Symmetric Solutions of a Nonlinear Elliptic Problem with Neumann Boundary Condition ()
1. Introduction
The maximum principle is one of the most used tools in the study of some differential equations of elliptic type. It is a generalization of the following well-known theorem of the elemental calculus “If f is a function of class 
in 
such that the second derivative is positive on (a, b) then the maximum value of f attains at the ends of
”. It is important to point out that the maximum principle gives information about the global behavior of a function over a domain from the information of qualitative character in the boundary and without explicit knowledge of the same function. The maximum principle allows us, for example, to obtain uniqueness of solution of certain problems with conditions of the Dirichlet and Neumann type. Also it allows to obtain a priori estimates for solutions. These reasons make interesting the study of the maximum principle on several forms and its generalizations and the Hopf lemma. For example a geometric version of the maximum principle allows us to compare locally surfaces that coincide at a point. On the other hand, the maximum principle and the Alexandrov reflection principle in [1] have been used to prove symmetries with respect to some point, some plane, symmetries of domain and to determine asymptotic-symmetric behavior of the solutions of some elliptic problems. (See Serrin [2], Gidas, Ni and Nirenberg [3], Gidas, Ni and Nirenberg [4], Caffarelli, Gidas and Spruck [5], Berestycki and Nirenberg [6]). The first person in use this technic was Serrin. Serrin proved that: “If 
is a positive solution of the problem
which is zero on the boundary and its outer normal derivative on the boundary is constant, then 
is a ball and 
is radially symmetric with respect to the center of
”. Using the ideas of Serrin and a version of the maximum principle for functions that do not change of sign, Gidas Ni and Nirenberg proved that: “If 
is a ball, 
and 
is a positive solution of the problem,
which is zero on the boundary, then u is radially symmetric with respect to the center of the ball”. Using the method of reflection and a version of maximum principle for thin domains Berestycki and Nirenberg made a generalization of this statement. Our proof shows that the technic used by Berestycki and Nirenberg for the study of symmetries of solutions of the elliptic problem with Dirichlet condition, can be applied in elliptic problems with Neumann conditions with nonlinear term
.
2. Maximum Principle and Hopf Lemma
Our result is based on the well known maximum principle and on the Hopf lemma for the differential operator of the form (see [7-9])
(1)
where 
is in 
We suppose that the coefficients 
and 
are bounded on 
and 
for all 
Theorem 2.1. (Maximum principle)
Let 
be such that 
Then 
cannot attain its maximum value in 
Lemma 2.2. (Hopf)
Suppose 
satisfies
Let 
be such that
• 
is continuous at 
• 
for all 
• 
existe.
Then 
3. Main Result
Theorem 3.3. Let 
be a solution of
where 
are bounded functions and symmetric with respect to the origin such that 
and 
for all 
is such that 
is strictly increasing in 
for all 
and is symmetric to 
for all 
and 
is a bounded function and odd. Then 
is symmetric with respect to the origin.
Proof: Define the reflected function of 
in 
by
Hence, 
Then v satisfies
Define
Then 
satisfies
Since 
is continuous in
, there are 
such that
Suppose that 
or 
then if
since 
Further
Therefore
Since 
is strictly increasing in 
Then
Therefore
If 
using a similar argue we demonstrate that 
and we obtain the same conclusion. Suppose that 
then 
since w(0) = 0 Further 
Therefore
Since 
is strictly increasing in 
Then
Therefore
We conclude
So 
is symmetric with respect to the origin.
We will prove that 
do not belong to 
Suppose now that 
and
for all 
then
and 
If 
and 
then 
and 
where 
are such that 
is the first zero of w and 
is the last. Since 
is strictly increasing in
, then
and
Applying maximum principle and Hopf lemma,
since 
is not constant. Which contradicts the fact that
Hence this case is impossible. It happens equally to 
and 
In conclusion we have that 
on 
and therefore 
is symmetric with respect to 
4. Example
Taking
in Theorem 3.3, we have the following system
following the steps of the demonstration, it follows that u is symmetric with respect to the origin.
5. Acknowledgements
The authors express their deep gratitude to CONACYT México, Programa de Mejoramiento del Profesorado (PROMEP)-México and Universidad de Cartagena for financial support.