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

and Applied Analysis 3 In 8 , Lair and Wood proved that 1.5 has entire blow-up radial solutions if and only if ∫ ∞


Introduction
We study the existence of entire blow-up positive solutions of the following elliptic system with quadratic gradient terms:   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 x 0 ∈ Ω and p x 0 0, then there exists a domain Ω 0 such that x 0 ∈ Ω 0 ⊂ Ω and p x > 0 for all x ∈ ∂Ω 0 .
ii A solution u 1 , u 2 , . . . , 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 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: 1.3

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| r p r dr < ∞.

1.6
Obviously, condition 1.6 is weaker than the requirements in 6 . 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
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 For the superlinear case α, β > 1, the fast decay conditions F are required to hold. Later, Cîrstea and Rȃdulescu 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 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 Rȃdulescu 11 to study the following elliptic system with gradient terms: 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: and the convexity conditions 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 Recently, Zhang and Liu 13 studied the following semilinear elliptic system with the magnitude of the gradient

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 Abstract and Applied Analysis 5 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
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 p 1 , . . . , 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.

Main Results
For convenience in presenting the results, we here define holds if and only if P i ∞ ∞.
The first result we obtained is the condition for nonexistence of entire positive blowup 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. 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.

Proofs of the Theorems
Firstly, via the change of variables φ i e u i , i 1, 2, . . . , d, we turn the system 1.1 to the following equivalent system with no gradient terms

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 φ 1 , φ 2 , . . . , φ d . Consider the spherical average of φ i defined by 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 ∞. Δφ i x dσ x , ∀r ≥ 0.

3.5
From 33 , it follows from 3.5 that for all r ≥ 0. It follows that Abstract and Applied Analysis 9 So, for all r ≥ r 0 ≥ 0, we have

3.11
Note that because of F , we can choose r 0 > 0 sufficiently large such that Since lim r → ∞ φ i t ∞, it follows that we can find r 1 ≥ r 0 such that U i r max r 0 ≤t≤r φ i t , ∀r ≥ r 1 .

3.13
Thus 3.11 and 3.13 yield 3.14 By 3.12 , we have that is,

10
Abstract and Applied Analysis 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

3.20
Obviously, for all r ≥ 0, we have φ The monotonicity of f i yields φ 1 r ≤ φ 2 r , r ≥ 0. Repeating the argument, we deduce that 3.24 So, we have As F −1 increases on 0, ∞ , from 3.26 , we have that ii Since F −1 is strictly increasing on 0, ∞ , we have 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

3.34
Abstract and Applied Analysis 13 This implies

3.35
Taking into account the monotonicity of 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 k i are increasing functions, it follows that the map L : 0, ∞ → 0, ∞ is nondecreasing and 3.37 Thus the sequences { φ k i k≥1 } are bounded from above on bounded sets. Let Then φ 1 , φ 2 , . . . , φ d is a positive solution of 3.18 . It follows from 3.33 and 3.35 that φ 1 , φ 2 , . . . , 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 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