Abstract and Applied Analysis

Volume 2014 (2014), Article ID 548301, 10 pages

http://dx.doi.org/10.1155/2014/548301

Research Article

Existence Results for a Perturbed Problem Involving Fractional Laplacians

Yan Hu

College of Mathematics and Econometrics, Hunan University, Changsha 410082, China

Received 25 October 2013; Accepted 20 February 2014; Published 27 March 2014

Academic Editor: Julio D. Rossi

Copyright © 2014 Yan Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We extend the results of Cabre and Sire (2011) to show the existence of layer solutions of fractional Laplacians with perturbed nonlinearity in with . Here is a positive periodic perturbation for , and is the derivative of a balanced well potential . That is, satisfies First, for odd nonlinearity and for every , we prove that there exists a layer solution via the monotone iteration method. Besides, existence results are obtained by variational methods for and for more general nonlinearities. While the case remains open.

1. Introduction

We consider the classical fractional Laplacian

where and

Here . stands for the Cauchy principle value and is a positive constant depending only on .

In the whole paper we suppose that is a positive function and for every .

It is well known that the fractional Laplacian is a nonlocal operator which can be localized by adding a variable

where , , and

which is called the -extension of . is a positive constant multiplier depending only on .

Obviously, (1) and (3) have their own variational structures. For nonlocal equation (1), its corresponding nonlocal energy functional on any open interval is defined by

with prescribed boundary data outside of .

As for elliptic equation (3), its local energy functional is given by

with an open domain. Here is the primitive function of . The details for the fractional Laplacian can be seen in [1–6].

In view of the De Giorgi conjecture [7–9], Cabré and Sire [1–3] consider layer solutions of the autonomous equation

The layer solution of (7) is a strictly increasing solution with limits at . In [2, 3] the necessary and sufficient conditions for existence of layer solutions were presented by a Hamiltonian equality and a Modica-type estimate via variational methods, that is, and for every . In other words, (7) is an Allen-Cahn-type equation.

Equation (7) does not depend on the variable explicitly; layer solutions were proved to be strictly increasing and are the unique local minimizer by the sliding method. Furthermore, if is odd and , the layer solution is proved odd symmetric about some point where it equals zero. Many properties about the fractional Laplacian were established such as the regularity, Hopf lemma, harnack, and maximum principle; for details see [2, 3].

We give the definition of layer solutions of (1).

Definition 1. A function is said to be a layer solution of (1) if it satisfies

Different from (7), (1) depends on the variable explicitly; the sliding method cannot be used anymore and layer solutions of (1) have no monotonicity. Therefore, methods in [2, 3] for existence of layer solutions cannot be used here; our problem becomes complicated. In addition, the fractional Laplacian is a nonlocal case. All these cause some difficulties for existence results of layer solutions of (1).

In our previous work [10], assuming that was odd and was even, we proved the existence of layer solutions of (1) by variational methods and a Liouville theorem for . The restriction on was due to no Liouville-type conclusion for . We also obtained a Hamiltonian equality and asymptotic estimates as for layer solutions. Furthermore, the asymptotic behavior of layer solutions as is also investigated.

In this paper, we still study existence results of layer solutions of (1). At first we assume that is odd and is even. For this case, we mainly consider the -extension of ; layer solutions are constructed by the subsupersolution technique for all .

Theorem 2. Suppose that and

(1) for every ;(2), in and in .

Obviously, if ,

If is even, then there exists a layer solution of (1) which is also odd,

For more general nonlinearities and , existence results are proved by variational methods and a careful energy comparison. This part needs some tricks.

Theorem 3. Let . Let be the primitive function of and

(a) for , ;(b) and .

Then, there exists a layer solution of (1) for some ,

Remark 4. From the assumption above, there exists a such that is strictly increasing in and decreasing in , for .

For , the global energy may be infinite. The desired asymptotic behavior of layer solutions at infinity, that is, as , cannot be obtained. This case is open.

The paper is organized as follows. In Section 2, we prove existence results of layer solutions of (1) under odd assumptions of . Section 3 is devoted to the Proof of Theorem 3. Finally, we give some results about regularity and gradient estimates and a Hopf lemma about (1) and (3) in the appendix.

For convenience, we give some notations.

One has

Let ,

where

is a seminorm.

2. Odd Nonlinearities

The main aim of this section is to solve existence problem for odd nonlinearities. We assume that satisfies assumptions in Theorem 2. For example, , the nonlinearity of Allen-Cahn equation satisfies this condition. Here we construct a layer solution of (1) by the subsupersolution method. For this purpose, let us recall results in [3].

Lemma 5. Let , where , such that . Then there exists a solution of

such that in and if and only if

If in addition and , then this solution is unique up to translations.

As a consequence, if is odd and , then the solution is odd with respect to some point. That is, for some .

Remark 6. Denote to be the odd solution of (15) such that . Define and ; they are a supersolution and a subsolution of (1) in . Indeed, by simple calculations,

Furthermore, in by monotonicity of and the fact that .

In order to prove Theorem 2, we give the following two lemmas.

Lemma 7. Let and be the -extensions of and , respectively, in ; then for and . Furthermore, one has

Proof. For every , , the -extension of can be expressed by

If , for some ,

Thus, and .

In addition, if ,

Thus, for and ,

since is increasing.

Equation (18) is the consequence of direct calculations; the proof of this lemma is complete.

Now let be a Lipschitz bounded domain; define the energy functional

where is a constant and is a nonnegative function.

Lemma 8. Given for some . Let , on . Then there exists a minimizer such that for some and satisfies the Euler-Lagrange equation:

Proof. One has

Obviously, ; we now extend it by zeroes outside of in , and by the trace theorem and a compact imbedding theorem [11–13], for every . Therefore, is well defined, bounded below, and coercive in ; there is a minimizer denoted again by . By variational calculations,

in weak sense. Furthermore, for some by regularity discussion.

Now we come to prove Theorem 2.

Proof of Theorem 2. By Lemma 7, and are a supersolution and a subsolution of (3) in , respectively. Let and . Consider the mixed-boundary problem

where . By Lemma 8, there is a solution for some , and further in by the strong maximum principle and Hopf lemma.

Denote ,

Again, by the strong maximum principle and Hopf lemma, in .

Now we start the iteration procedure. Given and such that in , the problem

has a solution for some by Lemma 8 and in by the strong maximum principle and Hopf lemma.

Let ,

Again by the strong maximum principle and Hopf lemma, in .

Assume that in for some . Let ,

in by maximum principle and Hopf lemma.

Thus there is a sequence , , by monotone convergence theorem and

in and where does not depend on by regularity discussion.

Up to a subsequence in , in as , and

in by the strong maximum principle and Hopf lemma. Therefore as from asymptotic behavior of and as .

We define for and ; then

and as . The desired solution is achieved. We complete the proof.

3. Nonsymmetric Nonlinearities

In this section, we deal with more general situations; that is, :

(1), for ;(2), .

For simplicity we can assume that by adding a constant.

Consider the nonlocal energy functional defined on an open interval ,

where with prescribed boundary data outside of .

Define a nondecreasing competitor

If , by direct calculations,

where is a positive constant depending only on (see [6]). In order to prove Theorem 3, we first give several lemmas.

Lemma 9. Let with . Let , for and for , . Then, for every , there is a minimizer of ,

and for some .

Proof. We can change the values of for such that has the linear growth there which is denoted by . Given the admissible set

Clearly and is an element of them.

Consider the energy functional

If the minimizer of in satisfies , then and ; that is, is also a minimizer of in .

Obviously, the energy is nonincreasing by cutting at and

is well defined, bounded below, and coercive in . Thus, there is a minimizer ,

by (37).

Lemma 10. Let . Let be the minimizer of with ; then

for any and , where as .

Proof. Denote . Clearly, as and as .

One has

Denote

here we use the periodic of ;

as .

Therefore,

Since is the minimizer of ,

Let

as .

By (48)-(49) and the fact that is the minimizer in ,

Therefore,

where as . The proof of Lemma 10 is completed.

It is noticed that for some independent of ; then by the canonical diagonal process, up to a subsequence, in as and

For , we have the following.

Lemma 11. is a local minimizer of the nonlocal energy; that is, for every and for any ,

Proof. Since in ,

For and , ,

Thus, by the Dominated convergence theorem and locally uniformly convergence,

By (54) and the above,

as . The same discussion leads to the fact that

as and for any .

Therefore, (53) follows from Lemma 10, and

Furthermore by the strong maximum principle.

Let be the minimizer of in . By continuity, there is a such that and for .

Lemma 12. One has as .

Proof. To check this, we suppose by contradiction that for some sequence . Up to a subsequence, as for some constant .

One has

for and large enough. This contradiction verifies our conclusion.

Now we start to prove our conclusion.

Proof of Theorem 3. Let for , where is denoted by the integer part of . And satisfies

Up to a subsequence which is denoted again by , in as and

Since and , for the above subsequence, there is a sub-subsequence such that , , and for .

One has

By Fatou’s lemma and Lemma 12, and .

We claim that as . Indeed, if there exists a sequence such that , by the regularity of , there must be a ball such that in for some ,

Therefore, or as . The proof as is similar. Since for , as .

In the following we exclude the case that as .

Assume that as by contradiction. Denote for some . by the strong maximum principle.

One has

. Therefore by Remark 4.

Choose a small positive constant such that . Denote and ; and are well defined since as .

Denote

If is large enough, , , , , , and are all well defined. By local uniform convergence, and :

for some positive constant independent of , .

For simplicity, denote , , , and ; then .

Define

We compare the energy of and of :

Since as , as .

One has

as .

Since in and in ,

Therefore, for large enough. This contradiction deduces the fact that as . is the desired solution.

Appendix

In this appendix we state several regularity results involving the nonlocal equation (1), the local equation (3), and a Hopf lemma; their proof can be found in [2, 3, 6].

Proposition A.1. Let and with . Then, any bounded solution of (1)

is for some depending only on and .

Furthermore, given there exists depending only on , , and and hence independent of —such that, for every ,

for some depending only on , , , , and , not on . In addition, the function , where is the Poisson kernel , satisfies

If , the constant is independent of and only dependent on the quantities as the above mentioned.

We remark that the above results are valid for solutions of (A.1) in bounded domains.

Proposition A.2. Let and . Let for some and be a weak solution of

Then, there exists depending only on , , , , and also (for ) and (for ), such that

Lemma A.3 (Hopf Principle). Let and consider the cylinder where is the ball of center and radius in . Let satisfy

Then,

In addition, if , then

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research has been supported by National Natural Science Foundation of China (Grant no. 11371128).

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