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

Abstract

We study the existence of entire positive solutions for the semilinear elliptic system with quadratic gradient terms, Δ𝑢_𝑖+|∇𝑢_𝑖|²=𝑝_𝑖(|𝑥|)𝑓_𝑖(𝑢₁,𝑢₂,…,𝑢_𝑑) for 𝑖=1,2,…,𝑑 on 𝑅^𝑁,𝑁≥3 and 𝑑∈{1,2,3,…}. We establish the conditions on 𝑝_𝑖 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 𝑓_𝑖 is simple and different to the Keller-Osserman condition.

1. Introduction

We study the existence of entire blow-up positive solutions of the following elliptic system with quadratic gradient terms: Δ𝑢₁+∇𝑢₁²=𝑝₁(|𝑥|)𝑓₁𝑢₁,𝑢₂,…,𝑢_𝑑,𝑥∈𝑅^𝑁,⋮⋮Δ𝑢_𝑑+∇𝑢_𝑑²=𝑝_𝑑(|𝑥|)𝑓_𝑑𝑢₁,𝑢₂,…,𝑢_𝑑,𝑥∈𝑅^𝑁, (1.1) where 𝑑≥1, 𝑁≥3, 𝑝_𝑖  (𝑖=1,2,…,𝑑) are 𝑐-positive functions and 𝑓_𝑖∶[0,∞)^𝑑→[0,∞) are nonnegative, continuous, and nondecreasing functions for each variable.

For convenience we recall the definitions about 𝑐-positive functions and entire blow-up positive solutions.(i)A function 𝑝 is 𝑐-positive (or circumferentially positive) in a domain Ω⊆𝑅^𝑁 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,…,𝑢_𝑑) 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 𝑅^𝑁 and 𝑢_𝑖(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 Δ𝑢=𝑓(𝑢),𝑥∈Ω, (1.2) where 𝑓(𝑢)=𝑒^𝑢. 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 𝑅^𝑛. They showed that blow-up solutions exist on Ω if and only if 𝑓 satisfies the following Keller-Osserman condition: ₁^+∞𝑡₀𝑓(𝑠)𝑑𝑠^−1/2𝑑𝑡<+∞. (1.3) Bandle and Marcus [4] later examined the equation Δ𝑢=𝑝(𝑥)𝑓(𝑢) (1.4) 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 Δ𝑢=𝑝(𝑥)𝑢^𝛾 (1.5) 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 ₀^+∞𝑟max_|𝑥|=𝑟𝑝(𝑟)𝑑𝑟<+∞. (1.6) 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 𝑟𝑝(₀^+∞𝑟)𝑑𝑟<+∞. (1.7) 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 ₀^+∞𝑟min_|𝑥|=𝑟𝑝(𝑟)𝑑𝑟=+∞. (1.8)

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 Δ𝑢=𝑝₁(|𝑥|)𝑣^𝛼,𝑥∈𝑅^𝑁,Δ𝑣=𝑝₂(|𝑥|)𝑢^𝛽,𝑥∈𝑅^𝑁. (1.9) 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 ^∞₀𝑡𝑝_𝑖(𝑡)𝑑𝑡<∞,𝑖=1,2, (𝐹) then the entire positive solutions are bounded, while if 𝑝 and 𝑞 satisfy the slow decay conditions ^∞₀𝑡𝑝_i(𝑡)𝑑𝑡=∞,𝑖=1,2, (𝐷) 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∈𝐶^0,𝛼_loc(𝑅^𝑁) (0<𝛼<1), the following semilinear elliptic system Δ𝑢=𝑝₁(𝑥)𝑓(𝑣),𝑥∈𝑅^𝑁,Δ𝑣=𝑝₂(𝑥)𝑔(𝑢),𝑥∈𝑅^𝑁 (1.10) has entire solutions if 𝑓 and 𝑔 satisfy lim_𝑡→∞𝑓𝑐𝑔(𝑡)𝑡=0 (1.11) 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: Δ𝑢+|∇𝑢|=𝑝₁𝑥𝑓(𝑣)||(||),𝑥∈Ω,Δ𝑣+∇𝑣=𝑝₂(|𝑥|)𝑔(𝑢),𝑥∈Ω, (1.12) 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 𝑝_𝑖, 𝑖=1,2 satisfy the decay conditions (𝐹). Peng and Song [12] also imposed on 𝑓 and 𝑔 the following Keller-Osserman conditions: ^∞₁^𝑠₀𝑓(𝑡)𝑑𝑡^−1/2𝑑𝑠<∞,^∞₁^𝑠₀𝑔(𝑡)𝑑𝑡^−1/2𝑑𝑠<∞, (1.13) and the convexity conditions 𝑓(()𝑏)(𝑎)+()𝑓(𝑏)(),𝑔(()𝑏)(𝑎)+()𝑔(𝑏)𝜆𝑎+1−𝜆≤𝜆𝑓1−𝜆,𝜆∈0,1𝜆𝑎+1−𝜆≤𝜆𝑔1−𝜆,𝑎,𝑏≥0. (1.14) Both papers [6, 12] considered system (1.10) where the nonnegative functions 𝑝_𝑖(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 𝑓(0)=𝑔(0)=0,𝑓(𝑠)>0,𝑔(𝑠)>0,for𝑠>0. (1.15) Recently, Zhang and Liu [13] studied the following semilinear elliptic system with the magnitude of the gradient Δ𝑢+|∇𝑢|=𝑝(|𝑥|)𝑓(𝑢,𝑣),𝑥∈𝑅^𝑁,Δ𝑣+|∇𝑣|=𝑞(|𝑥|)𝑔(𝑢,𝑣),𝑥∈𝑅^𝑁. (1.16) 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: Δ𝑢₁=𝑝₁(𝑥)𝑓₁𝑢₁,𝑢₂,…,𝑢_𝑑,𝑥∈𝑅^𝑁,⋮⋮Δ𝑢_𝑑=𝑝_𝑑(𝑥)𝑓_𝑑𝑢₁,𝑢₂,…,𝑢_𝑑,𝑥∈𝑅^𝑁. (1.17) Under some conditions on 𝑓_𝑖, 𝑝_𝑖, 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,…,𝑝_𝑑)-Laplacian in 𝑅^𝑁. 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 𝑃_𝑖(∞)=lim_𝑟→∞𝑃_𝑖(𝑟),𝑃_𝑖(𝑟)=^𝑟₀𝑡^1−𝑁𝑡₀𝑠^𝑁−1𝑝_𝑖(𝑠)𝐹(∞)𝑑𝑠𝑑𝑡,𝑟≥0,𝑖=1,2,…,𝑑,=lim_𝑟→∞𝐹(𝑟),𝐹(𝑟)=^𝑟_𝑎𝑑𝑠∑^𝑑_𝑖=1𝑠𝑓_𝑖(ln𝑠,ln𝑠,…,ln𝑠),𝑟≥𝑎>1,𝑖=1,2,…,𝑑. (2.1)

Remark 2.1. For any 𝑖∈{1,2,…,𝑑}, since 𝐹^′(𝑟)=1∑^𝑑_𝑖=1𝑟𝑓_𝑖(ln𝑟,ln𝑟,…,ln𝑟)>0,∀𝑟>𝑎, (2.2)𝐹admits the inverse function 𝐹⁻¹ on [0,𝐹(+∞)).

Lemma 2.2 (see [8, 23]). The slow decay condition ^∞₀𝑡𝑝_𝑖(𝑡)𝑑𝑡=∞,𝑖=1,2,…,𝑑 (2.3) holds if and only if 𝑃_𝑖(∞)=∞.

The first result we obtained is the condition for nonexistence of entire positive blow-up solution, which asserts that if both 𝑓_𝑖, 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. Suppose 𝑓_𝑖, 𝑖=1,2,…,𝑑 satisfy maxsup_{∑𝑑𝑖=1𝑢𝑖≥1}𝑓₁𝑢₁,𝑢₂,…,𝑢_𝑑,…,sup_{∑𝑑𝑖=1𝑢𝑖≥1}𝑓_𝑑𝑢₁,𝑢₂,…,𝑢_𝑑<+∞, (2.4) and each 𝑝_𝑖, 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 constant 𝑎>1 such that ^∞_𝑎𝑑𝑠∑^𝑑_𝑖=1𝑠𝑓_𝑖(ln𝑠,ln𝑠,…,ln𝑠)=∞, (2.5) then the system (1.1) has infinitely many classical positive entire solutions (𝑢₁,𝑢₂,…,𝑢_𝑑). If, in addition, 𝑝_𝑖, 𝑖=1,2,…,𝑑 satisfy the decay conditions (𝐷), then all the positive entire solutions of (1.1) are blow-up. Moreover, if 𝑝_𝑖, 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 constant 𝑎>1 such that ^∞_𝑎𝑑𝑠∑^𝑑_𝑖=1𝑠𝑓_𝑖(ln𝑠,ln𝑠,…,ln𝑠)<∞, (2.6) and 𝑝_𝑖, 𝑖=1,2,…,𝑑 satisfy the decay conditions (𝐹) and, in addition, there exist 𝑏_𝑖>𝑎, 𝑖=1,2,…,𝑑 such that ^𝑑_𝑖=1𝑃_𝑖(∞)<𝐹(∞)−𝐹^𝑑_𝑖=1𝑏_𝑖, (2.7) then the system (1.1) has a positive radial bounded solution (𝑢₁,𝑢₂,…,𝑢_𝑑) satisfying 𝑏_𝑖+𝑏_𝑖𝑓_𝑖ln𝑏₁,ln𝑏₂,…,ln𝑏_𝑑𝑃_𝑖(𝑟)≤𝑢_𝑖(𝑟)≤𝐹⁻¹𝐹^𝑑_𝑖=1𝑏_𝑖+^𝑑_𝑖=1𝑃_𝑖(𝑟),𝑟≥0. (2.8)

Theorem 2.6. (i)If 𝑝_𝑖, 𝑖=1,2,…,𝑑 satisfy the decay conditions (𝐷) and lim^𝑑_{𝑠→∞𝑖=1}𝑓_𝑖(ln𝑠,ln𝑠,…,ln𝑠)=0, (2.9) then the system (1.1) has infinitely many positive entire blow-up solutions.(ii)If 𝑝_𝑖, 𝑖=1,2,…,𝑑 satisfy the decay conditions (𝐹) and sup^𝑑_{𝑠≥0𝑖=1}𝑠𝑓_𝑖(ln𝑠,ln𝑠,…,ln𝑠)<∞, (2.10) then the system (1.1) has infinitely many positive entire bounded solutions.

3. Proofs of the Theorems

Firstly, via the change of variables 𝜙_𝑖=𝑒^𝑢𝑖, 𝑖=1,2,…,𝑑, we turn the system (1.1) to the following equivalent system with no gradient terms Δ𝜙₁=𝑝₁(|𝑥|)𝜙₁𝑓₁ln𝜙₁,ln𝜙₂,…,ln𝜙_𝑑,𝑥∈𝑅^𝑁,⋮⋮Δ𝜙_𝑖=𝑝_𝑖(|𝑥|)𝜙_𝑖𝑓_𝑖ln𝜙₁,ln𝜙₂,…,ln𝜙_𝑑,𝑥∈𝑅^𝑁,⋮⋮Δ𝜙_𝑑=𝑝_𝑑(|𝑥|)𝜙_𝑑𝑓_𝑑ln𝜙₁,ln𝜙₂,…,ln𝜙_𝑑,𝑥∈𝑅^𝑁. (3.1) Thus we only need to consider system (3.1).

Proof of Theorem 2.3. We use proof by contradiction to testify. We suppose that the system (3.1) has the positive entire blow-up solution (𝜙₁,𝜙₂,…,𝜙_𝑑). Consider the spherical average of 𝜙_𝑖 defined by 𝜙_𝑖(𝑟)=1𝑐_𝑁𝑟^𝑁−1_|𝑥|=𝑟𝜙_𝑖(𝑥)𝑑𝜎_𝑥,𝑟≥0, (3.2) where 𝑐_𝑁 is the surface area of the unit sphere in 𝑅^𝑁. Since 𝜙_𝑖 are positive entire blow-up solutions, it follows that 𝜙_𝑖 are positive and lim_𝑟→∞𝜙_𝑖(𝑟)=+∞. By the change of variable 𝑥=𝑟𝑦, we have 𝜙_𝑖(𝑟)=1𝑐_𝑁|𝑦|=1𝜙_𝑖𝑟𝑦𝑑𝜎_𝑦,𝑟≥0. (3.3) Then 𝜙_𝑖^′(𝑟)=1𝑐_𝑁|𝑦|=1∇𝜙_𝑖𝑟𝑦⋅𝑦𝑑𝜎_𝑦,𝑟≥0. (3.4) Thus by the divergence theorem and (3.4), we have 𝜙_𝑖^′(𝑟)=𝑟𝑐_𝑁|𝑦|<1Δ𝜙_𝑖1𝑟𝑦𝑑𝑦=𝑐_𝑁𝑟^𝑁−1_|𝑥|<𝑟Δ𝜙_𝑖(𝑥)=1𝑑𝑥𝑐_𝑁𝑟^𝑁−1𝑟₀𝑑𝜌_|𝑥|=𝜌Δ𝜙_𝑖(𝑥)𝑑𝜎_𝑥,∀𝑟≥0. (3.5) From [33], it follows from (3.5) that 𝜙_𝑖^′′(𝑟)=1𝑐_𝑁𝑟^𝑁−1_|𝑥|=𝑟Δ𝜙_𝑖(𝑥)𝑑𝜎_𝑥−𝑁−1𝑐_𝑁𝑟^𝑁𝑟₀𝑑𝜌_|𝑥|=𝜌Δ𝜙_𝑖(𝑥)𝑑𝜎_𝑥=1𝑐_𝑁𝑟^𝑁−1_|𝑥|=𝑟Δ𝜙_𝑖(𝑥)𝑑𝜎_𝑥−𝑁−1𝑟𝜙_𝑖^′(𝑟),∀𝑟≥0. (3.6) Set 𝑈_𝑖(𝑟)=max_{0≤𝑡≤𝑟}𝜙_𝑖(𝑡), (3.7) Then, obviously, 𝑈_𝑖 are positive and nondecreasing functions. Moreover 𝑈_𝑖≥𝜙_𝑖 and 𝑈_𝑖(𝑟)→+∞ as 𝑟→+∞. Note from (2.4) that there exists 𝑀>0 such that max𝑓₁𝑢₁,𝑢₂,…,𝑢_𝑑,…,𝑓_𝑑𝑢₁,𝑢₂,…,𝑢_𝑑≤𝑀,𝑢₁+𝑢₂+⋯+𝑢_𝑑≥0. (3.8) Now (3.6) and (3.8) lead to 𝜙_𝑖^′′+𝑁−1𝑟𝜙_𝑖^′≤1𝑐_𝑁𝑟^𝑁−1_|𝑥|=𝑟Δ𝜙_𝑖(𝑥)𝑑𝜎_𝑥=𝑝_𝑖(𝑟)1𝑐_𝑁𝑟^𝑁−1_|𝑥|=𝑟𝜙_𝑖(𝑥)𝑓_𝑖ln𝜙₁(𝑥),ln𝜙₂(𝑥),…,ln𝜙_𝑑(𝑥)𝑑𝜎_𝑥≤𝑀𝑝_𝑖(𝑟)1𝑐_𝑁𝑟^𝑁−1_|𝑥|=𝑟1+^𝑑_𝑖=1𝜙_𝑖(𝑥)𝑑𝜎_𝑥=𝑀𝑝_𝑖(𝑟)1+^𝑑_𝑖=1𝜙_𝑖(𝑟)≤𝑀𝑝_𝑖(𝑟)1+𝑈_𝑖(𝑟), (3.9) for all 𝑟≥0. It follows that 𝑟^𝑁−1𝜙_𝑖^′′≤𝑀𝑟^𝑁−1𝑝_𝑖(𝑟)1+𝑈_𝑖(𝑟),𝑟≥0. (3.10) So, for all 𝑟≥𝑟₀≥0, we have 𝜙_𝑖(𝑟)≤𝜙_𝑖𝑟₀+𝑀^𝑟_𝑟0𝑡^1−𝑁𝑡₀𝑠^𝑁−1𝑝_𝑖(𝑠)1+𝑈_𝑖(𝑠)≤𝑑𝑠𝑑𝑡𝜙_𝑖𝑟₀+𝑀^𝑟_𝑟0𝑡^1−𝑁(1+𝑈(𝑡))^𝑡₀𝑠^𝑁−1𝑝(𝑠)≤𝑑𝑠𝑑𝑡𝜙_𝑖𝑟₀+𝑀1+𝑈_𝑖(𝑟)^𝑟_𝑟0𝑡^1−𝑁𝑡₀𝑠^𝑁−1𝑝_𝑖(𝑠)≤𝑑𝑠𝑑𝑡𝜙_𝑖𝑟₀+𝑀1+𝑈_𝑖(𝑟)^𝑟_𝑟0𝑡^1−𝑁𝑡₀𝑠^𝑁−1𝑝_𝑖(𝑠)≤𝑑𝑠𝑑𝑡𝜙_𝑖𝑟₀+𝑀1+𝑈_𝑖(𝑟)𝑁−2^𝑟_𝑟0𝑡𝑝_𝑖(𝑡)1𝑑𝑡−𝑟₀^𝑁−1𝑟_𝑟0𝑡^𝑁−1𝑝_𝑖(𝑡)≤𝑑𝑡𝜙_𝑖𝑟₀+𝑀1+𝑈_𝑖(𝑟)𝑁−2^𝑟_𝑟0𝑡𝑝_𝑖(𝑡)𝑑𝑡. (3.11) Note that because of (𝐹), we can choose 𝑟₀>0 sufficiently large such that max^∞_𝑟0𝑟𝑝₁(𝑟)𝑑𝑟,…,^∞_𝑟0𝑟𝑝_𝑑(𝑟)𝑑𝑟<𝑁−2.4𝑀 (3.12) Since lim_𝑟→∞𝜙_𝑖(𝑡)=∞, it follows that we can find 𝑟₁≥𝑟₀ such that 𝑈_𝑖(𝑟)=max_{𝑟0≤𝑡≤𝑟}𝜙_𝑖(𝑡),∀𝑟≥𝑟₁. (3.13) Thus (3.11) and (3.13) yield 𝑈_𝑖(𝑟)≤𝜙_𝑖𝑟₀+𝑀1+𝑈_𝑖(𝑟)𝑁−2^𝑟_𝑟0𝑡𝑝_𝑖(𝑡)𝑑𝑡,∀𝑟≥𝑟₁. (3.14) By (3.12), we have 𝑈_𝑖(𝑟)≤𝜙_𝑖𝑟₀+1+𝑈_𝑖(𝑟)4,∀𝑟≥𝑟₁, (3.15) that is, 𝑈_𝑖(𝑟)≤𝐶_𝑖+𝑈_𝑖(𝑟)4,∀𝑟≥𝑟₁, (3.16) where 𝐶_𝑖=(1/4)+𝜙_𝑖(𝑟₀)>0, which implies ^𝑑_𝑖=1𝑈_𝑖(𝑟)≤43^𝑑_𝑖=1𝐶_𝑖,∀𝑟≥𝑟₁. (3.17) 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 Theorem 2.4. We start by showing that (1.1) has positive radial solutions. Towards this end we fix 𝑏_𝑖>𝑎, 𝑖=1,2,…,𝑑 and we show that the system 𝜙_𝑖^′′+𝑁−1𝑟𝜙_𝑖^′=𝑝_𝑖(𝑟)𝜙_𝑖(𝑟)𝑓_𝑖ln𝜙₁(𝑟),ln𝜙₂(𝑟),…,ln𝜙_𝑑(𝑟)𝜙,𝑖=1,2,…,𝑑_𝑖′≥0,on[0,∞),𝜙_𝑖(0)=𝑏_𝑖>𝑎 (3.18) has a solution (𝜙₁,𝜙₂,…,𝜙_𝑑). Thus (𝑈₁(𝑥),𝑈₂(𝑥),…,𝑈_𝑑(𝑥))=(𝜙₁(|𝑥|),𝜙₂(|𝑥|),…,𝜙_𝑑(|𝑥|)) are positive solutions of (3.1). Integrating (3.18), for any 𝑟≥0 and 𝑖=1,2,…,𝑑, we have 𝜙_𝑖(𝑟)=𝑏_𝑖+^𝑟₀𝑡^1−𝑁𝑡₀𝑠^𝑁−1𝑝_𝑖(𝑠)𝜙_𝑖(𝑠)𝑓_𝑖ln𝜙₁(𝑠),ln𝜙₂(𝑠),…,ln𝜙_𝑑(𝑠)𝑑𝑠𝑑𝑡. (3.19)
Let {𝜙_𝑖^(𝑘)}_𝑘≥0 be sequences of positive continuous functions defined on [0,∞) for 𝑖=1,2,…,𝑑 by 𝜙_𝑖⁽⁰⁾(𝑟)=𝑏_𝑖,𝜙_𝑖^(𝑘+1)(𝑟)=𝑏_𝑖+^𝑟₀𝑡^1−𝑁𝑡₀𝑠^𝑁−1𝑝_𝑖(𝑠)𝜙_𝑖^(𝑘)(𝑠)×𝑓_𝑖ln𝜙₁^(𝑘)(𝑠),ln𝜙₂^(𝑘)(𝑠),…,ln𝜙_𝑑^(𝑘)(𝑠)𝑑𝑠𝑑𝑡. (3.20) Obviously, for all 𝑟≥0, we have 𝜙_𝑖^(𝑘)(𝑟)≥𝑏_𝑖, 𝜙₀≤𝜙₁. The monotonicity of 𝑓_𝑖 yields 𝜙₁(𝑟)≤𝜙₂(𝑟), 𝑟≥0. Repeating the argument, we deduce that 𝜙_𝑖^(𝑘)(𝑟)≤𝜙_𝑖^(𝑘+1)(𝑟),𝑟≥0,𝑘≥1, (3.21) which means {𝜙_𝑖^(𝑘)}_𝑘≥0 are nondecreasing sequences on [0,∞). Since 𝜙_𝑖^′(𝑘+1)(𝑟)=𝑟^1−𝑁𝑟₀𝑠^𝑁−1𝑝_𝑖(𝑠)𝜙_𝑖^(𝑘)(𝑠)𝑓_𝑖ln𝜙₁^(𝑘)(𝑠),ln𝜙₂^(𝑘)(𝑠),…,ln𝜙_𝑑^(𝑘)(𝑠)𝑑𝑠≤𝜙_𝑖^(𝑘)(𝑟)𝑓_𝑖ln𝜙₁^(𝑘)(𝑟),ln𝜙₂^(𝑘)(𝑟),…,ln𝜙_𝑑^(𝑘)(𝑟)𝑃_𝑖^′(𝑟), (3.22) we have ^𝑑_𝑖=1𝜙_𝑖^′(𝑘+1)(𝑟)≤^𝑑_𝑖=1𝜙_𝑖^(𝑘+1)(𝑟)^𝑑_𝑖=1𝑓_𝑖ln𝜙₁^(𝑘)(𝑟),ln𝜙₂^(𝑘)(𝑟),…,ln𝜙_𝑑^(𝑘)(𝑟)𝑃_𝑖^′(𝑟)≤^𝑑_𝑖=1𝜙_𝑖^(𝑘+1)(𝑟)^𝑑_𝑖=1𝑓_𝑖ln^𝑑_𝑖=1𝜙_𝑖^(𝑘+1)(𝑟),ln^𝑑_𝑖=1𝜙_𝑖^(𝑘+1)(𝑟),…,ln^𝑑_𝑖=1𝜙_𝑖^(𝑘+1)(𝑟)×^𝑑_𝑖=1𝑃_𝑖^′(𝑟). (3.23) Let 𝑤^(𝑘+1)(𝑟)=∑^𝑑_𝑖=1𝜙_𝑖^(𝑘+1)(𝑟) which implies 𝑤^′(𝑘+1)(𝑟)𝑤^(𝑘+1)(𝑟)∑^𝑑_𝑖=1𝑓_𝑖ln𝑤^(𝑘+1)(𝑟),ln𝑤^(𝑘+1)(𝑟),…,ln𝑤^(𝑘+1)(𝑟)≤^𝑑_𝑖=1𝑃_𝑖^′(𝑟). (3.24) So, we have ^𝑟₀𝑤^′(𝑘+1)(𝑡)𝑤^(𝑘+1)(𝑡)∑^𝑑_𝑖=1𝑓_𝑖ln𝑤^(𝑘+1)(𝑡),ln𝑤^(𝑘+1)(𝑡),…,ln𝑤^(𝑘+1)(𝑡)𝑑𝑡≤^𝑑_𝑖=1𝑃_𝑖(𝑟), (3.25) that is 𝐹^𝑑_𝑖=1𝜙_𝑖^(𝑘+1)(𝑟)−𝐹^𝑑_𝑖=1𝑏_𝑖≤^𝑑_𝑖=1𝑃_𝑖(𝑟),∀𝑟≥0. (3.26) As 𝐹⁻¹ increases on [0,∞), from (3.26), we have that ^𝑑_𝑖=1𝜙_𝑖^(𝑘+1)(𝑟)≤𝐹⁻¹𝐹^𝑑_𝑖=1𝑏_𝑖+^𝑑_𝑖=1𝑃_𝑖(𝑟),∀𝑟≥0. (3.27) It follows from 𝐹(∞)=∞ that 𝐹⁻¹(∞)=∞. By (3.27), the sequences {𝜙_𝑖^(𝑘)} are bounded and increasing on [0,𝑐₀] for any 𝑐₀>0. Thus, {𝜙_𝑖^(𝑘)} have subsequences converging uniformly to 𝜙_𝑖 on [0,𝑐₀]. Consequently, (𝜙₁,𝜙₂,,…,𝜙_𝑑) is a positive solution of (3.18); that is, (𝑈₁,𝑈₂,…,𝑈_𝑑) is a entire positive solution of (3.1). By noticing 𝜙_𝑖(0)=𝑏_𝑖 and that 𝑏_𝑖∈(0,∞) was chosen arbitrarily, it follows that (1.1) has infinitely many positive entire solutions.(i)If 𝑃_𝑖(∞)=∞, since 𝜙_𝑖(𝑟)≥𝑏_𝑖+𝑏_𝑖𝑓_𝑖ln𝑏₁,ln𝑏₂,…,ln𝑏_𝑑𝑃_𝑖(𝑟),𝑟≥0, (3.28)we have lim_𝑟→∞𝜙_𝑖(𝑟)=∞, (3.29)which means that (𝑈₁,𝑈₂,…,𝑈_𝑑) are positive entire blow-up solutions of (1.1).(ii)If 𝑃_𝑖(∞)<∞, then ^𝑑_𝑖=1𝜙_𝑖(𝑟)≤𝐹⁻¹𝐹^𝑑_𝑖=1𝑏_𝑖+^𝑑_𝑖=1𝑃_𝑖(∞)<∞, (3.30)which implies that (𝑈₁,𝑈₂,…,𝑈_𝑑) are positive entire bounded solutions of (1.1). Proof of the theorem is now completed.

Proof of Theorem 2.5. If condition (2.7) holds, then we have 𝐹^𝑑_𝑖=1𝜙_𝑖^(𝑘+1)(𝑟)≤𝐹^𝑑_𝑖=1𝑏_𝑖+^𝑑_𝑖=1𝑃_𝑖(𝑟)≤𝐹^𝑑_𝑖=1𝑏_𝑖+^𝑑_𝑖=1𝑃_i(∞)<𝐹(∞)<∞. (3.31) Since 𝐹⁻¹ is strictly increasing on [0,∞), we have ^𝑑_𝑖=1𝜙_𝑖^(𝑘+1)(𝑟)≤𝐹⁻¹𝐹^𝑑_𝑖=1𝑏_𝑖+^𝑑_𝑖=1𝑃_𝑖(∞)<∞. (3.32) The last part of the proof is clear from that of Theorem 2.4. Thus we omit it.

Proof of Theorem 2.6. (i)  It follows from (3.20) that 𝜙_𝑖^(𝑘)(𝑟)≤𝜙_𝑖^(𝑘+1)(𝑟)≤𝑏_𝑖+𝜙_𝑖^(𝑘)(𝑟)𝑓_𝑖ln𝜙₁^(𝑘)(𝑟),ln𝜙₂^(𝑘),…,ln𝜙_𝑑^(𝑘)𝑃_𝑖(𝑟)≤𝑏_𝑖+𝜙_𝑖^(𝑘)(𝑟)𝑓_𝑖ln^𝑑_𝑖=1𝜙_𝑖^(𝑘)(𝑟),ln^𝑑_𝑖=1𝜙_𝑖^(𝑘)(𝑟),…,ln^𝑑_𝑖=1𝜙_𝑖^(𝑘)(𝑟)𝑃_𝑖(𝑟). (3.33) Let 𝑅>0 be arbitrary. From (3.33) we get, for 𝑘≥1, ^𝑑_𝑖=1𝜙_𝑖^(𝑘)(𝑅)≤^𝑑_𝑖=1𝑏_𝑖+^𝑑_𝑖=1𝜙_𝑖^(𝑘)(𝑅)×^𝑑_𝑖=1𝑓_𝑖ln^𝑑_𝑖=1𝜙_𝑖^(𝑘)(𝑅),ln^𝑑_𝑖=1𝜙_𝑖^(𝑘)(𝑅),…,ln^𝑑_𝑖=1𝜙_𝑖^(𝑘)(𝑅)^𝑑_𝑖=1𝑃_𝑖(𝑅). (3.34) This implies ∑1≤^𝑑𝑖=1𝑏_𝑖∑^𝑑_𝑖=1𝜙_𝑖^(𝑘)(𝑅)+^𝑑_𝑖=1𝑓_𝑖ln^𝑑_𝑖=1𝜙_𝑖^(𝑘)(𝑅),ln^𝑑_𝑖=1𝜙_𝑖^(𝑘)(𝑅),…,ln^𝑑_𝑖=1𝜙_𝑖^(𝑘)(𝑅)^𝑑_𝑖=1𝑃_𝑖(𝑅). (3.35) Taking into account the monotonicity of (∑^𝑑_𝑖=1𝜙_𝑖^(𝑘)(𝑅))_𝑘≥1, there exists 𝐿(𝑅)∶=lim_𝑘→∞^𝑑_𝑖=1𝜙_𝑖^(𝑘)(𝑅). (3.36) 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 𝜙_𝑖^(𝑘) are increasing functions, it follows that the map 𝐿∶(0,∞)→(0,∞) is nondecreasing and ^𝑑_𝑖=1𝜙_𝑖^(𝑘)(𝑟)≤^𝑑_𝑖=1𝜙_𝑖^(𝑘)(𝑅)≤𝐿(𝑅),∀𝑟∈[0,𝑅],∀𝑘≥1. (3.37) Thus the sequences {(𝜙_𝑖^(𝑘))_𝑘≥1} are bounded from above on bounded sets. Let 𝜙_𝑖(𝑟)∶=lim_𝑘→∞𝜙_𝑖^(𝑘)(𝑟),∀𝑟≥0. (3.38) Then (𝜙₁,𝜙₂,…,𝜙_𝑑) is a positive solution of (3.18).
In order to conclude the proof, it is sufficient to show that (𝜙₁,𝜙₂,…,𝜙_𝑑) is a blow-up solution of (3.18). Let us remark that (3.19) implies 𝜙_𝑖(𝑟)≥𝑏_𝑖+𝑏_𝑖𝑓_𝑖ln𝑏₁,ln𝑏₂,…,ln𝑏_𝑑𝑃_𝑖(𝑟),𝑟≥0. (3.39) Since 𝑓_𝑖 are positive functions and 𝑃_𝑖(∞)=∞, (3.40) we can conclude that (𝜙₁,𝜙₂,…,𝜙_𝑑) is a blow-up solution of (3.18), and so (𝑈₁,𝑈₂,…,𝑈_𝑑) 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 𝑏_𝑖∈(0,∞) was chosen arbitrarily, it follows that (1.1) has infinitely many positive entire blow-up solutions.
(ii) If sup^𝑑_{𝑠≥0𝑖=1}𝑠𝑓_𝑖(ln𝑠,ln𝑠,…,ln𝑠)<∞ (3.41) holds, then by (3.35), we have 𝐿(𝑅)∶=lim^𝑑_{𝑘→∞𝑖=1}𝜙_𝑖^(𝑘)(𝑅)<∞. (3.42) Thus ^𝑑_𝑖=1𝜙_𝑖^(𝑘)(𝑟)≤^𝑑_𝑖=1𝜙_𝑖^(𝑘)(𝑅)≤𝐿(𝑅),∀𝑟∈[0,𝑅],∀𝑘≥1. (3.43) So the sequences {𝜙_𝑖^(𝑘)}_𝑘≥1 are bounded from above on bounded sets. Let 𝜙(𝑟)∶=lim_𝑘→∞𝜙_𝑖^(𝑘)(𝑟),∀𝑟≥0. (3.44) Then (𝜙₁,𝜙₂,…,𝜙_𝑑) is a positive solution of (3.18).
It follows from (3.33) and (3.35) that (𝜙₁,𝜙₂,…,𝜙_𝑑) 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 Δ𝑢₁+∇𝑢₁^𝑎1=𝑝₁(|𝑥|)𝑓₁𝑢₁,𝑢₂,…,𝑢_𝑑,𝑥∈𝑅^𝑁,⋮⋮Δ𝑢_𝑑+∇𝑢_𝑑^𝑎𝑑=𝑝_𝑑(|𝑥|)𝑓_𝑑𝑢₁,𝑢₂,…,𝑢_𝑑,𝑥∈𝑅^𝑁, (3.45) where 𝑎_𝑖∈(0,+∞), 𝑎_𝑖≠1,2, 𝑓_𝑖∶[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 𝑞_𝑖-Laplacian as considered by Covei [14–17].