1. Introduction
We consider the Cauchy problem for the fractional Schrödinger equation with a combined non-linearity
Here and hereafter
,
refers to the attractive or repulsive regime.
provides an unbounded inhomogeneous term
. The exponents of the source terms are
and
. The fractional Laplacian operator is defined using the Fourier transform, as follows
The fractional Schrödinger problem was first discovered by Laskin [
1,
2] as a result of extending the Feynmann path integral, from the Brownian-like to L
vy-like quantum mechanical paths. It also appears in the continuum limit of discrete models with long-range interactions (see [
3]) and in the description of Boson stars as well as in water wave dynamics (see [
4,
5]). In nonlinear optics and plasma physics, the Cauchy problem for the NLS with the inhomogeneous nonlinearity model the beam propagation in an inhomogeneous medium (see [
6]). From a mathematical point of view, the problem (
1) has no scaling invariance. But the term
is the
-critical nonlinearity due to the fact that the
-norm and (
1) without the inhomogeneous term are invariant under the scaling symmetry
. The term
is the
-critical nonlinearity because the
-norm and (
1) are invariant under the scaling symmetry
when
. For simplification, the term
is called the
-critical nonlinearity when
and the term
is also called the
-critical nonlinearity when
.
Over the past decade, there has been a great deal of interest in studying the fractional Schrödinger problem
For the well-posedness of (
2), we refer the reader to [
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21] and the references therein. For the
-critical case, a series of studies dealing with problem (
2), we refer to [
22,
23,
24,
25,
26,
27] and the references therein. Recently, in paper [
28], we obtained a sharp threshold of global existence versus finite time blow-up dichotomy for mass-super-critical and energy sub-critical radial solutions for (
2) when
in conditions of both radial initial data and non-initial data. In addition, we also investigated the decay of global solutions by Morawetz estimate for radial initial data. For the limiting profile of infinite time blow-up solutions at the blow-up time, in [
25,
26,
29,
30,
31,
32], the authors investigated the
concentration phenomenon for different models.
Inspired by the above literature, in this paper, we study the limiting profile of the finite time blow-up solutions at the blow-up time for problem (
1). Specifically, we focus on the nonexistence of a strong limit for the non-radial initial data and
concentration phenomenon for the radial initial data. As two different nonlinear terms are included, the limiting profile of the finite time blow-up solutions at the blow-up time is different and interesting. For
and
, we investigate the nonexistence of a strong limit for the non-radial initial data. The difficulty in dealing with this problem stems from the effect of singular coefficient
, which is more complex than the classical equation (see [
26]). For the radial initial data, we can obtain better results, i.e., the
concentration phenomenon, which can be proven by the profile decomposition. For
and
, we find that the nonexistence of a strong limit for the non-radial initial data is the same as the first case, but the
concentration phenomenon for the radial initial data has changed. This is attributed to the effect of the singular coefficient
. We cannot use the profile decomposition and have to find some new estimates to deal with the problem. As a result, we only find that the mass of the solution is concentrated near the origin. These results fully illustrate the effect of the inhomogeneous term on the problem. Our main results can be represented in the following
Table 1.
The plan of this work is as follows: in the
Section 2 we provide some local well-posedness results, a Gagliardo–Nirenberg inequality, and a profile decomposition of the bounded sequence in
. In
Section 3, we provide the proof of nonexistence of a strong limit when
for a non-radial solution. For a radial solution, using a compactness lemma and profile decomposition, we prove the
concentration phenomenon in
Section 4. And when
and
, we investigate the nonexistence of a strong limit for a non-radial solution and
concentration phenomenon for a radial solution in
Section 5 and
Section 6.
2. Preliminaries
Next, we recall some preliminary results that will be used later. Firstly, let us recall the local theory for the problem (
1). Note that the parameter
is fractional admissible if
Using Strichartz estimates and the contraction mapping argument, we can prove the well-posedness for (
1) in the energy space
, as in [
28].
Proposition 1 (Radial
LWP)
. Let , , , and . LetThen, for any radial, there exist and a unique local solution to (1) satisfying Moreover, the following properties hold: for any fractional admissible pair .
If , then as .
The solution enjoys conservation of mass and energy, i.e., for all ,
In fact, the reader can find the proof of this result in [
28]. In this case, the Strichartz estimates have no loss of derivatives. We thus obtain a better local well-posedness result compared with the one in the following proposition.
Proposition 2 (Non-radial
LWP)
. Let , and be such thatThen, for any , there exist and a unique local solution to (1) satisfying for some when and some when . Moreover, the following properties hold: If , then as .
The solution enjoys conservation of mass and energy, i.e., and for all .
We refer the reader to [9] (or [10]) for the proof of this result. Note that in the case of non-radial initial data, Strichartz estimates have a loss of derivatives. However, the loss of derivatives can be compensated by using the Sobolev embedding. Secondly, we collect some classical results needed for this manuscript. We start with a sharp Gagliardo–Nirenberg-type inequality established in [
7,
16,
22].
Lemma 1. Let and , the Gagliardo–Nirenberg inequalityholds, and the constant is explicitly provided bywhere Q is the nonnegative, nontrivial solution of the equationMoreover, the solution Q satisfies the following relationsand Remark 1. When , we haveand the constant is explicitly provided bywhere R is the nonnegative, radially symmetric, decreasing solution of the equationMoreover, the solution R satisfies the following relationsand Remark 2. In fact, in this paper, we also use various other special cases of the Gagliardo–Nirenberg inequality, such as when and when . The details will not be described here.
Finally, we provide the profile decomposition of the bounded sequences in .
Lemma 2 (Profile Decomposition). Let and . If is a bounded sequence in , then there exists a subsequence of (also denoted by ), a family of sequence and a family of functions satisfying the following:
- (i)
- (ii)
For every and every , can be decomposed as
Moreover, as , the following relations hold,where is such that . Remark 3. The profile decomposition arguments were proposed by Gérard in [33], Hmidi and Keraani in [30], and Giampiero and Adriano in [34]. The proof of Lemma 2.6 is similar with Proposition 3.1 in [30]. Note that a spatial sequence
is called orthogonal if and only if (
12) is true for every
.
4. Radial Solution for , and
In this section, our purpose is to show the
concentration phenomenon for the problem (
15) if
is radial. In this case, we will obtain a stronger conclusion than Theorem (1).
Theorem 2. Let , . If the solution of (15) blows up in finite time . Let be a real-valued nonnegative function defined on , satisfying as . Then there exists such thatwhere R is the ground state solution of (8). To prove (2), we first prove the following compactness lemma. It implies that if a bounded sequence in some strong norm (e.g.,
) does not converge weakly to zero in a weaker norm, then this can be attributed to the sequence containing a profile of concentration. Although this lemma has been proven in many literature works (see [
29]), we provide the main steps of its proof for the sake of completeness.
Lemma 3 (Compactness Lemma)
. Assume and . Let be a bounded family of such thatThen, up to a subsequence, for some satisfying where R is the ground state solution of (8). Proof. Due to the profile decomposition (
12) and the sharp Gagliardo–Nirenberg inequality, we have
Applying the profile decomposition again, we obtain
Hence, it follows from (
18) and (
19) that
By the convergence of series
, there exists
, such that
and
Finally, a change of spatial translation and profile decomposition provides
Using the pairwise orthogonality of
, we have
, which converges weakly to 0 for every
in
. And
converges weakly to
in
, where
is the weak limit of
. However,
Following the uniqueness of the weak limit, we obtain
for all
and
converges weakly to
in
. This completes the proof. □
Remark 4. The lower bound on the -norm of V is optimal. In fact, if we take , then we obtain equality.
By applying the compactness lemma, we can prove Theorem 2.
Proof. In the sequel, we use the following notation:
Let
be an any time sequence such that
,
and
. Then, the family
satisfies
Observe that
Furthermore, by the Gagliardo–Nirenberg inequality (Lemma 2.2), we have
Note that
and
. Hence, we obtain
which implies
. Now, set
and
, we can apply the compactness lemma for the sequence
in
. And we obtain
and
such that, up to a subsequence,
converges weakly to
V in
. The weak limit
V satisfies
. Due to
as
, we have
Hence, for every
, there exists
such that for every
,
. So, we have
where we use the arbitrariness of
. And, as
is arbitrary, we can obtain
Using the continuity of this function
for every
and the fact
. There exists a function
such that for every
This and (
24) yield (
17). The proof is completed. □
Remark 5. Indeed, we can choose with . It is obvious that satisfies the condition in Theorem 2.
Next, using the uniqueness of radial solution of problem (
8), we study the limiting profile of the radial blow-up solutions of problem (
15).
Theorem 3. Let , , . If the solution of (15) blows up in finite time . Then, there exists and such thatwhere R is the ground state solution of (8) and . As the proof of the Theorem 3 is similar to the one in [
29,
30], we omit it.
Remark 6. We should note that when , there is a gap between the upper bound of p and the -critical index . Unfortunately, we do not have limit profile results when . This situation does not occur in the general fractional Schrödinger equation with a combined non-linearity, i.e., .
6. Radial Solution for , and
In this section, our purpose is to show
concentration phenomenon for the problem (
25) if
is radial. Unlike Theorem 2, due to the presence of inhomogeneous terms, the mass concentrates at the origin when
. In a sense, we could consider that the origin is a blow-up point for the solution of (
25).
Theorem 5. Let , . If the solution of (25) blows up in finite time . Let be a real-valued nonnegative function defined on satisfying as . Then,where Q is the ground state solution of (4) when . To prove (5), we first prove the following compactness lemma. Unlike Lemma 3, we must look for some weighted compactness lemma as the presence of inhomogeneous terms. And to prove this lemma, we have to apply some different methods.
Lemma 4 (Weighted compactness lemma)
. Assume and . Let be a radial bounded family of such thatThen, up to a subsequence, for some radial function satisfying where Q is the ground state solution of (4) when . Proof. Indeed, as
is a bounded radial sequence in
, there exists
, such that up to a sequence,
weakly in
. Firstly, we claim that
as
. Indeed, let
, since
is bounded in
, we have for any
,
where we use the Sobolev embedding
. By choosing
sufficiently large, we have
On the other hand, we have
provided that
. The term
is bounded if
. Thus
and
. Next, we bound
where
By the Sobolev embedding
for any
and the fact that
strongly in
for any
, we are able to choose
so that (
30) holds. Indeed, we can choose
, which is small but close to
. Furthermore, due to
we see that (
30) is satisfied. As a consequence, we obtain
for
n when it is sufficiently large, Collecting (
29) and (
31), we prove (
28).
Therefore, the sharp Gagliardo–Nirenberg inequality yields
As
weakly in
, the semi-continuity of weak convergence implies
Hence
This completes the proof of Lemma 4. □
Thanks to Lemma 4, next, we provide the proof of the -concentration result.
Proof. Set
Let
be an any time sequence, such that
,
and
. Therefore, the family
satisfies
Note that
Furthermore, from the Gagliardo–Nirenberg inequality (Lemma 2.2), we have
As
and
. We obtain
which implies
Set
and
. Then, it follows from the compactness lemma that there exist
V and
such that, up to a subsequence,
with
Observe that
Then for every
, there exists
such that for every
,
. Hence, using (
38), we get
for every
, which means that
Because the sequence
is arbitrary, it follows from (
41) that
This completes the proof. □
Remark 7. Comparing Theorem 2 and Theorem 5, we can find the effect of the inhomogeneous on the concentration phenomenon. Affected by the singular coefficients , when inhomogeneous term dominate the equation, the mass of the finite time blow-up solution can only be concentrated near the origin.
Remark 8. Unfortunately, we only know the existence but not the uniqueness of the solution to the following equationIf we can obtain the uniqueness of radial solution of above problem, we can study the limiting profile of radial blow-up solutions of problem (25) similar to Theorem 3. Remark 9. When and , it is possible that q is larger than the critical exponent of the inhomogeneous term, but this does not affect the concentration phenomenon of the solution at the blow-up time.