In this paper we are concerned with the existence of solutions to the problem
where , and is a smooth function such that . Here is the fractional Laplacian and it can be defined via Fourier transform by
for u belonging to the Schwartz space .
Problems like (1.1)) are motivated by the study of standing waves solutions of the fractional Schrödinger equation
This equation has been proposed by Laskin [23, 24] as a result of expanding the Feynman path integral, from the Brownian-like to the Lévy-like quantum mechanical paths. After that many papers appeared investigating existence, multiplicity and behavior of solutions to fractional Schrödinger equations; see [4, 5, 6, 17, 18, 19, 22, 26, 29] and references therein.
More in general, problems involving fractional operators are receiving a special attention in these last years; indeed, fractional spaces and nonlocal equations have great applications in many different fields, such as optimization, finance, continuum mechanics, phase transition phenomena, population dynamics, multiple scattering, minimal surfaces and game theory, as they are the typical outcome of stochastically stabilization of Lévy processes. The interested reader may consult [16, 27] and references therein, where a more extensive bibliography and an introduction to the subject are given.
Under general assumptions on g, they proved that there are no finite energy solutions to (1.2)) if , while if or , then it is possible to show the existence of a solution to (1.2)) via constraint minimization. The case is called null mass case and it is related to the Yang–Mills equation; see [20, 21].
Let us note that the case seems to be more intricate than , since unless g satisfies the condition , the energy functional associated to (1.2)) may be infinite on a dense set of points in and hence cannot be Fréchet-differentiable on .
The question that naturally arises is whether or not the above classical existence results for equation (1.2)) can be extended in the nonlocal setting. When (in the case of positive mass), the existence of a ground state has been proved in  by combining the Struwe–Jeanjean monotonicity trick and the Pohozaev identity for the fractional Laplacian. Now, our aim is to investigate problem (1.1)) when , and behaves like for u small and for u large, with .
In order to state our result, we introduce the basic assumptions on the nonlinearity g. Here we will assume that is an odd -function verifying the following conditions:
for any and for some .
. There exist with such that
The assumptions for all , and (g2) imply the existence of such that
As a model for g we can take the function
where and c are constants which make the function .
Let us point out that, when , the assumptions (g1) and (g2) have been introduced in  to study positive solutions to a nonlinear field equation set in exterior domain. The authors studied (1.2)) in the Orlicz space which seems to be the natural framework for studying “zero mass” problems. Subsequently, their approach has been also used in [7, 9, 10] to study nonlinear Schrödinger equations in with bounded or vanishing potentials. Further results concerning zero mass problems can be found in [2, 3, 30].
The main result of this paper is the following.
Let , and g satisfies (g1) and (g2). Then there exists a positive solution to (1.1)) which is spherically symmetric and decreasing in .
To deal with problem (1.1)), we develop an energy minimization argument on a Nehari manifold.
More precisely, solutions to (1.1)) will be obtained by minimizing
on the Nehari manifold
In order to obtain the smoothness of the functional I, we introduce the Orlicz space related to the growth assumptions of g at zero and at infinity. Then, we show that , and by proving the compactness of the subspace of nonnegative radial decreasing functions of in , we deduce that the infimum of I on is achieved at some .
As far as we know the result presented here is new.
In this section we collect some preliminary results which will be useful in the sequel.
We denote by the completion of with respect to
where is the fractional Sobolev exponent.
Let us denote by the standard fractional Sobolev space, defined as the set of satisfying with the norm
We recall the following embeddings.
Theorem 2 ().
Let and . There exists a constant such that
In particular, is continuously embedded in for any , and compactly embedded for any .
For more details about fractional Sobolev spaces, we refer to .
We remark that the symmetric-decreasing rearrangement of a measurable function that vanishes at infinity (that is, for all ) is given by
where and is such that its volume is that of A. For standard properties of rearrangements of functions one can see .
Now, we establish the following fractional Polya–Szegö inequality:
Let be a nonnegative measurable function that vanishes at infinity, and let us denote by its symmetric-decreasing rearrangement. Assume that . Then
for . Since u vanishes at infinity, . In particular, since
for any . By Monotone Convergence Theorem we have
Now, by using the result in , we know that
We also prove the following useful lemma.
Let , , be a nonnegative radial decreasing function (that is, if ). Then
where is the Lebesgue measure of the unit sphere in .
For all we have, setting ,
Given , we define the space as the set of functions such that
with and . We recall (see ) that is a Banach space with respect to the norm
Moreover, coincides with the dual of . Then
where and are the conjugate exponent of p and q, respectively. In particular, the norm
is equivalent to . Actually, is an Orlicz space with N-function (see )
The following statements hold.
If , the following inequalities hold:
where and .
Let and set . Then is bounded in if and only if the sequences and are bounded.
is a bounded map from into .
The following statements hold.
If are bounded in , then is bounded in .
is a bounded map from into .
is a continuous map from into .
The map from in is bounded.
The functional defined by
is of class .
If the sequence converges to u in , then the sequence converges to .
By Lemma 2 (a) we have when . In fact, by using we find for any ,
At this point we are ready to prove the following result.
Let , and . Let us denote by the space of nonnegative radial decreasing functions in . Then is compactly embedded in .
We proceed as in the proof of [8, Lemma 3]. Let be a sequence in such that as
By (2.4) it follows that there exists a positive constant such that
Fix . By using (2.6) and , for big enough we get
for all . Now, we observe that since . In particular, we have
If R is sufficiently large, by (2.6) it follows that for any ,
Then for all ,
Let us observe that
Here for . Since and in , we obtain
so, in particular,
3 Proof of Theorem 3
This section is devoted to the proof of the main result of this paper. In order to obtain a solution to (1.1), we will look critical points of the following functional:
By using the results in Section 2, we can see that I is well defined on and I is a -functional.
Proof of Theorem 1.
We divide the proof into several steps.
Step 1: is a -manifold. By using (g1) we have, for any ,
Step 2: Given , there exists a unique such that and is the maximum for for . Fix and let
for . Then
Let us observe that is a minimum for h since and . Moreover, if is a critical point of h, then by (g1), we obtain that is a maximum for h because of
By using (g2), we get
Step 3: The dependence of on u is of class . Let us define the following operator:
for . By Lemma 2 we can see that and if is a point such that and , then by (g1)
By invoking the Implicit Function Theorem, we obtain that is and
Step 4: . Let be a minimizing sequence in . We assume by contradiction that converges to zero in . We set , hence we can write , where . Since the embedding is continuous, we deduce that is bounded in . Then, by , and Remark 1, we get
that is, a contradiction. Therefore .
Since we are looking for positive solutions to (1.1), we can assume that for .
Step 5: The infimum
is achieved. Let be a minimizing sequence for (3.2). Then, by (g1), it follows that
that is, is bounded in .
We claim that . Indeed, if , the minimizing sequence is such that , and by (3.3) we deduce that . This gives a contradiction because of Step 4.
Now, by using Theorem 3, we can note that , where is the symmetric-decreasing rearrangement of . Moreover, by the boundedness of , we can see that is bounded in .
In virtue of Theorem 4, the embedding is compact, so, up to a subsequence, we may assume that strongly in , and weakly in .
Let us observe that thanks to Theorem 3, so we do not know if belongs to the Nehari manifold . Then, for any there exists a unique such that and converges to some . By Step 3 it follows that is the maximum for when , so we get
Since , we obtain . It is clear that . Otherwise in , and by (3.4) we deduce that , which provides a contradiction in virtue of Step 4.
Now, we argue by contradiction in order to prove . If we assume that , by and (3.4) we have
so we can find such that . As a consequence,
In view of (g1), the map
is strictly increasing, so by this and (3.5) we get
which is a contradiction. This concludes the proof of Step 5.
Then, by applying the Lagrange multiplier rule, there exists such that
for any .
By testing in (3.6), and keeping in mind that for all (see Step 1), we deduce that , that is, and is a nontrivial solution to (1.1). Actually, by the strong maximum principle , we argue that is positive. ∎
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Published Online: 2016-09-20
Published in Print: 2018-08-01