# Entire Blow-Up Solutions of Semilinear Elliptic Systems with Quadratic Gradient Terms

Authors: Yongju Yang; Xinguang Zhang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study the existence of entire positive solutions for the semilinear elliptic system with quadratic gradient terms,Δui+|∇ui|2=pi(|x|)fi(u1,u2,…,ud) for i=1,2,…,d on RN,N≥3 and d∈{1,2,3,…}. We establish the conditions on pi that ensure the existence of nonnegative radial solutions blowing up at infinity and also the conditions for bounded solutions on the entire space. The condition on fi is simple and different to the Keller-Osserman condition.


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## Body

## 1. Introduction

We study the existence of entire blow-up positive solutions of the following elliptic system with quadratic gradient terms:(1.1)Δu1+|∇u1|2=p1(|x|)f1(u1,u2,…,ud),x∈RN,⋮⋮Δud+|∇ud|2=pd(|x|)fd(u1,u2,…,ud),x∈RN,
where d≥1, N≥3, pi  (i=1,2,…,d) are c-positive functions and fi:[0,∞)d→[0,∞) are nonnegative, continuous, and nondecreasing functions for each variable.For convenience we recall the definitions aboutc-positive functions and entire blow-up positive solutions.(i)
A functionp is c-positive (or circumferentially positive) in a domain Ω⊆RN if p is nonnegative on Ω and satisfies the following condition: if x0∈Ω and p(x0)=0, then there exists a domain Ω0 such that x0∈Ω0⊂Ω and p(x)>0 for all x∈∂Ω0.(ii)
A solution(u1,u2,…,ud) of the system (1.1) is called an entire blow-up solution (or explosive solution) if it is a classical solution of the above problem on RN and ui(x)→∞,(i=1,2,…,d) as |x|→∞.Existence and nonexistence of blow-up solutions of semilinear elliptic equations and systems have received much attention worldwide. Bieberbach [1] is the first to study blow-up solutions to the semilinear elliptic problem
(1.2)Δu=f(u),x∈Ω,
where f(u)=eu. Following Bieberbach’s work, many authors have studied related problems for single equations and systems. In 1957, Keller [2] and Osserman [3] established the necessary and sufficient conditions for the existence of solutions to (1.2) on bounded domains in Rn. They showed that blow-up solutions exist on Ω if and only if f satisfies the following Keller-Osserman condition:
(1.3)∫1+∞‍[∫0t‍f(s)ds]-1/2dt<+∞.
Bandle and Marcus [4] later examined the equation
(1.4)Δu=p(x)f(u)
with f is nondecreasing on [0,+∞) and proved the existence of positive blow-up solutions under the condition that the function f satisfies the Keller-Osserman condition (1.3) and p is continuous and strictly positive on Ω¯. Lair [5] showed that the results also hold for (1.4) when p is allowed to vanish on a large part of Ω, including its boundary. In addition, many authors have examined some more specific forms of (1.4). The equation
(1.5)Δu=p(x)uγ
has been of particular interest. Cheng and Ni [6] considered the superlinear case γ>1 and proved that for this case (1.5) has blow-up solutions on bounded domains provided p is strictly positive on ∂Ω. Lair and Wood [7] generalized this to allow p to vanish on some portions of Ω including its boundary and also showed the existence of an entire blow-up solution to (1.5) provided that
(1.6)∫0+∞‍rmax|x|=rp(r)dr<+∞.
Obviously, condition (1.6) is weaker than the requirements in [6].In [8], Lair and Wood proved that (1.5) has entire blow-up radial solutions if and only if
(1.7)∫0+∞‍rp(r)dr<+∞.
They also demonstrated that for a bounded domain Ω, (1.5) has no positive blow-up solution when p is continuous in Ω. In addition, they proved that nonnegative, entire bounded solutions do not exist for (1.5) if
(1.8)∫0+∞‍rmin|x|=rp(r)dr=+∞.Although semilinear elliptic systems are the natural extension of single equations in many areas of applications, the results and methods for the study of single equations are often not applicable to the systems of equations. Recently, Lair and Wood [9] studied the existence of entire positive solutions of the system
(1.9)Δu=p1(|x|)vα,x∈RN,Δv=p2(|x|)uβ,x∈RN.
In the sublinear case 0<α≤β≤1, the authors proved that provided that the nonnegative functions p and q are continuous, c-positive, and satisfy the fast decay conditions
(F)∫0∞‍tpi(t)dt<∞,i=1,2,
then the entire positive solutions are bounded, while if p and q satisfy the slow decay conditions
(D)∫0∞‍tpi(t)dt=∞,i=1,2,
then the entire positive solutions blow up. For the superlinear case α,β>1, the fast decay conditions (F) are required to hold. Later, Cîrstea and Rădulescu [10] improved the results of Lair and Wood [9] and proved that for p,q∈Cloc0,α(RN)(0<α<1), the following semilinear elliptic system
(1.10)Δu=p1(x)f(v),x∈RN,Δv=p2(x)g(u),x∈RN
has entire solutions if f and g satisfy
(1.11)limt→∞f(cg(t))t=0
for all c>0 and has solutions that are bounded when (D) holds. Further, entire solutions exist and are blow-up when (F) holds. An analogous condition was also employed by Ghergu and Rădulescu [11] to study the following elliptic system with gradient terms:
(1.12)Δu+|∇u|=p1(|x|)f(v),x∈Ω,Δv+|∇v|=p2(|x|)g(u),x∈Ω,
where Ω is a bounded domain or the whole space. Peng and Song [12] also studied the existence of entire blow-up positive solutions of system (1.10) when the c-positive functions pi, i=1,2 satisfy the decay conditions (F). Peng and Song [12] also imposed on f and g the following Keller-Osserman conditions:
(1.13)∫1∞‍[∫0s‍f(t)dt]-1/2ds<∞,∫1∞‍[∫0s‍g(t)dt]-1/2ds<∞,
and the convexity conditions
(1.14)f(λa+(1-λ)b)≤λf(a)+(1-λ)f(b),λ∈(0,1),g(λa+(1-λ)b)≤λg(a)+(1-λ)g(b),a,b≥0.
Both papers [6, 12] considered system (1.10) where the nonnegative functions pi(i=1,2)∈C[0,+∞) satisfy (F) and the functions f,g∈C[0,+∞) are nondecreasing and satisfy the Keller-Osserman condition (1.13), and
(1.15)f(0)=g(0)=0,f(s)>0,g(s)>0,fors>0.
Recently, Zhang and Liu [13] studied the following semilinear elliptic system with the magnitude of the gradient
(1.16)Δu+|∇u|=p(|x|)f(u,v),x∈RN,Δv+|∇v|=q(|x|)g(u,v),x∈RN.
The results of nonexistence of entire positive solutions have been established if f and g are sublinear and p and q have fast decay at infinity, while if f and g satisfy some growth conditions at infinity, and p, q are of slow decay or fast decay at infinity, then the system has infinitely many entire solutions, which are large or bounded. In [14], Covei studied the existence of solution of the following semilinear elliptic system:
(1.17)Δu1=p1(x)f1(u1,u2,…,ud),x∈RN,⋮⋮Δud=pd(x)fd(u1,u2,…,ud),x∈RN.
Under some conditions on fi, pi, the system (1.17) has a bounded positive entire solution based on successive approximation. Furthermore, a nonradially symmetric solution also was obtained by using a lower and upper solution method. For more complicated Schrödinger systems, some nice work had been done by Covei in [15–17] with single equations or a system with (p1,…,pd)-Laplacian in RN. For further results on relevant work on single equations and/or systems as well as methods for the study of blow-up solutions of differential equations, see [8, 18–32] and the references therein.The authors in [13, 14] only studied the semilinear elliptic system with the magnitude of the gradient term or without the gradient term. For elliptic systems involving nonlinear quadratic gradient terms, no result has been obtained. Thus, motivated by [11–17], we study the more general systems case with indefinite number of equations involving a nonlinear quadratic gradient term. In our results, a simple condition (2.5) has been used instead of the Keller-Osserman condition (1.13) commonly used in previous results. The main results obtained are presented in Section 2 by Theorems 2.3 to 2.6, while the proofs of the theorems are given in Section 3.


## 2. Main Results

For convenience in presenting the results, we here define(2.1)Pi(∞)=limr→∞Pi(r),Pi(r)=∫0r‍t1-N∫0t‍sN-1pi(s)dsdt,r≥0,i=1,2,…,d,F(∞)=limr→∞F(r),F(r)=∫ar‍ds∑i=1dsfi(lns,lns,…,lns),r≥a>1,i=1,2,…,d.Remark 2.1.
For anyi∈{1,2,…,d}, since
(2.2)F′(r)=1∑i=1drfi(lnr,lnr,…,lnr)>0,∀r>a,Fadmits the inverse function F-1 on [0,F(+∞)).Lemma 2.2 (see [8, 23]).
The slow decay condition(2.3)∫0∞‍tpi(t)dt=∞,i=1,2,…,d
holds if and only if Pi(∞)=∞.The first result we obtained is the condition for nonexistence of entire positive blow-up solution, which asserts that if bothfi, i=1,2,…,d are bounded, then problem (1.1) does not have positive entire blow-up solution as detailed by the following theorem.Theorem 2.3.
Supposefi, i=1,2,…,d satisfy
(2.4)max{sup∑i=1dui≥1f1(u1,u2,…,ud),…,sup∑i=1dui≥1fd(u1,u2,…,ud)}<+∞,
and each pi, i=1,2,…,d satisfy the decay conditions (F). Then problem (1.1) does not have positive entire blow-up solution.The other main results we obtained are the conditions, respectively, for the existence of infinitely many positive entire blow-up solutions and infinitely many positive entire bounded solutions, which are summarized in the following three theorems.Theorem 2.4.
If there exists a constanta>1 such that
(2.5)∫a∞‍ds∑i=1dsfi(lns,lns,…,lns)=∞,
then the system (1.1) has infinitely many classical positive entire solutions (u1,u2,…,ud). If, in addition, pi, i=1,2,…,d satisfy the decay conditions (D), then all the positive entire solutions of (1.1) are blow-up. Moreover, if pi, i=1,2,…,d satisfy the decay conditions (F), then all the positive entire solutions of (1.1) are bounded.Theorem 2.5.
If there exists a constanta>1 such that
(2.6)∫a∞‍ds∑i=1dsfi(lns,lns,…,lns)<∞,
and pi, i=1,2,…,d satisfy the decay conditions (F) and, in addition, there exist bi>a, i=1,2,…,d such that
(2.7)∑i=1d‍Pi(∞)<F(∞)-F(∑i=1d‍bi),
then the system (1.1) has a positive radial bounded solution (u1,u2,…,ud) satisfying
(2.8)bi+bifi(lnb1,lnb2,…,lnbd)Pi(r)≤ui(r)≤F-1(F(∑i=1d‍bi)+∑i=1d‍Pi(r)),r≥0.Theorem 2.6.
(i)
Ifpi, i=1,2,…,d satisfy the decay conditions  (D) and
(2.9)lims→∞∑i=1d‍fi(lns,lns,…,lns)=0,
then the system (1.1) has infinitely many positive entire blow-up solutions.
(ii)
Ifpi, i=1,2,…,d satisfy the decay conditions (F) and
(2.10)sups≥0∑i=1d‍sfi(lns,lns,…,lns)<∞,
then the system (1.1) has infinitely many positive entire bounded solutions.


## 3. Proofs of the Theorems

Firstly, via the change of variablesϕi=eui, i=1,2,…,d, we turn the system (1.1) to the following equivalent system with no gradient terms
(3.1)Δϕ1=p1(|x|)ϕ1f1(lnϕ1,lnϕ2,…,lnϕd),x∈RN,⋮⋮Δϕi=pi(|x|)ϕifi(lnϕ1,lnϕ2,…,lnϕd),x∈RN,⋮⋮Δϕd=pd(|x|)ϕdfd(lnϕ1,lnϕ2,…,lnϕd),x∈RN.
Thus we only need to consider system (3.1).Proof of Theorem2.3.
We use proof by contradiction to  testify. We suppose that the system (3.1) has the positive entire blow-up solution (ϕ1,ϕ2,…,ϕd). Consider the spherical average of ϕi defined by
(3.2)ϕ¯i(r)=1cNrN-1∫|x|=r‍ϕi(x)dσx,r≥0,
where cN is the surface area of the unit sphere in RN. Since ϕi are positive entire blow-up solutions, it follows that ϕ¯i are positive and limr→∞ϕ¯i(r)=+∞. By the change of variable x=ry, we have
(3.3)ϕ¯i(r)=1cN∫|y|=1‍ϕi(ry)dσy,r≥0.
Then
(3.4)ϕ¯i′(r)=1cN∫|y|=1‍∇ϕi(ry)·ydσy,r≥0.
Thus by the divergence theorem and (3.4), we have
(3.5)ϕ¯i′(r)=rcN∫|y|<1‍Δϕi(ry)dy=1cNrN-1∫|x|<r‍Δϕi(x)dx=1cNrN-1∫0r‍dρ∫|x|=ρ‍Δϕi(x)dσx,∀r≥0.
From [33], it follows from (3.5) that
(3.6)ϕ¯i′′(r)=1cNrN-1∫|x|=r‍Δϕi(x)dσx-N-1cNrN∫0r‍dρ∫|x|=ρ‍Δϕi(x)dσx=1cNrN-1∫|x|=r‍Δϕi(x)dσx-N-1rϕ¯i′(r),∀r≥0.
Set
(3.7)Ui(r)=max0≤t≤rϕ¯i(t),
Then, obviously, Ui are positive and nondecreasing functions. Moreover Ui≥ϕ¯i and Ui(r)→+∞ as r→+∞. Note from (2.4) that there exists M>0 such that
(3.8)max{f1(u1,u2,…,ud),…,fd(u1,u2,…,ud)}≤M,u1+u2+⋯+ud≥0.
Now (3.6) and (3.8) lead to
(3.9)ϕ¯i′′+N-1rϕ¯i′≤1cNrN-1∫|x|=r‍Δϕi(x)dσx=pi(r)1cNrN-1∫|x|=r‍ϕi(x)fi(lnϕ1(x),lnϕ2(x),…,lnϕd(x))dσx≤Mpi(r)1cNrN-1∫|x|=r‍(1+∑i=1d‍ϕi(x))dσx=Mpi(r)(1+∑i=1d‍ϕ¯i(r))≤Mpi(r)(1+Ui(r)),
for all r≥0. It follows that
(3.10)(rN-1ϕ¯i′)′≤MrN-1pi(r)(1+Ui(r)),r≥0.
So, for all r≥r0≥0, we have
(3.11)ϕ¯i(r)≤ϕ¯i(r0)+M∫r0r‍t1-N∫0t‍sN-1pi(s)(1+Ui(s))dsdt≤ϕ¯i(r0)+M∫r0r‍t1-N(1+U(t))∫0t‍sN-1p(s)dsdt≤ϕ¯i(r0)+M(1+Ui(r))∫r0r‍t1-N∫0t‍sN-1pi(s)dsdt≤ϕ¯i(r0)+M(1+Ui(r))∫r0r‍t1-N∫0t‍sN-1pi(s)dsdt≤ϕ¯i(r0)+M(1+Ui(r))N-2(∫r0r‍tpi(t)dt-1r0N-1∫r0r‍tN-1pi(t)dt)≤ϕ¯i(r0)+M(1+Ui(r))N-2∫r0r‍tpi(t)dt.
Note that because of (F), we can choose r0>0 sufficiently large such that
(3.12)max{∫r0∞‍rp1(r)dr,…,∫r0∞‍rpd(r)dr}<N-24M.
Since limr→∞ϕ¯i(t)=∞, it follows that we can find r1≥r0 such that
(3.13)U¯i(r)=maxr0≤t≤rϕ¯i(t),∀r≥r1.
Thus (3.11) and (3.13) yield
(3.14)U¯i(r)≤ϕ¯i(r0)+M(1+U¯i(r))N-2∫r0r‍tpi(t)dt,∀r≥r1.
By (3.12), we have
(3.15)U¯i(r)≤ϕ¯i(r0)+1+U¯i(r)4,∀r≥r1,
that is,
(3.16)U¯i(r)≤Ci+U¯i(r)4,∀r≥r1,
where Ci=(1/4)+ϕ¯i(r0)>0, which implies
(3.17)∑i=1d‍U¯i(r)≤43∑i=1d‍Ci,∀r≥r1.
The inequality (3.17) means that U¯i are bounded and so ϕ¯i are bounded which is a contradiction. It follows that (1.1) has no positive entire blow-up solutions, and the proof is completed.Proof of Theorem2.4.
We start by showing that (1.1) has positive radial solutions. Towards this end we fix bi>a, i=1,2,…,d and we show that the system
(3.18)ϕi′′+N-1rϕi′=pi(r)ϕi(r)fi(lnϕ1(r),lnϕ2(r),…,lnϕd(r)),i=1,2,…,dϕi′≥0,on[0,∞),ϕi(0)=bi>a
has a solution (ϕ1,ϕ2,…,ϕd). Thus (U1(x),U2(x),…,Ud(x))=(ϕ1(|x|),ϕ2(|x|),…,ϕd(|x|)) are positive solutions of (3.1). Integrating (3.18), for any r≥0 and i=1,2,…,d, we have
(3.19)ϕi(r)=bi+∫0r‍t1-N∫0t‍sN-1pi(s)ϕi(s)fi(lnϕ1(s),lnϕ2(s),…,lnϕd(s))dsdt.
Let{ϕi(k)}k≥0 be sequences of positive continuous functions defined on [0,∞) for i=1,2,…,d by
(3.20)ϕi(0)(r)=bi,ϕi(k+1)(r)=bi+∫0r‍t1-N∫0t‍sN-1pi(s)ϕi(k)(s)bi+∫0r‍t1-N×fi(lnϕ1(k)(s),lnϕ2(k)(s),…,lnϕd(k)(s))dsdt.
Obviously, for all r≥0, we have ϕi(k)(r)≥bi, ϕ0≤ϕ1. The monotonicity of fi yields ϕ1(r)≤ϕ2(r), r≥0. Repeating the argument, we deduce that
(3.21)ϕi(k)(r)≤ϕi(k+1)(r),r≥0,k≥1,
which means {ϕi(k)}k≥0 are nondecreasing sequences on [0,∞). Since
(3.22)ϕi(k+1)′(r)=r1-N∫0r‍sN-1pi(s)ϕi(k)(s)fi(lnϕ1(k)(s),lnϕ2(k)(s),…,lnϕd(k)(s))ds≤ϕi(k)(r)fi(lnϕ1(k)(r),lnϕ2(k)(r),…,lnϕd(k)(r))Pi′(r),
we have
(3.23)∑i=1d‍ϕi(k+1)′(r)≤∑i=1d‍ϕi(k+1)(r)∑i=1d‍fi(lnϕ1(k)(r),lnϕ2(k)(r),…,lnϕd(k)(r))Pi′(r)≤∑i=1d‍ϕi(k+1)(r)∑i=1d‍fi(ln∑i=1d‍ϕi(k+1)(r),ln∑i=1d‍ϕi(k+1)(r),…,ln∑i=1d‍ϕi(k+1)(r))×∑i=1d‍Pi′(r).
Let w(k+1)(r)=∑i=1d‍ϕi(k+1)(r) which implies
(3.24)w′(k+1)(r)w(k+1)(r)∑i=1dfi(lnw(k+1)(r),lnw(k+1)(r),…,lnw(k+1)(r))≤∑i=1d‍Pi′(r).
So, we have
(3.25)∫0r‍w′(k+1)(t)w(k+1)(t)∑i=1dfi(lnw(k+1)(t),lnw(k+1)(t),…,lnw(k+1)(t))dt≤∑i=1d‍Pi(r),
that is
(3.26)F(∑i=1d‍ϕi(k+1)(r))-F(∑i=1d‍bi)≤∑i=1d‍Pi(r),∀r≥0.
As F-1 increases on [0,∞), from (3.26), we have that
(3.27)∑i=1d‍ϕi(k+1)(r)≤F-1(F(∑i=1d‍bi)+∑i=1d‍Pi(r)),∀r≥0.
It follows from F(∞)=∞ that F-1(∞)=∞. By (3.27), the sequences {ϕi(k)} are bounded and increasing on [0,c0] for any c0>0. Thus, {ϕi(k)} have subsequences converging uniformly to ϕi on [0,c0]. Consequently, (ϕ1,ϕ2,,…,ϕd) is a positive solution of (3.18); that is, (U1,U2,…,Ud) is a entire positive solution of (3.1). By noticing ϕi(0)=bi and that bi∈(0,∞) was chosen arbitrarily, it follows that (1.1) has infinitely many positive entire solutions.(i)
IfPi(∞)=∞, since
(3.28)ϕi(r)≥bi+bifi(lnb1,lnb2,…,lnbd)Pi(r),r≥0,we have
(3.29)limr→∞ϕi(r)=∞,which means that (U1,U2,…,Ud) are positive entire blow-up solutions of (1.1).(ii)
IfPi(∞)<∞, then
(3.30)∑i=1d‍ϕi(r)≤F-1(F(∑i=1d‍bi)+∑i=1d‍Pi(∞))<∞,which implies that (U1,U2,…,Ud) are positive entire bounded solutions of (1.1). Proof of the theorem is now completed.Proof of Theorem2.5.
If condition (2.7) holds, then we have
(3.31)F(∑i=1d‍ϕi(k+1)(r))≤F(∑i=1d‍bi)+∑i=1d‍Pi(r)≤F(∑i=1d‍bi)+∑i=1d‍Pi(∞)<F(∞)<∞.
Since F-1 is strictly increasing on [0,∞), we have
(3.32)∑i=1d‍ϕi(k+1)(r)≤F-1(F(∑i=1d‍bi)+∑i=1d‍Pi(∞))<∞.
The last part of the proof is clear from that of Theorem 2.4. Thus we omit it.Proof of Theorem2.6.
(i)  It follows from (3.20) that
(3.33)ϕi(k)(r)≤ϕi(k+1)(r)≤bi+ϕi(k)(r)fi(lnϕ1(k)(r),lnϕ2(k),…,lnϕd(k))Pi(r)≤bi+ϕi(k)(r)fi(ln∑i=1d‍ϕi(k)(r),ln∑i=1d‍ϕi(k)(r),…,ln∑i=1d‍ϕi(k)(r))Pi(r).
Let R>0 be arbitrary. From (3.33) we get, for k≥1,
(3.34)∑i=1d‍ϕi(k)(R)≤∑i=1d‍bi+∑i=1d‍ϕi(k)(R)×∑i=1d‍fi(ln∑i=1d‍ϕi(k)(R),ln∑i=1d‍ϕi(k)(R),…,ln∑i=1d‍ϕi(k)(R))∑i=1d‍Pi(R).
This implies
(3.35)1≤∑i=1dbi∑i=1dϕi(k)(R)+∑i=1d‍fi(ln∑i=1d‍ϕi(k)(R),ln∑i=1d‍ϕi(k)(R),…,ln∑i=1d‍ϕi(k)(R))∑i=1d‍Pi(R).
Taking into account the monotonicity of (∑i=1d‍ϕi(k)(R))k≥1, there exists
(3.36)L(R)∶=limk→∞(∑i=1d‍ϕi(k)(R)).
We claim that L(R) is finite. Indeed, if not, we let k→∞ in (3.35), and the assumption (2.9) leads us to a contradiction. Thus L(R) is finite. Since ϕi(k) are increasing functions, it follows that the map L∶(0,∞)→(0,∞) is nondecreasing and
(3.37)∑i=1d‍ϕi(k)(r)≤∑i=1d‍ϕi(k)(R)≤L(R),∀r∈[0,R],∀k≥1.
Thus the sequences {(ϕi(k))k≥1} are bounded from above on bounded sets. Let
(3.38)ϕi(r)∶=limk→∞ϕi(k)(r),∀r≥0.
Then (ϕ1,ϕ2,…,ϕd) is a positive solution of (3.18).
In order to conclude the proof, it is sufficient to show that(ϕ1,ϕ2,…,ϕd) is a blow-up solution of (3.18). Let us remark that (3.19) implies
(3.39)ϕi(r)≥bi+bifi(lnb1,lnb2,…,lnbd)Pi(r),r≥0.
Since fi are positive functions and
(3.40)Pi(∞)=∞,
we can conclude that (ϕ1,ϕ2,…,ϕd) is a blow-up solution of (3.18), and so (U1,U2,…,Ud) is a positive entire blow-up solution of (3.1). Thus any blow-up solution of (3.1) provides a positive entire blow-up solution of (1.1). Since bi∈(0,∞) was chosen arbitrarily, it follows that (1.1) has infinitely many positive entire blow-up solutions.
(ii) If(3.41)sups≥0∑i=1d‍sfi(lns,lns,…,lns)<∞
holds, then by (3.35), we have
(3.42)L(R)∶=limk→∞∑i=1d‍ϕi(k)(R)<∞.
Thus
(3.43)∑i=1d‍ϕi(k)(r)≤∑i=1d‍ϕi(k)(R)≤L(R),∀r∈[0,R],∀k≥1.
So the sequences {ϕi(k)}k≥1 are bounded from above on bounded sets. Let
(3.44)ϕ(r)∶=limk→∞ϕi(k)(r),∀r≥0.
Then (ϕ1,ϕ2,…,ϕd) is a positive solution of (3.18).
It follows from (3.33) and (3.35) that (ϕ1,ϕ2,…,ϕd) is bounded, which implies that (1.1) has infinitely many positive entire bounded solutions.In the end of this work we also remark on a system with different gradient exponent(3.45)Δu1+|∇u1|a1=p1(|x|)f1(u1,u2,…,ud),x∈RN,⋮⋮Δud+|∇ud|ad=pd(|x|)fd(u1,u2,…,ud),x∈RN,
where ai∈(0,+∞), ai≠1,2, fi:[0,∞)d→[0,∞) are nonnegative, continuous, and nondecreasing functions for each variable. For these cases, the problem is far more complex, and no analogous results have been established [9, 10, 13, 18, 21]. We also anticipate that the methods and concepts here can be extended to the systems with qi-Laplacian as considered by Covei [14–17].



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*Source: 697565-2012-12-02.xml*