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Layer solutions for a class of semilinear elliptic equations involving fractional Laplacians
Boundary Value Problems volume 2014, Article number: 41 (2014)
This paper is concerned with the nonlinear equation involving the fractional Laplacian: , , where , is a periodic, positive, even function and −f is the derivative of a double-well potential G. That is, (), , . We show the existence of layer solutions of the equation for and for some odd nonlinearities by variational methods, which is a bounded solution having the limits ±1 at ±∞. Asymptotic estimates for layer solutions as and the asymptotic behavior of them as are also obtained.
MSC:35B20, 35B40, 49J45, 82B26.
In this paper we study the fractional Laplacian
where , and is the fractional Laplacian defined by
Here P.V. stands for the Cauchy principle value and is a positive constant multiplier depending only on s.
The fractional Laplacian is a nonlocal operator which can be localized as
where , and . Moreover can be expressed by a Poisson kernel,
The necessary and sufficient conditions for the existence of one-dimensional layer solutions were given as
where . All these were obtained by a Hamiltonian equality and a Modica-type estimate for layer solutions. By the sliding method, the layer solution of (1.3) was proved to be the unique local minimizer which increases in x with values varying from −1 to 1. The regularity, Hopf principle, maximum principle as well as a Harnack inequality for (1.3) or for its extension equation (1.2) (in this case ) were given. Some of them will be used in our paper.
If b is not a constant and is periodic, the perturbed equation (1.1) becomes complicated. The aim of this paper is to study the layer solution of (1.1) with periodic perturbed nonlinearity.
Definition 1.1 A function () is said to be a layer solution of (1.1), if v solves (1.1),
Definition 1.2 A function is said to be a layer solution of (1.2), if u solves (1.2),
Namely, is the corresponding layer solution of (1.1).
Different from the unperturbed case (1.3), the inhomogeneous term depends explicitly on x in (1.1) and (1.2); the sliding method cannot be used and layer solutions of them have no monotonicity in the direction of x. The method for obtaining layer solutions in  and  cannot be used in our case directly; some difficulties need to be solved.
In the paper, we consider the extension problem (1.2). Obviously, (1.2) has a variational structure.
For , the norm is
The energy functional of u on Ω is given by
We state our main results in the following.
We show, via a Liouville result, the existence of layer solutions of (1.1) for and for some odd nonlinearities.
Theorem 1.1 Let . Assume that ():
is 1-periodic, even, not constant and positive; denote and ;
for any , , in and in .
Obviously, if ,
There exists a layer solution (for some ) of (1.1):
In addition, v is odd.
Furthermore we obtain asymptotic estimates of the layer solutions of (1.1) by comparing with a layer solution of the unperturbed equation (1.3).
Theorem 1.2 Let is positive. Let () satisfy
for , ;
If v is a layer solution of (1.1), then the following asymptotic estimates hold:
for some constants .
Finally we investigate the asymptotic behavior of as and obtain a local elliptic equation, which is stated as follows.
Theorem 1.3 Let . Let be a sequence of layer solutions of (1.1) in Theorem 1.1. Then, there exists a subsequence denoted again by converging locally uniformly to a function as , which is also a layer solution of the local elliptic equation
For convenience of the presentation we will use C for a general positive constant; such a C is usually different in different contexts.
2 Some preliminaries and properties
In this paper, we mainly study the extension equation (1.2). To make our problems clear, we present several properties of layer solutions.
Lemma 2.1 Let u be a bounded solution of (1.2),
with two constants . Then,
for every ;
Proof Our proof uses the invariance of the problem under periodic translations in x and a compactness argument.
Denote for . Since b is 1-periodic, still satisfies the equations
as . Then solves the equations
and it follows that for every .
Consider the Dirichlet problem
is the unique solution of (2.8) by Corollary 3.5 in . As a consequence, (2.2) and (2.5) are obvious. □
The following lemma is a necessary condition for a local minimizer of the energy functional ℰ.
Lemma 2.2 Let u be a local minimizer of the energy functional ℰ under perturbations in . That is, for any bounded Lipschitz domain and for any having compact support in such that ,
Proof To show (2.10), it is sufficient to prove that and for all . Suppose for some point by contradiction. For simplicity, assume that by adding a constant.
for some .
Let be a cut-off function with values in ,
where will be specified later, and .
Define . Let , then on and in . We have
We use (2.5) in the first inequality above.
Having chosen , by (2.11) and (2.12),
for large . This contradiction leads to for all . By the same discussion, for all . Thus we complete the proof. □
Lemma 2.3 Let be a solution of (1.2). Then for every , . In addition, the integral can be differentiated with respect to under the integral sign. We have
uniformly in . If u is a layer solution of (1.2),
Proposition 2.1 (Hamiltonian equality)
Let u be a layer solution of (1.2) for , i.e.,
For every , the Hamiltonian equality holds:
As a consequence,
Proof We note that the integral in (2.16) is well defined since and
where t is some point between and 1.
By Lemma 2.3, the left integral in (2.15) can be differentiated with respect to x,
In the second equality above we use the fact that (see ). We have
Let , the left of (2.17) converging to zero by (2.14); thus and (2.15) is proved. Letting , (2.16) is also obtained. □
To study asymptotic estimates of layer solutions of (1.1), we recall an explicit layer solution of the unperturbed problem (1.3).
Lemma 2.4 (, Theorem 3.1)
Let . For every , the function
is the layer solution to the fractional equation
for a nonlinearity which is odd and twice differentiable in and which satisfies
In addition, the following limits exist:
and, as a consequence,
3 Existence and asymptotic estimates
To prove the existence of layer solutions, we introduce a Liouville result where is required. This is the reason why we restrict ourselves to the case in Theorem 1.1.
Proposition 3.1 Let . Suppose u is a bounded nonnegative function which satisfies weakly the problem
Then a.e. in .
Proof Since , for . Let ξ be a smooth function with values in , in and outside of , . Multiplying (3.1) with and integrating by parts, we have that
Thus for some constant C independent of R. Let , . We deduce that and a.e. in . □
Next we prove an existence result about the local minimizer of ℰ.
Lemma 3.1 Let be a bounded Lipschitz domain. Let be a given function with ; b is a bounded positive function.
the energy functional admits a minimizer , which solves weakly
Moreover, u is a stable solution of (3.2), i.e.,
for every such that on in the weak sense.
Proof Consider the set , since . Denote
and . Up to an additive constant, in .
Consider the energy functional
If has an absolute minimizer u in , the statement of Lemma 3.1 is proved.
for if or for any if . Moreover, .
Since has linear growth at infinity, is well defined, bounded below and coercive in . There exists an absolute minimizer . By the first order variation, we have
Multiply with (3.5) and integrate in Ω,
Since , . Thus a.e. in Ω, i.e., a.e. in Ω. Similarly we also get a.e. in Ω. Hence . (3.2) follows from (3.5), and (3.3) comes from the second order variation of ℰ. □
Remark 3.1 Suppose that b is an even function, f and are odd with respect to x, with a slight modification we can also show that there is an odd minimizer in the admissible set .
Now we start to show the existence of layer solutions of (1.2).
Theorem 3.1 Let . Let and ():
is an even positive function, ,
for any , , in and in .
Then there exists a layer solution u of (1.2) in :
which is odd with respect to x, i.e., , and, for every ,
Furthermore, u is a local minimizer of the energy functional ℰ under odd perturbations in , and it is stable in the sense that
for every function with compact support in , and .
Proof The proof is divided into three parts. For simplicity, we make by adding a constant.
Step 1. We show that there exists a solution with values in of (3.6) which is odd with respect to the variable x for every .
Let and . Define the admissible set
By Remark 3.1, there is a minimizer in ,
and for . Thus for and . Obviously is still a minimizer of .
By the regularity results in , for some and the continuous module is uniform bounded. Up to a subsequence, , and in as for all . By the canonical diagonal procedure, u solves
and by the Hopf maximum principle .
Step 2. We show that there exists at least a subsequence such that .
First we claim that u is a local minimizer under odd perturbations in . That is,
for any and for any odd function with and on in the weak sense.
Let is odd with respect to x for every and . Since , for . We have
Let , and
for every and . Our claim is proved.
where . Therefore u is also a local minimizer of ℰ in with perturbations in , i.e.,
for any and for any with and on in weak sense.
Suppose for any sequence by contradiction. for some and . Hence for all and by the fact that .
Let . Let be a cut-off function with values 1 in and zeroes outside of , for some determined later.
Denote . Let , on . For ,
Here the constant C does not depend on R, we use the gradient estimates (see ) in the second line from the bottom.
On the other hand,
Choose , for large R. This contradiction leads to the result that there exists at least a sequence such that .
Step 3. We show that u is the layer solution, i.e., .
Let and . By the regularity results , up to a subsequence,
Define , we have
by Proposition 3.1, and or 1. Thus by step 2. That is, as . as is achieved by odd symmetry.
u is the desired layer solution. □
Proof of Theorem 1.1 It follows from Theorem 3.1; for the regularity of v see . □
Lastly we give asymptotic estimates for layer solutions of (1.1) as .
Proof of Theorem 1.2 Let v be a layer solution of (1.1),
where is some point between and 1.
Consider the layer solution of the unperturbed problem in Lemma 2.4,
with is some point between and 1.
Since , choose t large enough such that and choose such that for all .
Choose such that in , which can be done since , as .
. We have
Obviously, if , it is achieved at some point . Since in , from the first inequality of (3.16), which contradicts with the fact that
Therefore for given from above.
On the other hand, choose small such that and choose such that for all . Choose such that in .
and obviously .
If , it is only achieved at some point . Since in , from the first inequality of (3.17), which contradicts the fact that . Thus for some given from above.
by Lemma 2.4. Similarly,
Here c and C maybe different from above. □
4 Asymptotic as
In this section we prove Theorem 1.3, which consists of two lemmas.
Lemma 4.1 Let be a sequence of layer solutions of (1.1) in Theorem 1.1. Then there exists a subsequence denoted again by , converging locally uniformly to which solves the local elliptic equation
Proof Consider , the s-extension of , which solves
where and . Obviously as .
Let . Multiplying (4.2) with ξ and integrating in ,
Choose , and is a cut-off function which equals 1 in and 0 in , for some constant . Thus (4.3) can be rewritten as
By the regularity results in , the continuous module does not depend on s for . Up to a subsequence,
as (or equivalently ). Then
For the first integral in (4.4), we consider
for and small . Here we use the fact that locally uniformly in . We have
Since for and C independent of (see ),
Therefore, by (4.4), (4.5), (4.10), and (4.11),
in the weak sense (). By the regularity theory of elliptic equations, is also a classical solution of (4.13). □
Lemma 4.2 is also a layer solution of (4.1), i.e., as .
Proof Claim 1. is a local minimizer in under perturbations in . That is,
for any bounded open interval and for any such that , where
Indeed, for the test function ξ in Lemma 4.1 with the additional property that , we have
As in the discussions in Lemma 4.1, let , and we have
By (4.15), (4.19)-(4.22), our claim is proved.
Claim 2. as .
Define for , up to a subsequence, in as ,
Since and , in ℝ and or 1.
We show that or 1 as . Indeed, if there are two sequences and such that and as , there must exist such that .
Denote where is the integer part of . and up to a subsequence in , solves equation (4.23). Therefore or 1. For the above subsequence, there is a subsubsequence such that as and . This contradiction verifies or 1 as .
To check as , suppose that as by contradiction. Then,
for some .
Let , if and if where η will be determined later, . Define , then . We have
for large enough. Therefore as , by odd symmetry, as , i.e., is a layer solution of the local elliptic equation (4.13).
By the Hamiltonian equality (2.15),
Proof of Theorem 1.3 It follows from Lemmas 4.1 and 4.2. □
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This research has been supported by National Natural Science Foundation of China (Grant No. 11371128).
The author declares that she has no competing interests.
The author read and approved the final manuscript.