- Open Access
Layer solutions for a class of semilinear elliptic equations involving fractional Laplacians
© Hu; licensee Springer. 2014
- Received: 4 November 2013
- Accepted: 4 February 2014
- Published: 17 February 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.
- fractional Laplacian
- layer solutions
- local minimizers
Here P.V. stands for the Cauchy principle value and is a positive constant multiplier depending only on s.
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.
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.
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.
is 1-periodic, even, not constant and positive; denote and ;
for any , , in and in .
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).
for , ;
for some constants .
Finally we investigate the asymptotic behavior of as and obtain a local elliptic equation, which is stated as follows.
For convenience of the presentation we will use C for a general positive constant; such a C is usually different in different contexts.
In this paper, we mainly study the extension equation (1.2). To make our problems clear, we present several properties of layer solutions.
Proof Our proof uses the invariance of the problem under periodic translations in x and a compactness argument.
and it follows that for every .
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.
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 .
where will be specified later, and .
We use (2.5) in the first inequality above.
for large . This contradiction leads to for all . By the same discussion, for all . Thus we complete the proof. □
Proposition 2.1 (Hamiltonian equality)
where t is some point between and 1.
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)
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.
Then a.e. in .
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.
for every such that on in the weak sense.
and . Up to an additive constant, in .
If has an absolute minimizer u in , the statement of Lemma 3.1 is proved.
for if or for any if . Moreover, .
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).
is an even positive function, ,
for any , , in and in .
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 .
and for . Thus for and . Obviously is still a minimizer of .
and by the Hopf maximum principle .
Step 2. We show that there exists at least a subsequence such that .
for any and for any odd function with and on in the weak sense.
for every and . Our claim is proved.
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.
Here the constant C does not depend on R, we use the gradient estimates (see ) in the second line from the bottom.
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., .
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 .
where is some point between and 1.
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 .
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.
Here c and C maybe different from above. □
In this section we prove Theorem 1.3, which consists of two lemmas.
where and . Obviously as .
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 .
By (4.15), (4.19)-(4.22), our claim is proved.
Claim 2. 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 .
for some .
for large enough. Therefore as , by odd symmetry, as , i.e., is a layer solution of the local elliptic equation (4.13).
Proof of Theorem 1.3 It follows from Lemmas 4.1 and 4.2. □
This research has been supported by National Natural Science Foundation of China (Grant No. 11371128).
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