Existence Results for a Perturbed Problem Involving Fractional Laplacians
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.
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 .
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 , 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 .
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 .
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 ,