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|p−2∇u), 1<p<∞, λ1 and λ2 are positive parameters, Ω is the open unit ball in RN, N≥2.
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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Yichen Zhang, Meiqiang Feng .
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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|p−2∇u), 1<p<∞, λ1 and λ2 are positive parameters, Ω is the open unit ball in RN, N≥2.
It is well known that
{−Δpu=g(u) in Ω,u=0 on ∂Ω, | (1) |
where
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
{Δpu=−λf(u) in Ω,u=0 on ∂Ω. | (2) |
They obtained existence and uniqueness results to problem (2) for large
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
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
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
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
In order to study the existence of the positive radial solutions for system (3), let us firstly introduce the radial coordinates form of the
Δpz1(x)=r1−N(rN−1|u′(r)|p−2u′(r))′, |
Δpz2(x)=r1−N(rN−1|v′(r)|p−2v′(r))′. |
Therefore, the study of positive radial solutions of system (3) is reduced to the study of positive solutions to the following system:
{−(rN−1φp(u′))′=λ1rN−1g1(v) in 0<r<1,−(rN−1φp(v′))′=λ2rN−1g2(u) in 0<r<1,u′(0)=v′(0)=u(1)=v(1)=0, | (4) |
where
Lemma 2.1. Let
sφp(s)>0 if s≠0, φp(st)=φp(s)φp(t),φp(0)=0, φp(1)=1, φp(−1)=−1, |
φp(s+t)={2p−1(φp(s)+φp(t)), if p≥2, s,t>0,φp(s)+φp(t), if 1<p<2, s,t>0. |
On the other hand,
Next we mainly analyze the existence of positive solutions for system (4). In order to get our theorems, we let
limx→0+supxg′1(x)<∞, limx→0+supxg′2(x)<∞. |
limx→0+infg1(x)φp(xa)>0, limx→0+infg2(x)φp(xb)>0, |
limx→∞g1(x)φp(xa)=A, limx→∞g2(x)φp(xb)=D |
and for
g1(x)φp(xa1) and g2(x)φp(xb1) |
are nonincreasing for
A pair of functions
Let
E=C([0,1],R)×C([0,1],R). |
Then
Define a cone
P={(u,v)∈E:u≥0,v≥0}. |
Define an operator
F(u,v)(t)=(A(u,v)(t),B(u,v)(t)), t∈[0,1], |
where
A(u,v)(t)=∫1tφq(1sN−1∫s0λ1τN−1g1(v(τ))dτ)ds, |
and
B(u,v)(t)=∫1tφq(1sN−1∫s0λ2τN−1g2(u(τ))dτ)ds. |
It is easy to check that
F(u,v)=(u,v). |
Therefore, the task of the present paper is to search nonzero fixed points of
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
(a) There exist
y=Ty+θk, 0<θ<∞ |
satisfy
(b) There exists
z=θTz, 0<θ<1. |
Then
In this section, we analyze the uniqueness of fixed point of
Lemma 3.1. Let
limx→0+supxh′(x)<∞. |
Let
|h(γx)−φp(γr)h(x)|≤C(1−γ) |
for
Proof. Let
|H(γ)|=|H(γ)−H(1)|=(1−γ)|xh′(cx)−r(p−1)cr(p−1)−1h(x)|≤C(1−γ), |
where
C=sup{|yh′(y)|:0<y≤M}ε+r(p−1)max(εr(p−1)−1,1)sup{|h(y)|:y≤M}. |
Next we will check the upper and lower estimates for possible positive solutions of system (4).
Lemma 3.2. Let
M1(φq(λ1λa2))11−ab(1−t)≤u(t)≤M2(φq(λ1λa2))11−ab(1−t), 0<t<1, |
M3(φq(λ2λb1))11−ab(1−t)≤v(t)≤M4(φq(λ2λb1))11−ab(1−t), 0<t<1 |
for
Proof. Suppose that
u(t)=∫1tφq(1sN−1∫s0λ1τN−1g1(v(τ))dτ)ds,v(t)=∫1tφq(1sN−1∫s0λ2τN−1g2(u(τ))dτ)ds. |
Next, we can denote by
u(12)≥∫112φq(1sN−1∫120λ1τN−1g1(v(τ))dτ)ds=φq(λ1)∫112φq(1sN−1∫120τN−1g1(v(τ))dτ)ds≥φq(λ1)∫112φq(1sN−1)ds⋅φq(∫120τN−1dτ)⋅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
g1(x)≥φp(K1xa), g2(x)≥φp(K2xb) for x≥0. | (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)11−abφq(λ1λa2)11−ab=c2(φq(λ1λa2))11−ab. | (8) |
Similarly, we can get
v(12)≥(c1)11−abφq(λ2λb1)11−ab=c′2(φq(λ2λb1))11−ab. | (9) |
It follows from (7), (8) and (9) that for
−u′(t)=φq(λtN−1∫t0sN−1g1(v)ds)≥φq(λ1∫120sN−1g1(v)ds)≥φq(λ1N2Ng1(v(12)))=φq(λ1N2N)φq(g1(v(12)))≥φq(λ1N2N)K1ca2(φq(λ1μa))11−ab=c3(φq(λ1λa2))11−ab. |
Then by integrating, for
u(t)≥c3(φq(λ1λa2))11−ab(1−t). | (10) |
Similarly,
v(t)≥c4(φq(λ2λb1))11−ab(1−t). | (11) |
Because of
According to the formulas for
u(t)≤|u|∞≤∫10φq(1sN−1∫s0λ1τN−1g1(|v|∞)dτ)ds≤∫10φq(1sN−1∫10λ1τN−1g1(|v|∞)dτ)ds≤φq(λ1g1(|v|∞)) | (12) |
and
|v|∞≤φq(λ2g2(|u|∞)). | (13) |
By (7) and (11), for large
|v(t)|≥c4(φq(λ1λb2))11−ab(1−t)≫1 (i.e. |v|∞ is large) | (14) |
and
λ1g1(|v|∞)≥λ1φp(K1|v|a∞)≥λ1φp(K1[c4(φq(λ2λb1))11−ab]a)≥λ1φp(K1[ca4(φq(λ1λa2))11−ab])≫1. |
Note that from
|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|∞≤c11−ab5(φq(λ2λb1))11−ab. | (15) |
By combining the equation of
−u′(t)≤φq(λ1tN−1∫t0sN−1g1(|v|∞)ds)≤φq(λ1g1(|v|∞))≤φq(λ1f(c11−ab5(φq(λ2λb1))11−ab))≤φq(λ1)K1c11−ab5(φq(λ1λa2)11−ab)=c6(φq(λ1λa2)11−ab), | (16) |
and then it follows from integrating that
u(t)≤c6(φq(λ1λa2)11−ab)(1−t), 0<t<1. |
Similarly, we can get the upper estimate for
Theorem 3.3. Assume
Proof. We shall check the conditions of Lemma 2.2 to prove the existence of solution. Let
(u,v)=F(u,v)+θ(1,1) |
for some
u(12)≥c2(φq(λ1λa2))11−ab. |
So,
Next, set
(u,v)=θF(u,v) |
for some
|v|∞≤φq(λ2g2(φq(λ1g1(|v|∞)))), |u|∞≤φq(λ1g1(φq(λ2g2(|u|∞)))). |
In addition, if
1≤φq(λ2g2(φq(λ1g1(|v|∞))))|v|∞≤c5φq(λ2λb1)(|v|∞)ab|v|∞=0, |
which is impossible. So, there is a number
Next, we shall show that the solution is unique. Suppose that
M1M2u1≤u≤M2M1u1 on (0,1). |
Let
(tN−1φp(u′))′=−λ1tN−1g1(∫1tφq(1sN−1∫s0λ2τN−1g2(u(τ))dτ)ds), |
(tN−1φp(αu′1))′=−λ1tN−1φp(α)g1(∫1tφq(1sN−1∫s0λ2τN−1g2(u1(τ))dτ)ds), |
we can get
[tN−1(φp(u′)−φp(αu′1))]′≤−λ1tN−1[g1(∫1tφq(λ2sN−1∫s0τN−1g2(αu1(τ))dτ)ds)−φp(α)g1(∫1tφq(λ2sN−1∫s0τN−1g2(u1(τ))dτ)ds)]. | (17) |
Let
∫s0τN−1g2(αu1(τ))dτ≥φp(αb1)∫s0τN−1g2(u1(τ))dτ. | (18) |
According to
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))11−ab(1−T)≫1, |
where
Since
∫s0τN−1(g2(αu1)−φp(αb1)g2(u1))dτ≥(φp(αb2)−φp(αb1))∫s0τN−1g2(u1)dτ≥0. |
For
∫s0τN−1(g2(αu)−φp(αb1)g2(u))dτ=∫T0τN−1(g2(αu)−φp(αb1)g2(u))dτ+∫sTτN−1(g2(αu)−φp(αb1)g2(u))dτ≥(φp(αb2)−φp(αb1))∫s0τN−1g2(u)dτ−C(1−T)(1−α). |
Because
∫T0τN−1g2(u)dτ≥∫120τN−1g2(u)dτ≥g2(u(12))N2N≥K2N2N |
and since there is a positive number
(φp(αb2)−φp(αb1))≥l(1−αp−1) for α0≤α≤1. |
This follows that
∫s0τN−1(g2(αu)−φp(αb1)g2(u))dτ>0,s>T |
when
∫s0τN−1(g1(αu)−φp(αb1)g1(u))dτ>0, for s>T |
when
Substituting (18) into (17) and integrating gets
zN−1(u′−αu′1)(z)≤−λ1∫z0G(α,t)dt, |
where
G(α,t)=tN−1[g1(αb1∫1tφq(λ2sN−1∫s0τN−1g2(u1(τ))dτ)ds) −φp(α)g1(∫1tφq(λ2sN−1∫s0τN−1g2(u1(τ))dτ)ds)]. |
Applying (7) and Lemma 3.2, for
∫1rφq(1sN−1∫s0λ2τN−1g2(u1(τ))dτ)ds=φq(λ2)∫1rφq(1sN−1∫s0τN−1g2(u1(τ))dτ)ds≥φq(λ2)∫1Tφq(1sN−1∫T0τN−1g2(u1(τ))dτ)ds |
≥φq(g2(u1(T)))φq(λ2)∫1Tφq(1sN−1∫T0τN−1dτ)ds≥(1−T)φq(λ2TNN)φq(g2(u1(T)))≥(1−T)φq(λ2TNN)K2ub1(T)≥(1−T)φq(λ2TNN)K2[M1(φq(λ1λa2))11−ab(1−t)]b≥φq(λ2TNN)K2Mb1(φq(λ1λa2))b1−ab(1−T)b+1=h(T)(φq(λ2λb1))11−ab≫1, |
where
Because
g1(αb1x)≥φp(αa1b1)g1(x). |
Thus
G(α,t)=tN−1(φp(αa1b1)−φp(α))g1(∫1tφq(1sN−1∫s0μτN−1g2(u1(τ))dτ)ds)≥tN−1l0(1−αp−1)g1(h(T)(φq(λ2λb1))11−ab)≥H(T)tN−1φq(λ2λb1))a1−ab(1−αp−1)>0, t≤T, | (19) |
where
φp(αa1b1)−φp(α)≥l0(1−αp−1) for α0≤α≤1. |
This proves that
zN−1(u′−αu′1)(z)<0, 0<z≤T. |
On the other hand, if
∫z0G(α,r)dr≥∫120G(α,t)dt+∫zTG(α,t)dt ≥H(12)N2Nφq(λ2λb1))a1−ab(1−αp−1)−C(1−T)(1−α1q1) >0. |
Therefore, we have
(u′−αu′1)(z)<0 for 0<z≤1. |
This shows that there is a constant
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
Lemma 4.1. (See Theorem 2.3.6 of [8], on page 99) Suppose that
infx∈P∩∂DΓx>0, |
then
Let
‖x‖=maxt∈J|x(t)|. |
Let
P:={v∈E:v(t)≥0, t∈J, v(t)≥14‖v‖, t∈[14,34]}. | (20) |
It is easy to see that
For
(T1v)(t)=∫1tφq(1τN−1∫τ0sN−1g1(v(s))ds)dτ, | (21) |
(T2v)(t)=φq(λ2)∫1tφq(1τN−1∫τ0sN−1g2(v(s))ds)dτ. | (22) |
It follows from Lemma 3 in [1] that
Define a composite operator
Let
g∞1:=limv→∞g1(v)φp(v), g01:=limv→0g1(v)φp(v); |
g∞2:=limv→∞g2(v)φp(v), g02:=limv→0g2(v)φp(v), |
and
A=∫3414sN−1ds=3N−1N4N, B=∫134φq(1τN−1)dτ, B∗=∫10φq(1τN−1)dτ. | (23) |
Theorem 4.2. Suppose that
λ1Rλ2∈[λR,ˉλR], |
where
Proof. Since
l1φp(v)<g1(v)<l2φp(v), ∀v≥μ; |
l1φp(u)<g2(u)<l2φp(u), ∀u≥μ. |
Next, we verify that
ΩR={x∈E:‖x‖<R}, |
then
Since
u(t)≥14‖u‖=14R, v(t)≥14‖v‖=14R, t∈[14,34], |
and
u(t)≥14‖u‖>14β0=μ, v(t)≥14‖v‖>14β0=μ, t∈[14,34]. |
So, for any
(T1v)(t)≥∫134φq(1τN−1∫340sN−1g1(v(s))ds)dτ≥∫134φq(1τN−1∫3414sN−1g1(v(s))ds)dτ≥∫134φq(1τN−1∫3414sN−1l1φp(v(s))ds)dτ≥∫134φq(1τN−1∫3414sN−1l1φp(14‖v‖)ds)dτ=14‖v‖φq(l1A)B, ∀t∈J. |
Analogously, for
(T2u)(t)≥∫134φq(1τN−1∫340sN−1g2(u(s))ds)dτ≥∫134φq(1τN−1∫3414sN−1g2(u(s))ds)dτ≥∫134φq(1τN−1∫3414sN−1l1φp(u(s))ds)dτ≥∫134φq(1τN−1∫3414sN−1l1φp(14‖u‖)ds)dτ=14‖u‖φq(l1A)B, ∀t∈J. |
Therefore, we get
(Tu)(t)=(T1T2u)(t)≥14‖T2u‖φq(l1A)B≥116‖u‖(φq(l1A)B)2. |
This gives that
infu∈P∩∂ΩRTu≥116‖u‖(φq(l1A)B)2>0. |
For any
For operator
Let
TuR=μ1RuR=1φq(λ1R)uR. | (24) |
It follows from the proof above that, for any
uR(t)=φq(λ1R)TuR, |
and so
uR(t)=φq(λ1R)∫1tφq(1τN−1∫τ0sN−1g1(vR(s))ds)dτ, |
vR(t)=φq(λ2R)∫1tφq(1τN−1∫τ0sN−1g2(uR(s))ds)dτ |
with
On the one hand,
uR(t)=φq(λ1R)∫1tφq(1τN−1∫τ0sN−1g1(vR(s))ds)dτ≤φq(λ1R)∫10φq(1τN−1∫10sN−1g1(vR(s))ds)dτ≤φq(l2λ1R)‖vR‖∫10φq(1τN−1)dτ=φq(l2λ1R)B∗‖vR‖, ∀t∈J. |
Analogously,
vR(t)≤φq(l2λ2)B∗‖uR‖, ∀t∈J. |
This verifies that
‖uR‖=R≤φq(l22(B∗)2λ1Rλ2)‖uR‖, |
and so,
λ1Rλ2≥1l22φp((B∗)2)=λR. |
On the other hand,
(uR)(t)≥φq(λ1R)∫134φq(1τN−1∫340sN−1g1(vR(s))ds)dτ≥φq(λ1R)∫134φq(1τN−1∫3414sN−1g1(vR(s))ds)dτ≥φq(λ1R)∫134φq(1τN−1∫3414sN−1l1φp(vR(s))ds)dτ≥φq(λ1R)∫134φq(1τN−1∫3414sN−1l1φp(14‖vR‖)ds)dτ=14φq(λ1Rl1A)B‖vR‖, ∀t∈J. |
Analogously, we can show that
(vR)(t)≥14φq(λ2l1A)B‖uR‖, ∀t∈J. |
Therefore, we get
‖uR‖≥116φq(λ1Rλ2l21A2)B2‖uR‖, |
and so,
λ1Rλ2≤φp(16)l21A2φp(B2)=ˉλR. | (25) |
We hence get
If we define another composite operator
(T∗1v)(t)=φq(λ1)∫1tφq(1τN−1∫τ0sN−1g1(v(s))ds)dτ, | (26) |
(T∗2v)(t)=∫1tφq(1τN−1∫τ0sN−1g2(v(s))ds)dτ. | (27) |
Corollary 1. Let
λ1λ2R∈[λR,ˉλR], | (28) |
where
Proof. Similar to the proof of Theorem 4.2, we can prove Corollary 1.
Theorem 4.3. Suppose that
λ1rλ2∈[λr,ˉλr], |
where
Proof. Similar to the proof of Theorem 4.2, we can prove Theorem 4.3.
Theorem 4.4. Suppose that
λ1R∗λ2∈(0,λR∗], | (29) |
where
Proof. Since
g1(v)>l∗φp(v), ∀v≥μ∗; |
g2(u)>l∗φp(u), ∀u≥μ∗. |
Now, we show that
ΩR∗={x∈E:‖x‖<R∗}. |
Since
u(t)≥14‖u‖=14R∗, v(t)≥14‖v‖=14R∗, t∈[14,34], |
and
u(t)≥14‖u‖>14ˉβ0=μ∗, v(t)≥14‖v‖>14ˉβ0=μ∗, t∈[14,34]. |
So, for any
(T1v)(t)≥∫134φq(1τN−1∫340sN−1g1(v(s))ds)dτ≥∫134φq(1τN−1∫3414sN−1g1(v(s))ds)dτ≥∫134φq(1τN−1∫3414sN−1l∗φp(v(s))ds)dτ≥∫134φq(1τN−1∫3414sN−1l∗φp(14‖v‖)ds)dτ=14‖v‖φq(l∗A)B, ∀t∈J. |
Analogously, for
(T2u)(t)≥∫134φq(1τN−1∫340sN−1g2(u(s))ds)dτ≥∫134φq(1τN−1∫3414sN−1g2(u(s))ds)dτ≥∫134φq(1τN−1∫3414sN−1l∗φp(u(s))ds)dτ≥∫134φq(1τN−1∫3414sN−1l∗φp(14‖u‖)ds)dτ=14‖u‖φq(l∗A)B, ∀t∈J. |
Therefore, we get
(Tu)(t)=(T1T2u)(t)≥14‖T2u‖φq(l∗A)B≥116‖u‖(φq(l∗A)B)2. |
This gives that
infu∈P∩∂ΩR∗Tu≥116‖u‖(φq(l∗A)B)2>0. |
For any
For operator
Let
Theorem 4.5. Suppose that
λ1r∗λ2∈(0,λ∗∗], |
where
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
~T2=T2T∗1, |
which has the same meaning as
Theorem 5.1. Suppose that
limλ1→0+‖uλ1‖=∞, limλ2→0+‖vλ2‖=∞; |
limλ1→0+‖uλ1‖=0, limλ2→0+‖vλ2‖=0. |
Proof. We need only verify this theorem under condition
g1(v)≤1λ1φp(B∗)φp(v), ∀ 0≤v≤r, |
g2(u)≤1λ2φp(B∗)φp(u), ∀ 0≤u≤r, |
where
Thus, for
(T∗1v)(t)=φq(λ1)∫1tφq(1τN−1∫τ0sN−1g1(v(s))ds)dτ≤φq(λ1)∫10φq(1τN−1∫10sN−1g1(v(s))ds)dτ≤φq(λ1)∫10φq(1τN−1∫10sN−11λ1φp(B∗)φp(v(s))ds)dτ≤φq(λ1)∫10φq(1τN−1∫10sN−11λ1φp(B∗)φp(‖v‖)ds)dτ≤‖v‖, ∀t∈J, |
and
(T2u)(t)=φq(λ2)∫1tφq(1τN−1∫τ0sN−1g2(u(s))ds)dτ≤φq(λ2)∫10φq(1τN−1∫10sN−1g2(u(s))ds)dτ≤φq(λ2)∫10φq(1τN−1∫10sN−11λ2φp(B∗)φp(u(s))ds)dτ≤φq(λ2)∫10φq(1τN−1∫10sN−11λ2φp(B∗)φp(‖u‖)ds)dτ≤‖u‖, ∀t∈J. |
So
‖~T1u‖=‖T∗1T2u‖≤‖T2u‖≤‖u‖. | (30) |
Next, for
g1(v)≥εφp(v), ∀v≥ˆR, |
g2(u)≥εφp(u), ∀u≥ˆR, |
where
φq(λ1λ2A2ε2)B2≥1, | (31) |
where
Let
u(t)≥14‖u‖≥ˆR, v(t)≥14‖v‖≥ˆR, t∈[14,34], |
and then
(T∗1v)(t)=φq(λ1)∫1tφq(1τN−1∫τ0sN−1g1(v(s))ds)dτ≥φq(λ1)∫134φq(1τN−1∫340sN−1g1(v(s))ds)dτ≥φq(λ1)∫134φq(1τN−1∫3414sN−1g1(v(s))ds)dτ≥φq(λ1)∫134φq(1τN−1∫3414sN−1εφp(v(s))ds)dτ≥φq(λ1)∫134φq(1τN−1∫3414sN−1εφp(14‖v‖)ds)dτ=φq(λ1Aε)B‖v‖, ∀t∈J. |
Similarly, we get
(T2u)(t)≥φq(λ2Aε)B‖u‖, ∀t∈J. |
So, by (31), we have
(~T1v1)(t)=(T∗1T2u)(t)≥φq(λ1Aε)B‖T2u‖≥φq(λ1λ2A2ε2)B2‖u‖≥‖u‖. | (32) |
From the above estimate and the fixed point theorem of cone expansion and compression of norm type, we deduce that operator
Similarly, we can prove that
Next, for
‖uλ1m‖≤ς1, ‖vλ2m‖≤ς2 (m=1,2,3,⋯). |
Moreover, the sequence
If
1φq(λ1m)=‖∫1tφq(1τN−1∫τ0sN−1g1(v(s))ds)dτ‖‖uλ1m‖≤‖∫10φq(1τN−1∫10sN−1g1(v(s))ds)dτ‖‖uλ1m‖≤φq(D1)B∗‖uλ1m‖<2φq(D1)B∗η1 (m>M), |
and
1φq(λ2m)=‖∫1tφq(1τN−1∫τ0sN−1g2(u(s))ds)dτ‖‖vλ2m‖≤‖∫10φq(1τN−1∫10sN−1g2(u(s))ds)dτ‖‖vλ2m‖≤φq(D2)B∗‖vλ2m‖<2φq(D1)B∗η2 (m>M), |
where,
D1=max{g1(v), r≤‖v‖≤R}, |
D2=max{g2(u), r≤‖u‖≤R}. |
This gives a contradiction as
If
g1(vλ2m)≤εφp(v), ∀ 0≤vλ2m≤r∗, |
g2(uλ1m)≤εφp(u), ∀ 0≤uλ1m≤r∗. |
Then, for
1φq(λ1m)=‖∫1tφq(1τN−1∫τ0sN−1g1(v(s))ds)dτ‖‖uλ1m‖≤‖∫10φq(1τN−1∫10sN−1g1(v(s))ds)dτ‖‖uλ1m‖≤‖∫10φq(1τN−1∫10sN−1εφp(v(s))ds)dτ‖‖uλ1m‖≤φq(ε)B∗‖v‖‖uλ1m‖, |
and
1φq(λ2m)=‖∫1tφq(1τN−1∫τ0sN−1g2(u(s))ds)dτ‖‖vλ2m‖≤‖∫10φq(1τN−1∫10sN−1g2(v(s))ds)dτ‖‖vλ2m‖≤‖∫10φq(1τN−1∫10sN−1εφp(v(s))ds)dτ‖‖vλ2m‖≤φq(ε)B∗‖v‖‖vλ2m‖, |
where
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
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
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
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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