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Entire Blow-Up Solutions of Semilinear Elliptic Systems with Quadratic Gradient Terms
We study the existence of entire positive solutions for the semilinear elliptic system with quadratic gradient terms, for on and . We establish the conditions on 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 is simple and different to the Keller-Osserman condition.
We study the existence of entire blow-up positive solutions of the following elliptic system with quadratic gradient terms: where , , are -positive functions and are nonnegative, continuous, and nondecreasing functions for each variable.
For convenience we recall the definitions about -positive functions and entire blow-up positive solutions.(i)A function is -positive (or circumferentially positive) in a domain if is nonnegative on and satisfies the following condition: if and , then there exists a domain such that and for all . (ii)A solution 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 and as .
Existence and nonexistence of blow-up solutions of semilinear elliptic equations and systems have received much attention worldwide. Bieberbach  is the first to study blow-up solutions to the semilinear elliptic problem where . Following Bieberbach’s work, many authors have studied related problems for single equations and systems. In 1957, Keller  and Osserman  established the necessary and sufficient conditions for the existence of solutions to (1.2) on bounded domains in . They showed that blow-up solutions exist on if and only if satisfies the following Keller-Osserman condition: Bandle and Marcus  later examined the equation with is nondecreasing on and proved the existence of positive blow-up solutions under the condition that the function satisfies the Keller-Osserman condition (1.3) and is continuous and strictly positive on . Lair  showed that the results also hold for (1.4) when is allowed to vanish on a large part of , including its boundary. In addition, many authors have examined some more specific forms of (1.4). The equation has been of particular interest. Cheng and Ni  considered the superlinear case and proved that for this case (1.5) has blow-up solutions on bounded domains provided is strictly positive on . Lair and Wood  generalized this to allow to vanish on some portions of including its boundary and also showed the existence of an entire blow-up solution to (1.5) provided that Obviously, condition (1.6) is weaker than the requirements in .
In , Lair and Wood proved that (1.5) has entire blow-up radial solutions if and only if They also demonstrated that for a bounded domain , (1.5) has no positive blow-up solution when is continuous in . In addition, they proved that nonnegative, entire bounded solutions do not exist for (1.5) if
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  studied the existence of entire positive solutions of the system In the sublinear case , the authors proved that provided that the nonnegative functions and are continuous, -positive, and satisfy the fast decay conditions then the entire positive solutions are bounded, while if and satisfy the slow decay conditions then the entire positive solutions blow up. For the superlinear case , the fast decay conditions are required to hold. Later, Cîrstea and Rădulescu  improved the results of Lair and Wood  and proved that for , the following semilinear elliptic system has entire solutions if and satisfy for all and has solutions that are bounded when holds. Further, entire solutions exist and are blow-up when holds. An analogous condition was also employed by Ghergu and Rădulescu  to study the following elliptic system with gradient terms: where is a bounded domain or the whole space. Peng and Song  also studied the existence of entire blow-up positive solutions of system (1.10) when the -positive functions , satisfy the decay conditions . Peng and Song  also imposed on and the following Keller-Osserman conditions: and the convexity conditions Both papers [6, 12] considered system (1.10) where the nonnegative functions satisfy and the functions are nondecreasing and satisfy the Keller-Osserman condition (1.13), and Recently, Zhang and Liu  studied the following semilinear elliptic system with the magnitude of the gradient The results of nonexistence of entire positive solutions have been established if and are sublinear and and have fast decay at infinity, while if and satisfy some growth conditions at infinity, and , are of slow decay or fast decay at infinity, then the system has infinitely many entire solutions, which are large or bounded. In , Covei studied the existence of solution of the following semilinear elliptic system: Under some conditions on , , 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 -Laplacian in . 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
Remark 2.1. For any , since admits the inverse function on .
The first result we obtained is the condition for nonexistence of entire positive blow-up solution, which asserts that if both , are bounded, then problem (1.1) does not have positive entire blow-up solution as detailed by the following theorem.
Theorem 2.3. Suppose , satisfy and each , satisfy the decay conditions . 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 such that then the system (1.1) has infinitely many classical positive entire solutions . If, in addition, , satisfy the decay conditions , then all the positive entire solutions of (1.1) are blow-up. Moreover, if , satisfy the decay conditions , then all the positive entire solutions of (1.1) are bounded.
Theorem 2.5. If there exists a constant such that and , satisfy the decay conditions and, in addition, there exist , such that then the system (1.1) has a positive radial bounded solution satisfying
Theorem 2.6. (i)If , satisfy the decay conditions and then the system (1.1) has infinitely many positive entire blow-up solutions.(ii)If , satisfy the decay conditions and then the system (1.1) has infinitely many positive entire bounded solutions.
3. Proofs of the Theorems
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 . Consider the spherical average of defined by where is the surface area of the unit sphere in . Since are positive entire blow-up solutions, it follows that are positive and . By the change of variable , we have Then Thus by the divergence theorem and (3.4), we have From , it follows from (3.5) that Set Then, obviously, are positive and nondecreasing functions. Moreover and as . Note from (2.4) that there exists such that Now (3.6) and (3.8) lead to for all . It follows that So, for all , we have Note that because of , we can choose sufficiently large such that Since , it follows that we can find such that Thus (3.11) and (3.13) yield By (3.12), we have that is, where , which implies The inequality (3.17) means that are bounded and so 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 , and we show that the system
has a solution . Thus are positive solutions of (3.1). Integrating (3.18), for any and , we have
Let be sequences of positive continuous functions defined on for by Obviously, for all , we have , . The monotonicity of yields , . Repeating the argument, we deduce that which means are nondecreasing sequences on . Since we have Let which implies So, we have that is As increases on , from (3.26), we have that It follows from that . By (3.27), the sequences are bounded and increasing on for any . Thus, have subsequences converging uniformly to on . Consequently, is a positive solution of (3.18); that is, is a entire positive solution of (3.1). By noticing and that was chosen arbitrarily, it follows that (1.1) has infinitely many positive entire solutions.(i)If , since we have which means that are positive entire blow-up solutions of (1.1).(ii)If , then which implies that are positive entire bounded solutions of (1.1). Proof of the theorem is now completed.
Proof of Theorem 2.6. (i) It follows from (3.20) that
Let be arbitrary. From (3.33) we get, for ,
Taking into account the monotonicity of , there exists
We claim that is finite. Indeed, if not, we let in (3.35), and the assumption (2.9) leads us to a contradiction. Thus is finite. Since are increasing functions, it follows that the map is nondecreasing and
Thus the sequences are bounded from above on bounded sets. Let
Then is a positive solution of (3.18).
In order to conclude the proof, it is sufficient to show that is a blow-up solution of (3.18). Let us remark that (3.19) implies Since are positive functions and we can conclude that is a blow-up solution of (3.18), and so 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 was chosen arbitrarily, it follows that (1.1) has infinitely many positive entire blow-up solutions.
(ii) If holds, then by (3.35), we have Thus So the sequences are bounded from above on bounded sets. Let Then is a positive solution of (3.18).
It follows from (3.33) and (3.35) that 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 , , 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 -Laplacian as considered by Covei [14–17].
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Copyright © 2012 Yongju Yang and Xinguang Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.