 Applied Mathematics, 2010, 1, 351-356 doi:10.4236/am.2010.15046 Published Online November 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM Existence and Non-Existence Result for Singular Quasilinear Elliptic Equations* Mingzhu Wu1, Zuodong Yang1,2 1Institute of Mathematics, School of Mathematical Science, Nanjing Normal University, Nanjing, China 2College of Zhongbei, Nanjing Normal University, Nanjing, China E-mail: zdyang_jin@263.net Received June 1, 2010; revised August 11, 2010; accepted August 14, 2010 Abstract We prove the existence of a ground state solution for the qusilinear elliptic equation 2(| |)pdiv uu  =(,)fxu in NR, under suitable conditions on a locally Holder continuous non-linearity ),( txf , the non-linearity may exhibit a singularity as 0t. We also prove the non-existence of radially symmetric solutions to the singular elliptic equation 2(||)=( )[( )( )||],pqdivuudxg ur uu ()>0ux in NR, () 0ux  as || ,x where ()=(||)(,(0, )),NdxdxCR 3, 0.Nq10,(0,),(0,), ([0,),locgC rC  [0,)), 0<<1. Keywords: Quasilinear Elliptic Equations, Existence, Non-Existence, Singular 1. Introduction In this paper, we are concerned with the existence of ground state solution or positive solution for the follow- ing problem 2(||) =(,),>0,()=0,pdivuufx uxuxux x  (1) in which NR, () 0ux  when ||x . Let :[0,)[0,)f be a locally Holder continuous function which may be singular at =0t. The problem (1) appears in the study of non-Newto- nian fluids [1,2] and non-Newtonian filtration . The quantity p is a characteristic of the medium. Media with >2p are called dilatant fluids and those with <2p are called pseudoplastics. If =2p, they are Newtonian fluid. When 2p, the problem becomes more complicated since certain nice properties inherent to the case =2p seem to be lose or at least difficult to verify. The main differences between =2p and 2p can be found in [4-6]. In recent years the study of ground state solutions for =2p has received a lot of interest and gets numerous existence results (see [7-12]). For p-Laplacian equa- tions, in most papers, the focus has been on separable nonlinearities like (2). 2(||)=( )( ),>0,()0,| |pNNdivuubx g uxuxRuxas x R (2) We refer readers to the paper [13-19]. In this paper, we consider the situation of 1>p. In , the author extended results to the problem (1) for =2p where f is not necessarily separable. For 2p we can see [1-6]. Motivated by papers [4-6,20], we extend the results to >1p and get two theorems. But we still have many difficulties to get entire ground state solution of (1). The second purpose is to give a result for nonexistence of solution. To the best of our knowledge, there has been very less result for nonexistence of solution about singular elliptic equation. We solve an open problem in  for >1p, for the case when u is a radially sym- metric solution. The paper is organized as follows. In Section 2 we recall some facts and give many lemmas that will be needed in the paper. In Section 3, we give the proof of the main result of the paper. *Project Supported by the National Natural Science Foundation of China (No.10871060); the Natural Science Foundation of Educational Department of Jiangsu Province (No.08KJB110005) M. Z. WU ET AL. Copyright © 2010 SciRes. AM 352 2. Preliminaries Firstly we list the following assumptions and results that needed below. 2(||) =(),>0,()0,| |pNNdivWWbxxWxRWx asx R (3) 1[]>0Bb is a locally Holder continuous function on NR. 111* 1200[](()) 0t. 3[]B Problem (3) has a solution bW in NR when b satisfy some condition. It can be proved that condition 2[]B implies 3[]B. This follows from the observation that () 0vr  as r where 111* 10()=(())tNN prvrtsb sdsdt  and satisfies the following equation 2*(||) =().pdivvvb x  Thus ()= (||)wxvx is a supersolution of (3). Since 0w, and zero is clearly a sub-solution it follows from standard results that (3) admits a solution bW such that 0< bWw. The following eigenvalue problem 21(||) =(),0,| |ppNdivc xxas x R (4) where N R is a bounded smooth domain, and (,(0,))cC for some 0< <1. The first eigen- value of the problem (4) will be denoted by >0. It is easily noted from the variational characterization of eigenvalues that 12 where12 are smooth bounded domains in NR. Consider :(0, )(0, )g  satisfies the following conditions: 1[]Gg is 1C. 2()1[] 0 and 1> such that 1,,,0,pNfxsaxs xsR where 1 is the first eigenvalue of the problem (4) on the ball (0 , 1 )B of radius one and centered at the origin and where ()= ()cxax. Recall that the nonlinearity (,)fxt may exhibit sin- gularity as 0t. We will consider the following Dirichlet problem for a given smooth bounded domain N R. 2(||) =(),>0,()=0,pdivWWbxxWxWxx  (5) To establish the main theorem, from reference  we give the following lemmas. Lemma 1. (Weak Comparison Principle) Let  be a bounded domain in 2NNR with smooth boundary  and :(0, )(0,) is continuous and nonde- creasing. Let 1,12,()()plocuu WC  satisfy 21121222||()||()ppuudxudxuudxu dx    for all non-negative 1, ()plocW. Then the inequality 12sup( ()())0limxuxu x implies that 12 .uuin Before Lemma 2, we give the following equation 2(||)(,) = 0,pNdivuufx ux R (6) Lemma 2. Suppose that (,)fxu is defined on 1NR and is locally Holder continuous (with exponent (0,1)) in x. Suppose moreover that there exist functions 1,Nlocvw CR such that 22(||)( ,)0(||)( ,)0ppdivvvfx vdivwwfx w  and () ()vx wx and that (,)fxu is locally Lipschitz continuous in u on the set ,: ,Nxuxvx uwxR Then Equation (6) possesses an entire solution )(xu satisfying ,Nvx uxwxx R Lemma 3. Let b satisfy 1[]B, 3[]B and g sati- sfy both 1[]G, 2[]G. Then there is 1NvCR such M. Z. WU ET AL. Copyright © 2010 SciRes. AM 353that 21(||)( )(()),>0,()0||.ppdivvv bxgvxvvx asx  Proof. Since g satisfies 2[]G, we define ()ˆ():=sup:>, >0gsgtst ts (7) Note that ˆ()gt is non-increasing, positive and 1ˆ() ()gtgtt. Furthermore, by 2[]G we have 1ˆ()< bgtW  for sufficiently large t. Let 22ˆ()=( ),>0tthtgsdstt (8) Then h is 1C, non-increasing and ˆˆ()()( )2tgt ht g for all )(0,t. Since h is non- increasing, we note that 1()0()ttdsths (9) On using 1ˆ()< bgtW  in (8) for sufficiently large >0t, we see from (9) that ()> bW , (10) for a sufficiently large 1. Let 1= be the inverse function of . By direct calculation, we see that ()=( ())tht, ()>0t, for >0t and (0) = 0. Furthermore ()=(())(()), >0.thth tt By condition 3[]B, we take a solution bW of (3) with NR. Let us now set ():= (())bvxW x for all x. We note from (10) that ()= ()0s and 0011==lim limssgs gs; 11,iiigg are non-increasing on (0, ); 11()= ()=0lim limssivgsg s. Proof. 11100==,limlim 11pppssgsgs ssss 111==0.limlim 11pppssgsgs ssss  We set  >0 1=sup1ts pgtgst, then we have 1,>01pgtgssandtst  and )(sg is non-increasing on )(0,. Moreover, 0==0.lim limssgsand gs Now we can assume 10,gC. If not, we can replace it by  122=,>0ssg sgtdt ss Obviously,  1,>0;2sgs gs gs  and M. Z. WU ET AL. Copyright © 2010 SciRes. AM 354   12221 2'= 22sssgsgsg gtdtss   221 21)=( 0.22 22ss sgsggsgs gsss     11. .,((0,),(0,)).ie gC 1>0()()=inf (1)pstgtgs t Observe that 1()0<( ),>0;(1)pgtgs st and )( sg is non-increasing on )(0,. Moreover, 0()= ()=0.lim limssgsandgs Now we can assume 1(0, )gC. If not, we can replace it by 112()=() , >0ssgs gtdtss Obviously, 1(1)() (),>0gsg sgss and 1'()=(1)()0,> 0g sgsgss 11. .,((0,),(0,)).ie gC 3. Proof of Main Theorems In this section, we prove our main results. Theorem 1. Let N R be a bounded smooth domain that contains (0 , 1 )B, the ball of radius one centered at the origin, and let f satisfies 1[]F, 2[]F and 3[]F, where b satisfies 1[]B, 3[]B and g satisfies 1[]G, 2[]G. Then the problem 2(||) =(,),()=0,pdivuufx uxux x  (12) has a positive solution u in 1,() ()CC such that u where  is an eigenfunction of the eigenvalue problem (4) on  with ()= ()cxax normalized such that 0< on . Here  is the constant in condition 3[]F. Proof. Let W be the solution of (5) and set 1()= ()vx W where  and  are defined as in (9) and (10) respectively. Then =0v on , and proceeding as in the proof of Lemma 3, we note that 21(||)( )(),.ppdivvvb x gvx   (13) Therefore, by condition ][ 2F, we see that 2(||)( ,),.pdivvvfx vx  (14) We recall, by the above, that v also satisfies 21ˆ(||)( )(),.ppdivvvbx gvx  (15) Let  be a smooth bounded domain that contains (0, 1 )B the unit ball centered at the origin. Now, let  be the first eigenfunction of the problem (4) with ()= ()cxax such that 0<, where  is the positive constant in 3[]F. Invoking conditions 2[]F and 3[]F, we get 2111(| |)=()()(,),pppdivaxaxf xx    (16) Moreover, since 1<0, we also note that, 211(||)( ,)ˆ ()()()(),pppdivf xbxg bxgx    Therefore, we get 21ˆ(||)()(),ppdivb x gx    (17) Recalling that ˆg is non-increasing, by Lemma 1 we note, from (15) and (17), that v. Then by the elliptic regularity theory and Lemma 2, (14) and (16), we conclude that (12) has a solution u such that uv and 1,() ()uC C. Let W be as in the proof of the above Theorem. Then we note that bWW, and hence vv where v is as in Lemma 3. Then we deduce a non-singular case. Theorem 2. If f satisfies 1[]F, (,0)=0fx, where b satisfies 1[]B, 3[]B and g satisfies 1[]G, 2[]G, then problem (1) has a solution 1, NlocuCR. Proof. For each positive integer k, let )(0,=kBBk be the ball of radius k centered at the origin. By Theorem 1, for each positive integer k we let )(1,kkBCu be a solution of 2(||) =(,),()=0,pkkdivuufx uxBuxx B  (18) Then 0()()kkux vx in kB, where kv are as in Theorem 1. Corresponding to the ball kB. It is easy to see that 0 is a subsolution. Recall the above, kvv on kB for all 1k, and hence 0()()kux vx for all 2Bx. By a standard procedure, we conclude that {}klu has a subsequence that converges uniformly to a function in 2, ()C. By a diagonalization process it follows that {}ku has a subsequence that converges uniformly on open bounded subsets of NR to 2,NlocuCR and that u is a solution of (1). Since 0uv, it follows M. Z. WU ET AL. Copyright © 2010 SciRes. AM 355that () 0ux as ||x. In the last part of the paper, we prove a nonexistence result for the following problem, 2(| |) =()[()() ||],>0,()0,| |pqNdiv uudx guruuuinuxas x R(19) The result solves an open problem in  for >1p, 0q for the case when u is a radially symmetric solution. Before the proof, we state some conditions which we needed at the below. 110,, 0,GgC; 201=limupguGu; 31=0limupguGu . Theorem 3. Suppose 13()()GG are fulfilled and d is a positive radial function, r is a nonnegative radial function that is continuous on NR and satisfies ,=))(( 111010 ddd pNN then the problem (19) has no positive radial solution that decays to zero near infinity. Proof. Suppose (19) has such a solution )(ru . Then 2(|()|( ))=( )[(( ))(( ))|( )|],pqdivu rurdr gurrurur  or, equivalently, ()ur is a solution to the problem 12 1(||)()[ (())(())]Np Nruurdr gurrur   (20) Integrating (20) from 0 to r, we have 12 10| |()[(())(())].rNp Nru udguru d   Hence ()<0ur; i.e., ()ur is non-increasing. We put ln1=>0urur for all >0r. Then we have 212(|()|())11=()(|()|())(1)||,1(1)ppp ppdivu ru rdivu ru rpuuuand ur satisfies 2211(|' |')|'|'1(())(())(1)| |()(1) (1)pppppNuu uurgur rurpudruu   (21) Multiplying Equation (21) by 1Nr and integrating on (0,) yields 1120110(1)|'|'| |(1)(( ))(( ))() (1)NNp ppNppuu udugu ruddu   (22) If p is even, we can deduce that dduupuru pppNNr111010)||1)(1)(((0)~)(~  111 1100(( ))(( ))(() )(1)rNN ppgu rudddu    (23) We observe that ()< (0)ur u for all >0r and ()< (0)uru for all >0r. Since ()ur is positive, then (23) implies 111 1100(( ))(( ))(() )(0)(1)rNN ppgu ruddduu    (24) for all 0>r. Now, using Lemma 4 in (24), we have 111 1100111 1100(()(()))(( ))(() )(( )1)rNN prNN ppdgu ddgudddu     111 1100(( ))(( ))(( ))(0).(( )1)rNN ppgu ruddduu    If p is odd, we deduce that 111100(1)() (0)(||)(1)NrNppppuruu ddu  111 1100(( ))(( ))(() )(1)rNN ppgu rudddu   (25) then (25) implies 111 1100(( ))(( ))(() )(1)()< (0)rNN ppgu ruddduur u   (26) for all 0>r. As the same as the above we get 111 1100(()(()))<(0).rNN pdgudd u    But, since 1g is non-increasing on )(0,, we have M. Z. WU ET AL. Copyright © 2010 SciRes. AM 356 111 110011(0)=(() )<<,((0))rNN ppudd dgu    which is a contradiction. 4. References  G. Astrita and G. Marrucci, “Principles of Non-New- tonian Fluid Mechanics,” McGraw-Hill, Rochester, 1974.  L. K. Martinson and K. B. 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