Uniqueness of Radial Solutions for Elliptic Equation Involving the Pucci Operator ()
1. Introduction
We study the nonlinear elliptic equation
(1)
where
is Pucci maximal operator, the potential f is super linear with some further constraints. Using
to denote the eigenvalues of
then explicitly, the Pucci operator
is given by
![](https://www.scirp.org/html/9-5300259\414ee22c-21f4-4d91-a423-450d227a18ec.jpg)
For more detailed discussion, see for example [1,2]. This equation has been extensively studied, see [3-5], etc. and the references therein.
Normalize
to be
for simplicity. We will in this paper investigate the uniqueness of
positive radial solution of (1) in the annulus
![](https://www.scirp.org/html/9-5300259\10a14c7a-5f32-469f-b5fa-1fb65897528d.jpg)
with Dirichlet boundary condition. In this case, Equation (1) reduces to
(2)
where
![](https://www.scirp.org/html/9-5300259\a4ccce98-b76f-4845-9eaa-7c84ceb9655d.jpg)
Throughout the paper, we assume
Note that
Now we could state our main results.
Theorem 1. Suppose
is small enough and
![](https://www.scirp.org/html/9-5300259\964db5ec-de3d-4950-a282-a8c7f2508bc3.jpg)
Then (2) has at most one positive solution with Dirichlet boundary condition.
If instead of the smallness of
we assume further growing condition on
then we have the following Theorem 2. Suppose that for
,
![](https://www.scirp.org/html/9-5300259\fc63ef06-e6c0-4da8-aa6b-4cbb885fe878.jpg)
where
![](https://www.scirp.org/html/9-5300259\6678db47-6fcb-4376-80e9-f6e0ddf6c22a.jpg)
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
which also depends on
We will not pursue this further in this paper.
2. Lane-Emden Transformation and Uniqueness of the Radial Solutions
2.1. Proof of Theorem 1
We shall perform a Lane-Emden type transformation to Equation (2). Let us introduce a new function
![](https://www.scirp.org/html/9-5300259\29a7a13c-7f7c-4fa6-9ca2-15d96e3af15f.jpg)
where
with
![](https://www.scirp.org/html/9-5300259\a88f38fe-5773-4421-95f8-5cb87b2428ff.jpg)
Then
satisfies
(3)
where we have denoted
![](https://www.scirp.org/html/9-5300259\5e908622-dbe6-4190-b216-f0ddb0870f30.jpg)
and
Note that m may not be continuous at the points where
or
Additionally, if
and
then ![](https://www.scirp.org/html/9-5300259\f1a7fbdf-cffc-4d11-893a-540bbcc1c6cd.jpg)
Lemma 3. Let w be a positive solution of (3) with
Then there exists
such that
and
![](https://www.scirp.org/html/9-5300259\9982a287-b024-4e05-91b0-02cf55216dff.jpg)
![](https://www.scirp.org/html/9-5300259\442e14e2-0642-493e-a5fe-51adaf0c1078.jpg)
Proof. If
for some
then
![](https://www.scirp.org/html/9-5300259\51b96b83-0e18-4b23-bb7d-04f0a694fa39.jpg)
The conclusion of the lemma follows immediately from this inequality. ■
Given
the solution of (3) with
and
will be denoted by
. Let
![](https://www.scirp.org/html/9-5300259\6d3d2772-d01c-4469-b7a9-e6f3a3d2001d.jpg)
By standard argument, we know that positive solution of (3) with Dirichlet boundary condition is unique if we could show that
![](https://www.scirp.org/html/9-5300259\d4c1e6d0-72df-4b27-bfd6-f4d7b6b4a380.jpg)
whenever
is a positive solution to (3) with ![](https://www.scirp.org/html/9-5300259\5c5b43b0-9857-4e26-bd32-ab5655b67c6d.jpg)
The functions
and
satisfy the following equations:
![](https://www.scirp.org/html/9-5300259\5e7d5a77-c046-4e38-ae56-0b01c7e51ff5.jpg)
![](https://www.scirp.org/html/9-5300259\c6cb26ab-4f33-4df5-9856-d0cd99f13bcf.jpg)
The initial condition satisfied by
is:
,
.
Now let
be a positive constant such that
is a positive solution to (3) with
. To show that
, let us first prove that
must vanish at some point in the interval
In the following, we write
simply as ![](https://www.scirp.org/html/9-5300259\61af553e-6ecb-4f17-8b15-2cf30345b8c5.jpg)
Lemma 4. There exists
such that
.
Proof. Let us consider the function
![](https://www.scirp.org/html/9-5300259\83e65621-cafb-4448-9fb1-1cc6f9a57705.jpg)
We have
![](https://www.scirp.org/html/9-5300259\fb8d487e-9895-410b-8e0c-e7e918d036da.jpg)
We remark that
is indeed not everywhere differentiable, since m is not continuous. It however could be shown that the jump points of m are isolated. Here by
, we mean the derivative of
at the point where it is differentiable. The same remark applies to the functions
and
below.
Now if
for
then
![](https://www.scirp.org/html/9-5300259\68f56656-8777-4122-a0b3-2835641ea0dc.jpg)
Since
we infer that
![](https://www.scirp.org/html/9-5300259\7d393d86-6d9a-45ee-bf1f-f00e71102ec1.jpg)
It follows that
![](https://www.scirp.org/html/9-5300259\017e7802-90ce-4ff4-a2f8-adfc86e0801e.jpg)
This is a contradiction, since
and
. ■
With the above lemma at hand, we wish to show that in the interval
vanishes at only one point ξ. For this purpose, let us define functions
and
Put
![](https://www.scirp.org/html/9-5300259\fe47ee8c-2a12-4846-817d-da7f852f5533.jpg)
![](https://www.scirp.org/html/9-5300259\384042df-8a59-442d-91d5-a4a582531a3a.jpg)
and
![](https://www.scirp.org/html/9-5300259\0a3fc3ec-d9a9-4212-af23-6c353fc30304.jpg)
Lemma 5. We have
(4)
(5)
Proof. Differentiate the Equation (3) with respect to s gives us
(6)
Hence
![](https://www.scirp.org/html/9-5300259\a1ccac33-1d3c-4e93-9af2-d367c72c7016.jpg)
As to the function h, there holds
![](https://www.scirp.org/html/9-5300259\4c65d99d-cef2-4a87-92b7-628ef87322d2.jpg)
Combining this with (3) and (6) we get
![](https://www.scirp.org/html/9-5300259\9043cb04-bc66-46ce-813a-7187864f1576.jpg)
It follows that
![](https://www.scirp.org/html/9-5300259\3ebdaaf4-85f0-4c2e-aa71-a037245fc275.jpg)
■
Now we are ready to prove Theorem 1.
Proof of Theorem 1. We need to show that
.
We first of all claim that the first zero
of
in
must stay in the interval
where
is given by Lemma 3. Suppose to the contrary that
By (5) using the fact that
we find that if
is small enough, then in the interval ![](https://www.scirp.org/html/9-5300259\da7dea9f-b900-4041-9edf-874aeb1265cd.jpg)
![](https://www.scirp.org/html/9-5300259\ca463b1d-2e34-4e34-8db7-2f98e8c32810.jpg)
Since
we find that
![](https://www.scirp.org/html/9-5300259\a694e0d4-68ca-423a-a951-2d7376f40f98.jpg)
Therefore
![](https://www.scirp.org/html/9-5300259\2c0ef1b3-7948-4b97-ad30-a436c0a00363.jpg)
This is a contradiction, since
and ![](https://www.scirp.org/html/9-5300259\a9b3e504-22b4-4fa7-9a61-35287d311f5d.jpg)
Now the first zero
of
lies in
If
then the second zero
of
lies in
Note that in
Therefore, by identity (4)
![](https://www.scirp.org/html/9-5300259\660e3268-2dc4-4087-9cab-c90244197ddd.jpg)
This together with
![](https://www.scirp.org/html/9-5300259\99b616bb-0ce8-4871-8002-8528c27c3e49.jpg)
implies that
![](https://www.scirp.org/html/9-5300259\dbe7d496-0add-488d-ac4f-c2997128f0ec.jpg)
but this contradicts with
,
, and
This finishes the proof. ■
2.2. Proof of Theorem 2
Similar arguments as that of Theorem 1 could be used to prove Theorem 2. In this case, we shall make the following transform:
![](https://www.scirp.org/html/9-5300259\e3456ec9-b736-4d6f-9263-259318f63979.jpg)
where
![](https://www.scirp.org/html/9-5300259\4ed8cd11-e447-4f0d-97a8-836971f24509.jpg)
and
Then
(7)
With this transformation, in the interval
, w satisfies
(8)
where
![](https://www.scirp.org/html/9-5300259\77a86609-9fcd-45f4-a59b-b9c5905c9e9a.jpg)
![](https://www.scirp.org/html/9-5300259\d2abb2ab-8499-46df-9eea-39cec9914e26.jpg)
By the definition of
one could verify that
Note that
and
are step functions and not continous.
Let
be the solution of (8) with
and
. Now similar as in the proof of Theorem 1, we suppose
is a positive solution with Dirichlet boundary condition and
. We have the following lemma, whose proof will be omitted.
Lemma 6. There exists
such that
, and
![](https://www.scirp.org/html/9-5300259\1595263a-9251-41fc-9ecf-699ff087f11b.jpg)
![](https://www.scirp.org/html/9-5300259\91c77b14-153e-49c2-a413-be3d08dfe3e0.jpg)
With this lemma at hand, we observe that by (8)
![](https://www.scirp.org/html/9-5300259\22832bce-f0c7-42f6-90e9-910bc52cc4cb.jpg)
This combined with (7) tells us that
Then it is not difficult to show that for
and
while for
![](https://www.scirp.org/html/9-5300259\b0d0cb73-6707-4a8b-8fd8-603b1d51156d.jpg)
Recall that
satisfies
![](https://www.scirp.org/html/9-5300259\aec1e091-c923-4815-872a-793010183219.jpg)
Consider the function
then
![](https://www.scirp.org/html/9-5300259\9eefb7bd-a513-49a8-952f-dabb8ed31fcb.jpg)
From this we infer that the function
must change sign in the interval
similar as that of Theorem 1.
Now let us define
![](https://www.scirp.org/html/9-5300259\a55a7b72-6beb-4e97-b3a3-54bdde478237.jpg)
and
![](https://www.scirp.org/html/9-5300259\e9516ecd-2f12-4b10-a873-af17484ff11a.jpg)
where
and
Moreover, denote
![](https://www.scirp.org/html/9-5300259\aae9e7c8-ea7a-4ac1-8233-2203e03783cc.jpg)
Lemma 7. There holds
![](https://www.scirp.org/html/9-5300259\2be4019c-f6fd-41f2-ae23-cc6e4ed94810.jpg)
![](https://www.scirp.org/html/9-5300259\c87a4161-f20b-42fb-8405-07592ea9a94e.jpg)
Proof. Direct calculation shows
![](https://www.scirp.org/html/9-5300259\dc6380e3-21e7-4997-ada9-a9189a54f75a.jpg)
and
![](https://www.scirp.org/html/9-5300259\ce338355-2e9f-4791-882d-a881e3cbbd0e.jpg)
This then leads to the desired identity. ■
Now with the help of this lemma, we could prove Theorem 2.
Proof of Theorem 2. First we show the first zero
of
is in the interval
Otherwise, since
![](https://www.scirp.org/html/9-5300259\523a055a-acd7-4f16-90b4-153e73e8ff8a.jpg)
one could then use the fact that
in
and
to deduce that in ![](https://www.scirp.org/html/9-5300259\d71fe556-2b37-4ddf-be82-66931808e759.jpg)
![](https://www.scirp.org/html/9-5300259\a98d86b6-d141-4d3d-b151-b9a371f33bb7.jpg)
But this contradicts with
and
.
Now if the second zero
of
is in
Then since
![](https://www.scirp.org/html/9-5300259\6798824f-98a8-49d4-83e8-d3384334a347.jpg)
one could use
in
to deduce that
in
which contradicts with
and
■
3. Acknowledgements
The author would like to thank Prof. P. Felmer for useful discussion.
NOTES