Nondegeneracy of Solution to the Allen-Cahn Equation with Regular Triangle Symmetry ()
1. Introduction
We are interested in the following Allen-Cahn equation
(1.1)
There have been a lot of work on this equation during the last two decays (see for example [1] -[5] and the references there in). An important class of solutions to (1.1) is the multiple-end solutions. By definition, a solution 
to (1.1) is called a multiple-end solution, if outside a large ball, the nodal curves of 
are asymptotic to finitely many half straight lines. One knows that these solutions have finite Morse index, and one also expects that any solution with finite Morse index should be a multiple-end solution. The most simple example of a multiple-end solution is
, where
, 
, and 
is the heteroclinic solution:
It is also well known ([6] ) that for each
, 
, (1.1) has a solution 
with regular polygon symmetry. More precisely, the nodal set of 
consists of 
straight lines
In the polar coordinate 
(we abuse the notation and use the same symbol for the function 
in the
or 
coordinate):
We could also assume
. In the special case
, the solution 
is called saddle solution. It turns out that 
is not isolated in the set of 4-end solutions. Actually, there is a family of 4-end solutions to (1.1) containing 
and the solutions 
whose nodal lines are two almost parallel curves which are close to the solutions of the Toda system, see [7] [8] . In [9] , it is shown that for each
, there is a family of 2k-end solutions whose nodal lines are close to suitable solutions of the classical Toda system. Intuitively, these solutions are in some sense on the boundary of the moduli space of 2k-end solutions and it is natural to expect that they are on the same connected component as
. In particular, one expects that around
, there should be a family of 2k-end solutions to (1.1). While this is true for
, in this paper, we will focus our attention on the solution
.
To state our result in a precise way, let us introduce some notations.
We will use 
to denote
. Let 
be the linearized operator around
:
Our main theorem is the following nondegeneracy result:
Theorem 1.1 Assume 
is a 
solution of the equation 
and
.
Suppose furthermore that in the polar coordinate
,
,
.
Then
.
With the help of this theorem and the moduli space theory developed in [10] , we have the following Corollary 1.1 There is a family of 6-end solutions
, different from
, such that 
in 
as 
and in the polar coordinate,
2. Proof of Theorem 1.1
To prove our main theorem, we will use the ideas developed in [11] . Assume to the contrary that 
is not identically zero. As a first step, we show that 
has the same symmetry as the function
.
Lemma 2.1 Under the assumption of Theorem 1.1,
.
Proof. The crucial observation is that since
,
(2.1)
Note that the Laplacian operator is taken with respect to the 
variable and in the Equation (2.1) the function is expressed in the polar coordinate.
Consider the function
It follows from (2.1) that 
is also in the kernel of
, that is,
(2.2)
It will then suffice to show 
By the definition of 
and the symmetric of
, we have
On the other hand, the solution 
itself satisfies
Moreover, in the region
, 
is positive and is a supersolution of the operator
:
(2.3)
Denoting
, then it is well defined in
. From (2.2) and (2.3), we get
(2.4)
Following similar arguments as that of Lemma 2.1 in [11] , with slight modification (one should take care of the fact that the right hand side of (2.4) is not identically zero), we could obtain 
for some constant
, which implies that
Since 
decays to zero at infinity, we conclude 
and then 
This completes the proof of this lemma.
Let 
be the nodal set of
. We proceed to show that
is bounded.
Lemma 2.2 There exists a constant 
such that
Proof. We first show that for each connected component 
of
, there exists a constant 
such that 
is contained in the ball of radius
:
.
We argue by contradiction and assume that 
is unbounded.
Case 1. 
In this case, we will consider
. By the symmetry property of
, 
is still connected and unbounded. Therefore, one could find a continuous and piecewise smooth curve contained in
:
such that
, as
. One then could consider the nodal domain 
of 
contained in 
whose boundary is the image 
of 
(one could assume without loss of generality that 
does not have self intersection, otherwise, the presence of the supersolution 
yields a contradiction). Since 
is a positive supersolution of 
in 
similar arguments as Lemma 2.1 implies that 
in 
for some constant
, which is a contradiction.
Case 2. 
In this case, consider the region 
Using symmetries of
, we could assume
But in 
(This follows from a moving plane argument, see for example [12] ) and
, in particular, it is a positive supersolution. Hence similar arguments as in Lemma 2.1 imply that
, a contradiction.
Hence each connected component 
is contained in a ball. To prove the assertion of this lemma, it will be suffice to show that there are only finite many connected components. We will assume to the contrary that there are infinite many of them. Then for each 
large, one could find a nodal domain of 
which is contained in
for some 
But when 
is large, since multiple-end solutions have finite Morse index, there is a positive supersolution 
in
:
This contradicts with the maximum principle. The proof is thus completed. □
Now we are ready to prove our main theorem.
Proof of Theorem 1. Suppose 
is not identically zero. By the previous lemmas, we could assume that
when 
is large enough, say for
.
Let us consider the projection of 
onto
. That is, we define the function
Note that 
and hence
.
We compute
Since 
(This could be seen from the construction of
), we find that 
for 
large. On the other hand, by the assumption, 
, as
. Therefore as 
goes to infinity, 
goes to zero. This contradicts with the fact that 
and the proof is thus completed. □
A simple application of Theorem 1.1 is Corollary 1.1.
Proof of Corollary 1.1 By Theorem 1.1, the solution 
is nondegenerated in the class of functions having regular triangle symmetry. Then we could use the moduli space theory developed in [10] , in suitable functional spaces having these symmetry. This theory tells us that locally around 
(in certain natural topology), the solution set has a structure of one dimensional real analytic manifold. Hence we get a family of 6-end solution 
satisfying the property stated in Corollary 1.1. □
Remark 2.1 By the moduli space theory, the angles of consecutive ends of 
will be close to
. We conjecture that these angles should be exactly equal to
. But this seems to be a difficult problem.
Acknowledgements
Y. Liu is partially supported by NSFC grant 11101141 and the Fundamental Research Funds for the Central Universities 13MS39.