4) admits a pair of positive solutions uλ1,vλ2 with

    limλ10+uλ1=, limλ20+vλ2=;

    (C4) If g0i= and gi=0, then for every λi>0 system (4) admits a pair of positive solutions uλ1,vλ2 with

    limλ10+uλ1=0, limλ20+vλ2=0.

    Proof. We need only verify this theorem under condition (C3) because the proof is similar when (C4) is satisfied. For i{1,2}, let λi>0. Since g0i=0, there exists r>0 such that

    g1(v)1λ1φp(B)φp(v),    0vr,
    g2(u)1λ2φp(B)φp(u),    0ur,

    where B is defined in (23).

    Thus, for i={1,2} and u,vPΩr, we get

    (T1v)(t)=φq(λ1)1tφq(1τN1τ0sN1g1(v(s))ds)dτφq(λ1)10φq(1τN110sN1g1(v(s))ds)dτφq(λ1)10φq(1τN110sN11λ1φp(B)φp(v(s))ds)dτφq(λ1)10φq(1τN110sN11λ1φp(B)φp(v)ds)dτv, tJ,

    and

    (T2u)(t)=φq(λ2)1tφq(1τN1τ0sN1g2(u(s))ds)dτφq(λ2)10φq(1τN110sN1g2(u(s))ds)dτφq(λ2)10φq(1τN110sN11λ2φp(B)φp(u(s))ds)dτφq(λ2)10φq(1τN110sN11λ2φp(B)φp(u)ds)dτu, tJ.

    So

    ~T1u=T1T2uT2uu. (30)

    Next, for i={1,2}, considering gi=, there exists ˆR satisfying 0<r<ˆR so that

    g1(v)εφp(v),  vˆR,
    g2(u)εφp(u),  uˆR,

    where ε>0 satisfies

    φq(λ1λ2A2ε2)B21, (31)

    where A and B are respectively defined in (23).

    Let R>4ˆR. Then, for u,vPΩR, we get

    u(t)14uˆR, v(t)14vˆR, t[14,34],

    and then

    (T1v)(t)=φq(λ1)1tφq(1τN1τ0sN1g1(v(s))ds)dτφq(λ1)134φq(1τN1340sN1g1(v(s))ds)dτφq(λ1)134φq(1τN13414sN1g1(v(s))ds)dτφq(λ1)134φq(1τN13414sN1εφp(v(s))ds)dτφq(λ1)134φq(1τN13414sN1εφp(14v)ds)dτ=φq(λ1Aε)Bv, tJ.

    Similarly, we get

    (T2u)(t)φq(λ2Aε)Bu, tJ.

    So, by (31), we have

    (~T1v1)(t)=(T1T2u)(t)φq(λ1Aε)BT2uφq(λ1λ2A2ε2)B2uu. (32)

    From the above estimate and the fixed point theorem of cone expansion and compression of norm type, we deduce that operator ~T1 has a fixed point uP(ˉΩRΩr). Denote v=T2u, then u and v are the desired solution of system (4).

    Similarly, we can prove that ~T2 has a fixed point vP(ˉΩRΩr).

    Next, for i{1,2}, we prove that uλ1+, vλ2+ as λi0+. In deed, if not, there are a number ςi>0 and a sequence λim+ such that

    uλ1mς1, vλ2mς2 (m=1,2,3,).

    Moreover, the sequence {uλ1m} and {vλ2m} respectively contain a subsequence that converges to a number ηi(0ηiςi). For simplicity, we suppose that {uλ1m} itself converges to η1, and {vλ2m} itself converges to η2.

    If η1>0,η2>0, then uλ1m>η12, vλ2m>η22 for sufficiently large m (m>M, M denotes a natural number), and so

    1φq(λ1m)=1tφq(1τN1τ0sN1g1(v(s))ds)dτuλ1m10φq(1τN110sN1g1(v(s))ds)dτuλ1mφq(D1)Buλ1m<2φq(D1)Bη1 (m>M),

    and

    1φq(λ2m)=1tφq(1τN1τ0sN1g2(u(s))ds)dτvλ2m10φq(1τN110sN1g2(u(s))ds)dτvλ2mφq(D2)Bvλ2m<2φq(D1)Bη2 (m>M),

    where,

    D1=max{g1(v), rvR},
    D2=max{g2(u), ruR}.

    This gives a contradiction as λim0+ for i{1,2}.

    If η1=0 and η2=0, then uλ1m0,vλ2m0 for sufficiently large m (m>M), and so it follows from (C3) that for any ε>0 there is r>0 so that

    g1(vλ2m)εφp(v),  0vλ2mr,
    g2(uλ1m)εφp(u),  0uλ1mr.

    Then, for uλ1m, vλ2mPΩr, we have

    1φq(λ1m)=1tφq(1τN1τ0sN1g1(v(s))ds)dτuλ1m10φq(1τN110sN1g1(v(s))ds)dτuλ1m10φq(1τN110sN1εφp(v(s))ds)dτuλ1mφq(ε)Bvuλ1m,

    and

    1φq(λ2m)=1tφq(1τN1τ0sN1g2(u(s))ds)dτvλ2m10φq(1τN110sN1g2(v(s))ds)dτvλ2m10φq(1τN110sN1εφp(v(s))ds)dτvλ2mφq(ε)Bvvλ2m,

    where B is defined in (23). Because ε is arbitrary, for i{1,2}, we get λim+ (m+), which contradicts λim0+. The proof of Theorem 5.1 is finished.

    In this section, we offer some remarks and applications on the associated system (4).

    Remark 6.1. The present research extends the study in Hai [11] from Laplacian system to p-Laplacian system. Meanwhile, we obtain some new existence results by defining composite operators and using the eigenvalue theory in cones. Moreover, we also analyze the asymptotic behavior of positive solutions to system (4).

    Remark 6.2. In this paper, we also generalize the study in Guo [6], Guo and Webb [7], Hai and Shivaji [11], Shivaji, Sim and Son [20], and Chu, Hai and Shivaji [3] from single p-Laplacian equation to coupled p-Laplacian system. Here, we not only get the uniqueness results, but also we obtain some existence results, and we consider the asymptotic behavior of positive solutions.

    Remark 6.3. The approaches to prove Theorem 3.3, Theorem 4.2-Theorem 4.5 and Theorem 5.1 can be applied to the single equation case

    {pz=λg(z)  in  Ω,z=0  on  Ω,

    where pu=div(|u|p2u), 1<p<N, λ is a positive parameter, Ω is the open unit ball in RN.

    The authors also would like to thank the anonymous referees for their valuable comments which has helped to improve the paper.



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