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Research ArticleFractional Hardy–Sobolev Inequalities with Magnetic Fields
Min Liu ,1 Fengli Jiang,1 and Zhenyu Guo 2
1School of Sciences, Liaoning Shihua University, Fushun 113001, China2School of Mathematics, Liaoning Normal University, Dalian 116029, China
Correspondence should be addressed to Zhenyu Guo; [email protected]
Received 12 October 2019; Accepted 18 November 2019; Published 12 December 2019
Academic Editor: Jacopo Bellazzini
Copyright © 2019 Min Liu et al. �is 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.
A fractional Hardy–Sobolev inequality with a magnetic �eld is studied in the present paper. Under appropriate conditions, the achievement of the best constant of the fractional magnetic Hardy–Sobolev inequality is established.
1. Introduction
During the last decades, researchers paid more and more attention to the study of Sobolev inequalities and Hardy–Sobolev inequalities, including the fractional case and mag-netic case, see e.g. [1–9].
It is well known that the sharp constant of the embedding �1,2(ℝ�) �→ �2
∗(ℝ�) is attained (see [10]), where 2∗ := 2�/(� − 2)
is the Sobolev critical exponent and �1,2(ℝ�) ={� ∈ �2
∗(ℝ�) : |∇�| ∈ �2(ℝ�)}. �at is,
is achieved by the so-called Aubin–Talenti instanton (cf. [11, 12]) ��,�0 ∈ �
1,2(ℝ�) de�ned by
where � > 0, �0 ∈ ℝ�. Moreover, ��,�0 is a positive solution of −Δ� = |�|2
∗−2� in ℝ�,
and {��,�0 : � > 0, �0 ∈ ℝ�} contains all positive solutions of
−Δ� = |�|2∗−2� in ℝ�.
For the best constant of the Hardy–Sobolev inequality
where 0 < � < 2 and 2∗(�) := 2(� − �)/(� − 2) is the Hardy–Sobolev critical exponent, it follows from [13] that �� is achieved by functions of the form
where � > 0. �e function �� is a positive solution of −Δ� = |�|2
∗(�)−2�/|�|� in ℝ�, and moreover,
For the fractional Sobolev inequality, consider the Hilbert space ��(ℝ�) de�ned as Gagliardo seminorm
where 0 < � < 1, 2∗� := 2�/(� − 2�) is the fractional Sobolev critical exponent and the norm
(1)� := inf
�∈�1,2(ℝ�)� ̸=0
∫ℝ� |∇�|2d�
(∫ℝ� |�|2∗d�)2/2
∗
(2)��,�0(�) := (�(� − 2))(�−2)/4( �
�2 + ����� − �0����2)(�−2)/2
,
(3)∫ℝ������∇��,�0�����2d� = ∫
ℝ��������,�0�����2∗d� = ��/2,
(4)�� := inf
�∈�1,2(ℝ�)� ̸=0
∫ℝ� |∇�|2d�
(∫ℝ�(|�|2∗(�)/|�|�)d�)
2/2∗(�) ,
(5)��(�) := ((� − �)(� − 2))(�−2)/2(2−�)�(�−2)/2(�2−� + |�|2−�)
−(�−2)/(2−�),
(6)∫ℝ�����∇������2d� = ∫
ℝ�
����������2∗(�)
|�|�d� = (��)
(�−�)/(2−�).
(7)��(ℝ�) := {� ∈ �2∗� (ℝ�) :
�����(�) − �(�)����
����� − �����(�/2)+� ∈ �
2(ℝ2�)},
(8)‖�‖2�� :=��,�2 ∫ℝ2������(�) − �(�)
����2
����� − ������+2� d�d�
HindawiAdvances in Mathematical PhysicsVolume 2019, Article ID 6595961, 5 pageshttps://doi.org/10.1155/2019/6595961
Advances in Mathematical Physics2
is induced by the scalar product
Here ��,� is a dimensional constant precisely given by
De�ne the best constant for the fractional Sobolev inequality as
From [14–16], we see that �� is achieved by �̃(�) =�(�2 + |� − �0|)
−(�−2�)/2, that is, �� = ‖�̃‖2��/|�̃|22∗�. Normali zing
�̃ by |�̃|2∗�, we get that � = �̃/|�̃|2∗� ful�lls
and � is a positive ground state solution of (−Δ)�� = ��|�|2∗� −2�
in ℝ� (see Lemma 2.12 in [17]). Denote
where �̂ = �1/(2∗� −2)
� � is a positive ground state solution of (−Δ)�� = |�|2
∗� −2� in ℝ� and
�en, Lemma 2.12 in [17] yields that ��,� solves (−Δ)�� = |�|2∗� −2�
in ℝ�. �e fractional Hardy–Sobolev inequality
was considered in [18, 19], where 2∗� (�) := 2(� − �)/(� − 2�)is the fractional Hardy–Sobolev critical exponent. Marano and Mosconi [18] proved that ��,� is achieved by a optimizer � ∈ ��(ℝ�), whose asymptotic behavior is
For the magnetic Hardy–Sobolev inequality, we regard the range of function as ℂ, that is, � : ℝ� → ℂ, � ≥ 3, � = (�1, . . . , ��) : ℝ� → ℝ� is a magnetic vector potential. Setting −Δ� := (−�∇ + �)2, ∇� := ∇ + ��, and
then −Δ�� = −Δ� − ��div� − �� ⋅ ∇� + |�|2� and �1,2� (ℝ�,ℂ) is the Hilbert space obtained as the closures of �∞� (ℝ�,ℂ) with respect to scaler product
(9)⟨�, v⟩�� :=��,�2 ∫ℝ2�(�(�) − �(�))(v(�) − v(�))
����� − ������+2� d�d�.
(10)��,� = (∫ℝ�1 − cos �1����������+2� ��)
−1
.
(11)�� := inf
�∈��(ℝ�)� ̸=0
‖�‖2��
(∫ℝ� |�|2∗� d�)2/2
∗�.
(12)�� = inf
�∈��(ℝ�)(�)2∗� =1
‖�‖2�� = ‖�‖2��
(13)��,�(�) = �−(�−2�)/2�̂(�� ),
(14)‖�̂‖2�� = ∫ℝ�|�̂|2
∗� d� = (��)
�/2�.
(15)��,� := inf
�∈��(ℝ�)� ̸=0
‖�‖2��
(∫ℝ�(|�|2∗� (�)/|�|�)d�)
2/2∗� (�),
(16)�(�) ≃ |�|2�−� as |�|→ +∞.
(17)�1,2� (ℝ�,ℂ) := {� ∈ �2∗(ℝ�,ℂ) : ����∇��
���� ∈ �2(ℝ�)},
where the bar denotes complex conjugation. De�ne
where 2∗(�) := 2(� − �)/(� − 2) is the Hardy–Sobolev critical exponent. �en, by �eorem 1.1 in [20], we see that if � ∈ ��
loc(ℝ�,ℝ�), then ��,� is attained by some � ∈ �1,2�
(ℝ�, ℂ)\{0} if and only if curl � ≡ 0, where curl � is the usual curl operator for � = 3 and the � ×� skew-symmetric matrix with entries ��� = ���� − ��� � for � ≥ 4.
In our paper, we consider a fractional Hardy–Sobolev ine-quality with a magnetic �eld. To show our question, we �rst intro-duce the fractional magnetic Sobolev space ���(ℝ�,ℂ), which is the completion of �∞� (ℝ�,ℂ) with respect to the so-called frac-tional magnetic Gagliardo seminorm [⋅]��� given by
where ��,� is given by (10), � : ℝ� → ℂ, 0 < � < 1,� ≥ 3, and � = (�1, . . . , ��) : ℝ� → ℝ� is a magnetic vector potential. �e scalar product in ���(ℝ�,ℂ) is
Although [⋅]��� is a seminorm, by fractional magnetic Sobolev embeddings (see Lemma 3.5 in [21]), we can view [⋅]��� as a norm ‖ ⋅ ‖��� := [⋅]��� in the space ���(ℝ�,ℂ). As in Propositions 2.1 and 2.2 in [21], we can verify that ���(ℝ�,ℂ) is a Hilbert space. ��(ℝ�, d�/|�|�) denotes the space of ��-integrable functions with respect to the measure d�/|�|�, endowed with norm
�e aim of the present paper is to investigate the following fractional magnetic Hardy–Sobolev inequality
where 0 < � < 1, 0 < � < 2� and 2∗� (�) := 2(� − �)/(� − 2�) is the fractional Hardy–Sobolev critical exponent. Problem (23) relates to the fractional magnetic Laplacian de�ned by
(18)Re(∫ℝ�∇�� ⋅ ∇�vd�),
(19)��,� := inf
�∈�1,2� (ℝ�)� ̸=0
∫ℝ�����∇������2d�
(∫ℝ�(|�|2∗(�)/|�|�)d�)
2/2∗(�) ,
(20)
[�]2��� :=��,�2 ∫ℝ2�
�������−�(�−�)⋅∫10�((1−�)�+��)d��(�) − �(�)
������2
����� − ������+2� d�d�,
(21)
⟨�, v⟩��� :=��,�2 Re∫ℝ2�
(�−�(�−�)⋅∫10�((1−�)�+��)d��(�) − �(�))����� − ������+2�
⋅ (�−�(�−�)⋅∫10�((1−�)�+��)d�v(�) − v(�))d�d�.
(22)|�|�,� := (∫ℝ�|�|�
|�|�d�)1/�
.
(23)��,�,� := inf
�∈���(ℝ�)� ̸=0
‖�‖2���|�|22∗� (�),�,
(24)(−Δ)���(�) = ��,� lim�→0+ ∫���(�)
�(�) − ��(�−�)⋅∫10�((1−�)�+��)d��(�)����� − ������+2� d�, � ∈ ℝ�.
3Advances in Mathematical Physics
For � = 3 and with mid-point prescription, the fractional mag-netic Laplacian was studied in [21]. In particular, d’Avenia and Squassina [21] considered the operator
Obviously, (24) can be regarded as an extension of the above-mentioned operator involving mid-point prescription. �e fractional magnetic Laplacian (−Δ)�� : ���(ℝ�,ℂ)→�−�� (ℝ�, ℂ) can also be de�ned by duality as
Denote
�en (23) can be characterized as:
Our main result is:
Theorem 1. If � : ℝ� → ℝ� is a continuous function with locally bounded gradient, then ��,�,� is achieved by a nonzero element ��,�,� ∈ ���(ℝ�, ℂ).
2. Proof of the Main Result
To prove �eorem 1, we need the following Lemma, which is obtained in [22].
Lemma 1 (Diamagnetic inequality). For any � ∈ ���(ℝ�, ℂ), we have
and
which means |�| ∈ ��(ℝ�, ℝ).By the fractional Hardy–Sobolev inequality (15) and
Lemma 1, we have the following lemma.
Lemma 2 (Fractional magnetic Sobolev embeddings). �e embedding
(25)
(−Δ)���(�) = ��,� lim�→0+ ∫���(�)�(�) − ��(�−�)⋅�((�+�)/2)�(�)
����� − ������+2� d�, � ∈ ℝ3.
(26)
⟨(−Δ)���, v⟩ :=��,�2 Re∫ℝ2�
(�−�(�−�)⋅∫10�((1−�)�+��)d��(�) − �(�))����� − ������+2�
⋅ (�−�(�−�)⋅∫10�((1−�)�+��)d�v(�) − v(�))d�d�
=��,�2 Re∫ℝ2�
(�(�) − ��(�−�)⋅∫10�((1−�)�+��)d��(�))����� − ������+2�
⋅ (v(�) − ��(�−�)⋅∫10�((1−�)�+��)d�v(�))d�d�.
(27)J = {� ∈ ���(ℝ�,ℂ) : |�|2∗� (�),� = 1}.
(28)��,�,� = inf�∈J ‖�‖2��� .
(29)
||�(�)| − |�(�)||
≤�������−�(�−�)⋅∫10�((1−�)�+��)d��(�) − �(�)
������ ��� �.�. �, � ∈ ℝ�
(30)‖|�|‖�� ≤ ‖�‖��� ,
is continuous.
Proof of �eorem 1. Since the best constant ��,� of fractional Hardy–Sobolev inequality (15) is achievable, we only need to show that
In fact, for any � > 0, there exists � ∈ �∞� (ℝ�,ℝ) such that
Similarly to Lemma 4.6 in [21], for any � > 0, consider the scaling
Substituting � = �� and � = ��, we derive that
and the following invariance of scaling:
Noticing that |� − v|2 = |�|2 + |v|2 − 2Re(�v) for �, v ∈ ℂ, we have
(31)���(ℝ�, ℂ) �→ �2∗� (�)(ℝ�, ℂ)
(32)��,�,� = ��,�.
(33)‖�‖2�� ≤ ��,� + � and |�|2∗� (�),� = 1.
(34)��(�) := �(2�−�)/2�(�� ), � ∈ ℝ
�.
(35)
����������2��� =��,�2 ∫ℝ2��������−�(�−�)⋅∫10�((1−�)�+��)d���(�) − ��(�)
������2
����� − ������+2�d�d�
=��,�2 ∫ℝ2��2�−��������
−��(�−�)⋅∫10�[�((1−�)�+��)]d��(�) − �(�)������2
��+2�|� − �|�+2��2�d�d�
=��,�2 ∫ℝ2��������−��(�−�)⋅∫10�[�((1−�)�+��)]d��(�) − �(�)������
2
����� − ������+2�d�d�
(36)
����������2�� =��,�2 ∫ℝ2�������(�) − ��(�)����2����� − ������+2�
d�d�
=��,�2 ∫ℝ2��2�−�������(�) − ��(�)����2��+2�|� − �|�+2�
�2�d�d� = ‖�‖2�� ,
(37)
����������2∗� (�)2∗� (�),�= ∫ℝ��((2�−�)/2)⋅(2(�−�)/(�−2�))|�(�/�)|2
∗� (�)
|�|�d�
= ∫ℝ���−�|�(�)|2
∗� (�)
��|�|���d� = |�|2
∗� (�)2∗� (�),�.
(38)
����������2��� − ‖�‖2��
=��,�2 ∫ℝ2��������−��(�−�)⋅∫10�[�((1−�)�+��)]d��(�) − �(�)������
2
����� − ������+2�d�d�
−��,�2 ∫ℝ2������(�) − �(�)����2����� − ������+2�
d�d�
=��,�2 ∫ℝ2�2Re((1 − �−��(�−�)⋅∫10�[�((1−�)�+��)]d�)�(�)�(�))
����� − ������+2�d�d�
= ��,�∫ℝ2�
(1 − cos (�(� − �) ⋅ ∫10[�((1 − �)� + ��)]d�))����� − ������+2��(�)�(�)d�d�
=: ��,�∫ℝ2�Υ�(�, �)d�d� = ��,�∫
�×�Υ�(�, �)d�d�,
Advances in Mathematical Physics4
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where � is compact support of �. Obviously, Υ�(�, �) → 0 a.e. in ℝ2� as �→ 0. Since � is locally bounded, for �, � ∈ �, we have
�en, there exists � > 0 such that for �, � ∈ �,
De�ne
�en, |Υ�(�, �)| ≤ �(�, �) for �, � ∈ �. Since
where � may be di¯erent from line to line, we get that � ∈ �1(� ×�). By Lebesgue’s Dominated Convergence �eorem, we see that lim�→0‖��‖2��� = ‖�‖
2��0. �en, it follows
from (33) that
which means that ��,�,� ≤ ��,�. �e opposite inequality holds also because of Lemma 1. �us, (32) holds, which completes the proof of �eorem 1. ☐
Data Availability
�e [inequality] data used to support the �ndings of this study have been deposited in the [references 21 and 22] repository.
Conflicts of Interest
�e authors declare that they have no con¨icts of interest.
Acknowledgments
Supported by NSFC (11701248), NSFLN (20180540028) and DSRFLNNU (BS2018L004).
(39)1 − cos (�(� − �) ⋅ ∫
1
0�[�((1 − �)� + ��)]d�) ≤ ������ − �
����2.
(40)����Υ�(�, �)���� ≤{{{
�|�−�|�−2+2� , if
����� − ����� < 1,�|�−�|�+2� , if
����� − ����� ≥ 1.
(41)�(�, �) := {{{
�|�−�|�−2+2� , if
����� − ����� < 1,�|�−�|�+2� , if
����� − ����� ≥ 1.
(42)
∫�×��(�, �)d�d� = ∫
(�×�)∩{|�−�|<1}�(�, �)d�d�
+ ∫(�×�)∩{|�−�|≥1}
�(�, �)d�d�
≤ �∫{|�|<1}
1|�|�−2+2
d� + �∫{|�|≥1}
1|�|�+2
d�
= �∫1
0
1��−2+2 ��−1d� + �∫
+∞
1
1��+2 ��−1d�
< +∞,
(43)��,�,� ≤ lim�→0����������2��� = ‖�‖2�� ≤ ��,� + �,
5Advances in Mathematical Physics
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