 Advances in Pure Mathematics, 2011, 1, 351-358 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 p-Laplacian 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 E-mail: 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  22=,=,, 3ap paq qdivxuubxfu vdivxvvdxgu 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 p-Laplacian 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 p-laplacian elliptic system: 22=,,=,, ,ap p ,Naq qNdivxuubxfuvxdivxvvdxguvx  RR (1) where , and are continuous, posi- tive and nondecreasing functions in 3,> 0Nab dNR, are positive, nondecreasing and continuous functions. , :)fg  [0,)[0,) [0,When , the following semi-linear elliptic system: =0,==2apq,,=0, in ,,,=0, in ,ufxuvvgxuv   has been studied extensively over the years, for example see [1-4]. If =,=fbxv gdxu, the above system becomes =, ,=, ,NNubxvxvdxuxRR for which existence results for boundary blow-up posi- tive solution can be found in a recent paper by Lair and Wood . The authors established that all positive entire radial solutions of system above are boundary blow-up provided that  00d=, d=.tbt ttdt t On the other hand, if  00d<, d<,tbt ttdt t then all positive entire radial solutions of this system are bounded. F. Cìrstea and V. Rădulescu  extended the above results to a larger class of systems =, ,=,N ,NubxgvxvdxfuxRR *Project Supported by the National Natural Science Foundation of 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  extended the above results to a class of quasi-linear elliptic system: 22=, ,(||)=(||), ,pNqdivuubxg vxdivvvdxf ux  RRN where , , and 3N,>1pq,Nbd CR are positive functions, 0,)1[C,fgf are positive and non-decreasing. When and g satisfy (H1)  0=0=0,= >0;lim infufufg gu (H2) The Keller-Osserman condition,   10d<, =d,tptGtgs sGt then there exists an entire positive radial solution, and in addition, the function satisfy ,bd(H3) 111100 dd=tpNNtsbss t  ; (H4) 111100 dd=tqNNtsdss t  , then all entire positive radial solutions are large. On the other hand, if and satisfy b d(H5) 111100 dd1q=0,ap=0,apFor convenience, we need the following definition: Definition. A solution of system (,)uv22=,,, ,=,,, ,ap paq qdivxuufx uvxdivxvvgx uvx  (2) is called an entire large solution(or explosive solution, or blow-up solution), if it is classical solution of (2) on and NRux and vx as x. Now we give our main theorem: Theorem 1. Suppose f and g satisfy 1111,,max,< ,sup supmmst stfst gstst st  (3) and satisfy the decay conditions ,bd1/(1)1/( 1)00d<, d<,pqap aqtbtttdt t (4) where , then problem (1) has no positive entire radial large solution. =min{, }mpqIn order to state our results conveniently, let us write  1/( 1)1100:= ,lim=ddrprtap NNBBrBrtsbsstr  ,0,  1/( 1)1100:= ,lim=drqrtaq NNDDrDrtsds str  d,0 and   1/(1)0:= ,limd=,,,rrmFFrsFr rfss gss>0, where satisfies 0m0min,, if 1,=max,, if <1,pqf gmpqf g we see that   1/(1)01=>,,mFr rfrr grr0,> so, F has the inverse function 1F on [, ), and F and 1F 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  and > such that   0 as . It follows from (3) that there exists such that 10max,,,, for 1,mfst gstC stst (8) and 0max,,,, for 1.fst gstCst (9) Then by (8) and (9), we have 10max,,,1, for 0.mfst gstCstst (10) Then we can get    1010,1 1, for 0.mmfurvrCurvrCUrVrr  then the system (1) has infinitely many positive entire bounded solutions. So, for all , we obtain 00rr    1/(1)110001/( 1)11100011/( 1)11100011/( 1)100=,d 1dd 1dd 1d prtap NNrprt map NNrmprtap NNprmprapprururtsbs fusvssturCtsbsUs VssturCUrVrtsbssturCUrVrt btt     1/(1)00 1d,praprurCUrVrtbtt dq 1/(1)1/( 1)001maxd ,d<.4pqap aqrrrbrr rdrrC (11) where is a positive constant. As , we have , so the last inequality above is valid. Notice that (4), we choose such that C0<=min{ ,}mp>01<1mp0r D. J. LEI ET AL. 354 =0It follows that , we can find such that  =lim limrrur vr 1rr 100= ,=, maxmax rtrrtrUrutVrvtrr  . (12) Thus, we have  1/(1)0011.praprUrurCUr Vrtbttrrd, By (11), we get  01, .4Ur VrUrurr r 1 that is   11, .4Ur VrUrCr r  where 101=4Cur>0. Similarly,  21, .4Ur VrVrCr r  which implies  12 12, 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. uvRemark. In Theorem 1, if , and ,>2pq fg satisfy 11,,max , 0C3max,,,, for 1fst gstCstst and 3max,,,, for 1fstg stCst so, we can get 3max,,,1, for 0.fst gstCstst Thus, we can get   33,1 1, for 0,furvrCurvrCUrVr 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)110001/(1)1104 0011/(1)11104 0011/(104 0=,d ()(1()())dd 1dd 1prtap NNrprtaq NNrprtap NNprrapprururtsbs fusvssturCts bsUsVs sturCUrVrtsbssturCUrVrt bt    1) dptd where is a positive constant. 4Notice the condition (4), we choose such that C0>0r1/(1)1/( 1)00 41maxd ,d<,pqap aqrrrbrrrdrrC together with (12), we get  1101pUrurUr Vr Similarly,we have  1101qVrvrUr Vr Set , we get =min{ , }mpq    11001111001 1 21pqmUr VrurvrUr VrUr VrurvrUr Vr  Copyright © 2011 SciRes. APM D. J. LEI ET AL.355 that is,   11001211, 0mUr VrUr Vrur vrr   We claim the above inequality is invalid. In fact, set a function    111:=12 1mTUrVrUr VrUr Vr  then   2121=11 >01asis large enough.mmT UrVrUrVrmr , 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    1111=,=,0=>0, 0=>0,,0, on [0,),Nap NpNaq Nqrurbrfurvrrrurdrgurvrruvuv  , >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=ppsss). Thus =,=Ux uxvxd,0,d,0.d, 0,d, 0.Vx are positive solutions of (1). Integrating (14) we have  1/(1)1100=,dprtap NNurtsbs fusvsstr  1/( 1)1100=,dqrtaq NNvrtsdsgusvsst r  Let and be the sequences of posi- tive continuous functions defined on [0 by 0{}nnu0{}nnv,)  001/( 1)111001/( 1)11100=,=,=,d=,dprtap NNnnnqrtaq NNnnnurvrurt sbsfusvsstrvrt sbsgusvsstr   (15) Obviously, for all , we have 0r 0101, , , .nnurvru uv v The monotonicity of f and g yield  1212, , 0.urur vrvr rRepeating such arguments we deduce that 11, , 0,1,nn nnurur vrvr rn  and we obtain that sequences and are nondecreasing on 0{}nnu0{}nnv[0, ). Notice        1/( 1)1/( 1)11101/(1)11111/( 1)1111=,d, (,(, ,pprap NNnnnnnpnnnn nnnnpnnnnurrsbsfus vssfur vrBrfur vrurvrBrfur vrur vrgu rv ru rv rBr    and    1/( 1)1/( 1)11101/( 1)11111/( 1)1111=,d, (,(, ,,qqraq NNnnnnnqnnnn nnnnqnnnnvrrsdsgusvssgurvrD rgurvrurvrDrfu rv ru rv rgu rv ru rv rDr  Copyright © 2011 SciRes. APM D. J. LEI ET AL. 356 which implies   1/(1)0,,,nnmnnnn nnnnur vrBr Drfurvrurvr gurvrurvr where has been defined before. And then integrat- ing on we obtain 0m(0, )r     1/(1)00d.,,rnnmnnnn nnnnut vttBr Drfut vtut vtgut vtutvt So  () ()1/(1)0d,,,urvrnnmBr Drfg  that is  ,0nnFu rv rFBrDrr. (16) It follows from 1F is increasing on [0 and (16) that , ) 1,0nnur vrFFBr Drr. th By (17) It follows from ()Fat F (17), the sequences {}nu {}nv bounded and in- creasing on 0[0 or arbitrary 0cThus, and have subsequences converging uniformly to and on 0[0 . By the arbitrariness of 0c, 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. ,cuv={}nu>0{}nvv)= ,Uu(,0)V(,UV(0) (and (, )(0,)(0,) was chosen arbitrarily, it follows that (1) has infinitely many positive entire solutions. 1) If and , then ()0R        1/( 1)1/( 1)1/( 1)1, , , ,, pnn nnnnqnnnnnnnnmnnnnuRvRf uRvRuRvRBRg uRvRuRvRDRfuRvRuRvRgu Rv Ru Rv RBRDRn   1. Copyright © 2011 SciRes. APM D. J. LEI ET AL.357 This implies      1/( 1)1[, ,1 , 1.mnnnnnnnnnn nnfuRvRuRvR guRvRuRvRuR vRuR vRBR DRn ] Taking into account the monotonicity of , there exists  1nnnuRvR := .lim nnnLRuRvR  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 ,nnuv:(0,L)(0, )    ,0,, 1.nn nnur vruR vRLRrRn   Thus the sequences are bounded from above on bounded sets. Let 1{} ,{}nn nnuv1. :=, :=, for 0.lim limnnnnururvrv rr  Then is a positive solution of (14). (,)uvIn order to conclude the proof, it is enough to show that is a large solution of (14). We see (,)uv1/(1)1/( 1),, ,, 0pqur fBrvrgDr r Since f and g 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)10,, 0Nab dNR, 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)1100:= ,lim=edd, 0rprttapNNBBrBrtsbs str ,  1/( 1)1100:= ,lim=edd, 0rqrttaqN NDBrDrtsds str , Copyright © 2011 SciRes. APM D. J. LEI ET AL. 358 and    1/(1)0:= ,limd=,,,rrmFFrsFr rfss 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  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. 919-932. doi:10.1016/S0362-546X(98)00143-6  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. 827-846. doi:10.1016/S0021-7824(02)01265-5  C. Yarur, “Existence of Continuous and Singular Ground States for Semilinear Elliptic Systems,” Electronic Jour-nal of Differential Equations, Vol. 1, 1998, pp. 1-27.  R. 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