Existence and Non-Existence Result for Singular Quasilinear Elliptic Equations

We prove the existence of a ground state solution for the qusilinear elliptic equation 2 (| | ) p div u u     = ( , ) f x u in N   R , under suitable conditions on a locally Holder continuous non-linearity ) , ( t x f , the non-linearity may exhibit a singularity as 0 t   . We also prove the non-existence of radially symmetric solutions to the singular elliptic equation 2 (| | ) = ( )[ ( ) ( ) | | ], p q div u u d x g u r u u        ( ) > 0 u x in N R , ( ) 0 u x  as | | , x  where ( ) = (| |) ( , (0, )), N d x d x C   R 3, 0. N q     1 0, (0, ), (0, ) , ([0, ), loc g C r C       [0, )),0 < < 1   .

[0, ) [0, ) f      be a locally Holder continuous function which may be singular at = 0 t .The problem (1) appears in the study of non-Newtonian fluids [1,2] and non-Newtonian filtration [3].The quantity p is a characteristic of the medium.Media  , they are Newtonian fluid.When 2 p  , the problem becomes more complicated since certain nice properties inherent to the case = 2 p seem to be lose or at least difficult to verify.The main differences between = 2 p and 2 p  can be found in [4][5][6].
In recent years the study of ground state solutions for = 2 p has received a lot of interest and gets numerous existence results (see [7][8][9][10][11][12]).For p -Laplacian equations, in most papers, the focus has been on separable nonlinearities like (2).
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 [13] for > 1 p , 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.

Preliminaries
Firstly we list the following assumptions and results that needed below.


. The first eigenvalue of the problem (4) will be denoted by > 0   .It is easily noted from the variational characterization of eigenvalues that , where .
The nonlinearity f in problem ( 1) is assumed to be a real function that satisfies the following conditions: There is a continuous function 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 ( ) = ( ) c x a x .Recall that the nonlinearity ( , ) f x t may exhibit singularity as 0 t   .We will consider the following Dirichlet problem for a given smooth bounded domain To establish the main theorem, from reference [2] we give the following lemmas.

R
and is locally Holder continuous (with exponent (0,1)

 
) in x .Suppose moreover that there exist functions Note that ˆ( ) g t is non-increasing, positive and Then h is 1 C , non-increasing and for some [0, ) , we see from ( 9) that for a sufficiently large 1    be the inverse function of  .
By direct calculation, we see that A simple computation shows that v has the desired properties.
Indeed, on recalling We have used (11) in the last inequality.Since

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 has a positive solution u in 1, ( ) ( ) where  and  are defined as in ( 9) and ( 10) respectively.Then = 0 v  on  , and proceeding as in the proof of Lemma 3, we note that Therefore, by condition ] [ 2 F , we see that We recall, by the above, that v also satisfies 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 ( ) = ( ) , where  is the positive constant in Moreover, since , we also note that, Therefore, we get Recalling that ĝ 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) Then 0 ( ) ( ) R and that u is a solution of (1).Since 0 u v   , it follows that ( ) 0 u x  as | | x   .In the last part of the paper, we prove a nonexistence result for the following problem, The result solves an open problem in [4] for > 1 p , 0 q  for the case when u is a radially symmetric solution.Before the proof, we state some conditions which we needed at the below.
are fulfilled and d is a positive radial function, r is a nonnegative radial function that is continuous on N R and satisfies , = ) ) ( ( then the problem (19) has no positive radial solution that decays to zero near infinity.
Proof.Suppose (19) has such a solution ) (r u .Then  g is non-increasing on ) (0, , we have

References
f x u in N   R , under suitable conditions on a locally Holder continuous non-linearity ) , ( t x f, the non-linearity may exhibit a singularity as 0 t   .We also prove the non-existence of radially symmetric solutions to the singular elliptic equation2In this paper, we are concerned with the existence of ground state solution or positive solution for the follow-

2
has a solution u such that u v be as in the proof of the above Theorem.Then we note that b W W   , and hence v v   where v is as in Lemma 3. Then we deduce a non-singular case. the ball of radius k centered at the origin.By Theorem 1, for each positive integer k we let )