Advances in Pure Mathematics, 2011, 1, 351358 doi:10.4236/apm.2011.16063 Published Online November 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Existence and Nonexistence of Entire Positive Solutions for a Class of Singular pLaplacian Elliptic System* Daojin Lei1, Zuodo ng Yang1,2 1School of Mathematical Sciences, Nanjing Normal University, Nanjing, China 2College of Zhongbei, Nanjing Normal University, Nanjing, China Email: zdyang_jin@263.net Received August 7, 2011; revised October 2, 2011; accepted October 10, 2011 Abstract In this paper, we show the existence and nonexistence of entire positive solutions for a class of singular el liptic system 2 2 =, =,, 3 ap p aq q divxuubxfu v divxvvdxgu vxRN N. We have that entire large positive solutions fail to exist if f and g are sublinear and b and d have fast decay at infinity. However, if f and g satisfy some growth conditions at infinity, and b, d are of slow decay or fast de cay at infinity, then the system has infinitely many entire solutions, which are large or bounded. Keywords: Singular pLaplacian Elliptic System, Entire Positive Solution, Large Solution, Bounded Solution, Entire Large Positive Radial Solution 1. Introduction In this paper, we mainly consider the existence and nonexistence of positive solutions for the following singular plaplacian elliptic system: 2 2 =,, =,, , ap p , aq q divxuubxfuvx divxvvdxguvx R R (1) where , and are continuous, posi tive and nondecreasing functions in 3,> 0Nab d R, are positive, nondecreasing and continuous functions. , :)fg [0,)[0,) [0, When , the following semilinear elliptic system: =0,==2apq ,,=0, in , ,,=0, in , ufxuv vgxuv has been studied extensively over the years, for example see [14]. If =,= bxv gdxu , the above system becomes =, , =, , ubxvx vdxux R R for which existence results for boundary blowup posi tive solution can be found in a recent paper by Lair and Wood [5]. The authors established that all positive entire radial solutions of system above are boundary blowup provided that 00 d=, d=.tbt ttdt t On the other hand, if 00 d<, d<,tbt ttdt t then all positive entire radial solutions of this system are bounded. F. Cìrstea and V. Rădulescu [2] extended the above results to a larger class of systems =, , =, N , ubxgvx vdxfux R R *Project Supported by the National Natural Science Foundation o China (No.11171092); the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No.08KJB110005).
D. J. LEI ET AL. 352 Z. D.Yang [6] extended the above results to a class of quasilinear elliptic system: 2 2 =, , ()=(), , pN q divuubxg vx divvvdxf ux R RN where , , and 3N,>1pq , bd CR are positive functions, 0,) 1[C,fg f are positive and nondecreasing. When and satisfy (H1) 0=0=0,= >0; lim infu fu fg gu (H2) The KellerOsserman condition, 10 d<, =d, t p tGtgs s Gt then there exists an entire positive radial solution, and in addition, the function satisfy ,bd (H3) 1 1 11 00 dd= tp NN tsbss t ; (H4) 1 1 11 00 dd= tq NN tsdss t , then all entire positive radial solutions are large. On the other hand, if and satisfy b d (H5) 1 1 11 00 dd< tp NN tsbss t ; (H6) 1 1 11 00 dd< tq NN tsdss t , then all entire positive radial solutions are bounded. While in [7], the author got the relevant results of the same system only on the following conditions (H7) are continuous; , , ,:[0,)[0,)bdg f (H8) and are nondecreasing functions on [0, ). However, when , there are few results about the existence and nonexistence of singular pLa placian elliptic system (1). And as to the single equation, we can refer to [8]. The present results are complements and extensions of some results in [7,9], which to be more precise, if , you can get the relevant existence and nonexistence results for a class of semi linear elliptic system with gradient term in [9]; Mean while, if , you can get the relevant existence results for a class of quasilinear elliptic system in [7]. 0, 2ap = =2q , >1 q =0,ap =0,ap For convenience, we need the following definition: Definition. A solution of system (,)uv 2 2 =,,, , =,,, , ap p aq q divxuufx uvx divxvvgx uvx (2) is called an entire large solution(or explosive solution, or blowup solution), if it is classical solution of (2) on and N R ux and vx as x. Now we give our main theorem: Theorem 1. Suppose and satisfy 11 11 ,, max,< , sup sup mm st st fst gst st st (3) and satisfy the decay conditions ,bd 1/(1)1/( 1) 00 d<, d<, pq ap aq tbtttdt t (4) where , then problem (1) has no positive entire radial large solution. =min{, }mpq In order to state our results conveniently, let us write 1/( 1) 11 00 := , lim =dd r p rt ap NN BBr Brtsbsstr ,0, 1/( 1) 11 00 := , lim =d r q rt aq NN DDr Drtsds str d,0 and 1/(1) 0 := , lim d =, ,, r r m FFr s Fr r fss gss >0, where satisfies 0 m 0 min,, if 1, =max,, if <1, pqf g mpqf g we see that 1/(1) 0 1 => ,, m Fr r frr grr0,> so, has the inverse function 1 on [, ) , and and 1 are both increasing functions on [, ) . Theorem 2. Assume =.F Then the system (1) has infinitely many positive entire solutions 1 ,[0,uv C) . Moreover, the following hold: 1) If <B and , then u and are bounded; <D v 2) If ==BD , then = lim rr ur vr = lim , that is all positive entire solutions of (1) are large. Theorem 3. If Copyright © 2011 SciRes. APM
D. J. LEI ET AL. Copyright © 2011 SciRes. APM 353 2. Proof of Theorem 1 <, <,<,FBD and there exist > and > such that <BDFF , (5) In this section, we consider the proof of Theorem 1 by contradictions. Assume that the system (1) has the positive entire radial large solution . From (1), we know that (,)uv the system (1) has a positive radial bounded solution satisfying 1 ,[0,uv C 1/( 1)p ) ; . 1 , ,0 fBrur FFBrDr r 1/( 1) 1 , ,0 q gDrvr FFBrDr r Theorem 4. If satisfies 1 m 1 min,, if ,1, =max,, f ,<1, pqf g mpqi fg then we have 1) If == ,BD and 1/( 1) 1 ,, =0, lim m s fssgss s (6) then the system (1) has infinitely many positive entire large solutions; 2) If <, <,BD and 1/( 1) 1 0 ,, < sup m sfss gss , 0, 0. (7) 1 11 =(,), p Nap N tuttbtfutvtt 1 11 =,, q Naq N tvttdtgutvtt Now we set 00 =,= max max tr tr Urut Vrvt , it is easy to see that are positive and nonde creasing functions. Moreover, we have and (,)UV , Uu Vv , UrVr 0 C r >0 as . It follows from (3) that there exists such that 1 0 max,,,, for 1, m fst gstC stst (8) and 0 max,,,, for 1.fst gstCst (9) Then by (8) and (9), we have 1 0 max,,,1, for 0. m fst gstCstst (10) Then we can get 1 0 1 0 ,1 1, for 0. m m furvrCurvr CUrVrr then the system (1) has infinitely many positive entire bounded solutions. So, for all , we obtain 00rr 1/(1) 11 00 0 1/( 1) 1 11 00 0 11/( 1) 11 1 00 0 11/( 1) 1 00 =,d 1dd 1dd 1d p rt ap NN r p rt m ap NN r mp rt ap NN pr mp rap pr ururtsbs fusvsst urCtsbsUs Vsst urCUrVrtsbsst urCUrVrt btt 1/(1) 00 1d, p rap r urCUrVrtbtt d q 1/(1)1/( 1) 00 1 maxd ,d<. 4 pq ap aq rr rbrr rdrrC (11) where is a positive constant. As , we have , so the last inequality above is valid. Notice that (4), we choose such that C 0< =min{ ,}mp >0 1<1mp 0 r
D. J. LEI ET AL. 354 = 0 It follows that , we can find such that = lim lim rr ur vr 1 rr 1 00 = ,=, maxmax rtrrtr UrutVrvtrr . (12) Thus, we have 1/(1) 00 1 1 . p rap r UrurCUr Vrtbtt rr d, By (11), we get 0 1, . 4 Ur Vr Ururr r 1 that is 11 , . 4 Ur Vr UrCr r where 10 1 = 4 Cur>0 . Similarly, 21 , . 4 Ur Vr VrCr r which implies 12 1 2, UrVrCCr r. (13) (13) means that U and V are bounded and so and are bounded which is a contradiction. It follows that (1) has no positive entire radial large solutions and the proof is now completed. u v Remark. In Theorem 1, if , and ,>2pq f satisfy 11 ,, max , < sup sup st st fst gst st st and satisfy the same decay conditions (4), we can also get the same result that problem (1) has no positive entire radial large solution. ,bd In the following,we will give the detailed proof. Proof. We also consider the proof by contradiction. If using the same process in Theorem 1, we will omit that items here. Assume that the system (1) has the positive entire radial large solution , we can get from the given condition above that there exists such that (,)uv 3>0C 3 max,,,, for 1fst gstCstst and 3 max,,,, for 1 stg stCst so, we can get 3 max,,,1, for 0.fst gstCstst Thus, we can get 3 3 ,1 1, for 0, furvrCurvr CUrVr r here ,UrVr are the same functions defined in Theorem 1. As the proof of Theorem 1, we omit the same process here, for all , we obtain 00rr 1/(1) 11 00 0 1/(1) 11 04 0 0 11/(1) 11 1 04 0 0 11/( 1 04 0 =,d ()(1()())dd 1dd 1 p rt ap NN r p rt aq NN r p rt ap NN pr rap pr ururtsbs fusvsst urCts bsUsVs st urCUrVrtsbsst urCUrVrt bt 1) d pt d where is a positive constant. 4 Notice the condition (4), we choose such that C 0>0r 1/(1)1/( 1) 00 4 1 maxd ,d<, pq ap aq rr rbrrrdrrC together with (12), we get 1 1 01p UrurUr Vr Similarly,we have 1 1 01q VrvrUr Vr Set , we get =min{ , }mpq 1 1 00 1 1 1 1 00 1 1 21 p q m Ur VrurvrUr Vr Ur Vr urvrUr Vr Copyright © 2011 SciRes. APM
D. J. LEI ET AL.355 that is, 1 1 00 121 1, 0 m Ur VrUr Vr ur vrr We claim the above inequality is invalid. In fact, set a function 1 1 1 :=12 1m TUrVr Ur VrUr Vr then 2 1 2 1=11 >0 1 asis large enough. m m T UrVrUrVr m r , So, 1TUrVr has no positive entire radial large solution and the proof of the remark is completed. 3. Proofs of Theorem 2 and Theorem 3 Proof of Theorem 2. We start by showing that (1) has positive radial solutions. On this purpose we fix > and > and we show that the system 11 11 =, =, 0=>0, 0=>0, ,0, on [0,), Nap N p Naq N q rurbrfurvrr rurdrgurvrr uv uv , >0, , >0, (14) is an increasing function on ,and it can not be always controlled by a fixed constant, which is a contraction. It follows that system (1) [0, ) has positive solution (, (where )uv 2 =p p ss ). Thus =,=Ux uxvx d,0, d,0. d, 0, d, 0. Vx are positive solutions of (1). Integrating (14) we have 1/(1) 11 00 =,d p rt ap NN urtsbs fusvsstr 1/( 1) 11 00 =,d q rt aq NN vrtsdsgusvsst r Let and be the sequences of posi tive continuous functions defined on [0 by 0 {} nn u0 {} nn v,) 0 0 1/( 1) 11 100 1/( 1) 11 100 =, =, =,d =,d p rt ap NN nnn q rt aq NN nnn ur vr urt sbsfusvsstr vrt sbsgusvsstr (15) Obviously, for all , we have 0r 0101 , , , . nn urvru uv v The monotonicity of and yield 1212 , , 0.urur vrvr r Repeating such arguments we deduce that 11 , , 0,1, nn nn urur vrvr rn and we obtain that sequences and are nondecreasing on 0 {} nn u0 {} nn v [0, ) . Notice 1/( 1)1/( 1) 11 10 1/(1) 1111 1/( 1) 1111 =,d, (,(, , pp r ap NN nnnnn p nnnn nnnn p nnnn urrsbsfus vssfur vrBr ur vrurvrBrfur vrur vr gu rv ru rv rBr and 1/( 1)1/( 1) 11 10 1/( 1) 1111 1/( 1) 1111 =,d, (,(, ,, qq r aq NN nnnnn q nnnn nnnn q nnnn vrrsdsgusvssgurvrD r urvrurvrDrfu rv ru rv r gu rv ru rv rDr Copyright © 2011 SciRes. APM
D. J. LEI ET AL. 356 which implies 1/(1) 0, ,, nn m nnnn nnnn ur vrBr Dr furvrurvr gurvrurvr where has been defined before. And then integrat ing on we obtain 0 m(0, )r 1/(1) 00d. ,, rnn m nnnn nnnn ut vttBr Dr fut vtut vtgut vtutvt So () () 1/(1) 0 d, ,, urvr nn mBr Dr fg that is ,0 nn Fu rv rFBrDrr . (16) It follows from 1 is increasing on [0 and (16) that , ) 1,0 nn ur vrFFBr Drr . th By (17) It follows from ()Fat F (17), the sequences {} n u {} n v bounded and in creasing on 0 [0 or arbitrary 0 cThus, and have subsequences converging uniformly to and on 0 [0 . By the arbitrariness of 0 c, we see that is a positive solution of (15), that is, is an entire positive solution of (1). Notice = and ] f , ]c ) 1()=. are >0. ,c uv = {} n u >0 {} n vv ) = ,U u (, 0)V (,UV (0) ( and (, )(0,)(0,) was chosen arbitrarily, it follows that (1) has infinitely many positive entire solutions. 1) If and , then ()<B ()<D 1<,ur vrF FBD which implies that are the positive entire bounded solutions of (1). (,)UV 2) If , since ()== ()BD 1/(1) 1/( 1) ,, ,, 0. p q urfBr vr gDrr Thus, we have == lim lim rr ur vr , which yield (, are the positive entire large solutions of (1). The proof of theorem is now completed. )UV Proof of Theorem 3. If condition (5) holds, then we have <. nn ur vrFBrDr FBDF Since 1 is strictly increasing on , we have [0,) 1<. nn ur vr FFB D The last part of the proof is clear from the proof of Theorem 2. The proof of Theorem 3 is now finished. 4. Proof of Theorem 4 1) It follows from the proof of Theorem 3, we have 1/(1) 1 1/( 1) , ,, p nn nn pnnnn ur urfurvrBr ur vrurvrBr (18) and 1/( 1) 1 1/( 1) , ,. q nn nn qnnnn vr vrgurvrDr ur vrur vrDr (19) Let be arbitrary. From (18) and (19) we get >0R 1/( 1) 1/( 1) 1/( 1) 1 , , , ,, p nn nnnn qnnnn nnnn m nnnn uRvRf uRvRuRvRBR g uRvRuRvRDR fuRvRuRvR gu Rv Ru Rv RBRDRn 1. Copyright © 2011 SciRes. APM
D. J. LEI ET AL.357 This implies 1/( 1) 1 [, , 1 , 1. m nnnnnnnn nn nn fuRvRuRvR guRvRuRvR uR vRuR vR BR DRn ] Taking into account the monotonicity of , there exists 1 nn n uRvR := . lim nn n LRuRvR We claim that is finite. Indeed, if not, we let and the assumption (6) leads us to a con tradiction,thus LR n LR is finite. Since are increas ing functions, it follows that the map is nondecreasing and , nn uv :(0,L)(0, ) , 0,, 1. nn nn ur vruR vRLR rRn Thus the sequences are bounded from above on bounded sets. Let 1 {} ,{} nn nn uv 1 . :=, :=, for 0. lim lim nn nn ururvrv rr Then is a positive solution of (14). (,)uv In order to conclude the proof, it is enough to show that is a large solution of (14). We see (,)uv 1/(1) 1/( 1) ,, ,, 0 p q ur fBr vrgDr r Since and are positive functions and == =BD, we can conclude that is a large solution of (14) and so is a positive entire large solution of (1). Thus any large solution of (14) provide a positive entire large solution of (1) with (,)uv ) (,)UV (,UV (0) =,(0) =UV . Since ( , )(0,)(0,) was chosen arbitrarily, it follows that (1) has infinitely many positive entire large solutions. 2) If 1/( 1) 1 0 ,, < sup m sfss gss holds, then we have :=< . lim nn n LRuRvR Thus , 0,, 1. nn nn ur vruR vRLR rRn So the sequences 1 {} are bounded from above on bounded sets. Let 1 ,{} nn nn uv :=, :=, for 0. lim lim nn nn ururvrv rr Then is a positive solution of (14). (,)uv It follows from (18) and (19) that is bounded, which implies that (1) has infinitely many positive entire bounded solutions. The proof is completed. (,)uv 5. The Existence and Nonexistence of Entire Positive Solutions of the Corresponding Singular Elliptic Systems with Gradient Term In this section, we consider the following singular elliptic systems with gradient term: 21 21 =,, , =,, . ap pp aq qqN divxuuubxfu vx divx v v vdxguvx R R (20) where , and are continuous, posi tive and nondecreasing functions in 3, > 0Na b d R, are positive,nondecreasing and continuous functions. , :)fg [0,) [0,) [0, We can get the same four theorems under the same conditions in the foregoing items. In the detailed proofs, only a few modifications should be noticed. Such as, we note 1/(1) 11 00 := , lim =edd, 0 r p rt tapNN BBr Brtsbs str , 1/( 1) 11 00 := , lim =edd, 0 r q rt taqN N DBr Drtsds str , Copyright © 2011 SciRes. APM
D. J. LEI ET AL. 358 and 1/(1) 0 := , lim d =, ,, r r m FFr s Fr r fss gss >0, where 0 is defined as before and other changes are similar, so we omit here. m 6. Acknowledgements The authors are grateful to the editor and anonymous referees for their constructive comments and suggestions, which led to improvement of the present paper. 7. References [1] S. Chen and G. Lu, “Existence and Nonexistence of Posi tive Solutions for a Class of Semilinear Elliptic Systems,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 38, No. 7, 1999, pp. 919932. doi:10.1016/S0362546X(98)001436 [2] F. Cìrstea and V. Rădulescu, “Entire Solutions Blowing up at Infinity for Semilinear Elliptic Systems,” Journal de Mathématiques Pures et Appliquées, Vol. 81, No. 9, 2002, pp. 827846. doi:10.1016/S00217824(02)012655 [3] C. Yarur, “Existence of Continuous and Singular Ground States for Semilinear Elliptic Systems,” Electronic Jour nal of Differential Equations, Vol. 1, 1998, pp. 127. [4] R. Dalmasso, “Existence and Uniqueness of Positive Solutions of Semilinear Elliptic Systems,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 39, No. 5, 2000, pp. 559568. doi:10.1016/S0362546X(98)002211 [5] A. V. Lair and A. W. Wood, “Existence of Entire Large Positive Solutions of Semilinear Elliptic Systems,” Jour nal of Differential Equations, Vol. 164, No. 2, 2000, pp. 380394. doi:10.1006/jdeq.2000.3768 [6] Z. D. Yang, “Existence of Entire Explosive Positive Ra dial Solutions for a Class of Quasilinear Elliptic Sys tems,” Journal of Mathematical Analysis and Applica tions, Vol. 288, No. 2, 2003, pp. 768783. doi:10.1016/j.jmaa.2003.09.027 [7] H. X. Qing and Z. D. Yang, “A Remark on the Existence of Entire Positive Solutions for a Class of Quasilinear El liptic Problem,” Oriental Journal of Mathematics, Vol. 3, No. 1, 2010, pp. 4350. [8] C. S. Chen, Z. Q. Wang and F. P. Wang,” Existence and Nonexistence of Positive Solutions for Singular pLaplacian Equation in Rn,” Boundary Value Problems, Vol. 2010, 2010, Article ID 607453, pp. 117. [9] X. G. Zhang and L. S. Liu, “The Existence and Nonexis tence of Entire Positive Solutions of Semilinear Elliptic Systems with Gradient Term,” Journal of Mathematical Analysis and Applications, Vol. 371, No. 1, 2010, pp. 300308. doi:10.1016/j.jmaa.2010.05.029 Copyright © 2011 SciRes. APM
