Uniqueness of Radial Solutions for Elliptic Equation Involving the Pucci Operator *

The solution of a nonlinear elliptic equation involving Pucci maximal operator and super linear nonlinearity is studied. Uniqueness results of positive radial solutions in the annulus with Dirichlet boundary condition are obtained. The main tool is Lane-Emden transformation and Koffman type analysis. This is a generalization of the corresponding classical results involving Laplace operator.


Introduction
We study the nonlinear elliptic equation where  is Pucci maximal operator, the potential f is super linear with some further constraints.Using i , 1, , i n For more detailed discussion, see for example [1,2].This equation has been extensively studied, see [3][4][5], etc. and the references therein.
Normalize  to be 1 for simplicity.We will in this paper investigate the uniqueness of positive radial solution of (1) in the annulus 2 C   : : with Dirichlet boundary condition.In this case, Equation (1) reduces to , for > 0, for 0. where Throughout the paper, we assume Note that Now we could state our main results.we assume further growing condition on f then we have the following Theorem 2. Suppose that for Then (2) has at most one positive solution with Dirichlet boundary condition.
In the case  the Pucci operator reduces to the usual Laplace operator, and the corresponding unique results are proved by Ni and Nussbaum in [6].
We also remark that the above theorems could be generalized to nonlinearities , f which also depends on We will not pursue this further in this paper. .r

Lane-Emden Transformation and
Uniqueness of the Radial Solutions

Proof of Theorem 1
We shall perform a Lane-Emden type transformation to Equation (2).Let us introduce a new function f w s w s ms w s u r where we have denoted Note that m may not be continuous at the points where or Additionally, if and then m s Lemma 3. Let w be a positive solution of (3) with Then there exists The conclusion of the lemma follows immediately from this inequality.■ Given the solution of (3) with will be denoted by .Let


By standard argument, we know that positive solution of (3) with Dirichlet boundary condition is unique if we could show that  and satisfy the following equations: , .some point s s following, we write In the The initial condition satisfied by We have  below.
As to the function h, there holds get Combining this with (3) and ( 6) we  ■ Now we are ready to prove Theorem 1. Proof of Theorem 1.We nee hat We first of all claim that the first zero  of  in   Therefore, by iden = 0 tity (4) This together with With this transformation, in the interval   By the definition of , ,   one could verify at We have the following lemma, whose proof will be omitted.
Lem here exists ma 6. T   With this lemma at hand rve that by (8 , we obse ) Then it is not difficult to show that for From this we infer that the function  must change sign in the interval   where g w and sired identity.■ is lemma, we could prove Theorem 2. First we show the first zero has at most one positive solution with Dirichlet boundary condition.If instead of the smallness of b a ,


lemma at hand, we wish to show that in the interval   vanishes at only one point ξ.For this purpos s efine functions   e, let u d

2 ,
This then leads to the de Now with the help of th Proof of Theorem 2. of  is in the interval   the second zero  of  is in  