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Research Article Fractional Hardy–Sobolev Inequalities with Magnetic Fields Min Liu , 1 Fengli Jiang, 1 and Zhenyu Guo 2 1 School of Sciences, Liaoning Shihua University, Fushun 113001, China 2 School 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 field 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 ) defined by where > 0, 0 ∈ℝ . Moreover, , 0 is a positive solution of -Δ = || 2 * -2 in , and { , 0 : > 0, 0 ∈ℝ } contains all positive solutions of -Δ = || 2 * -2 in . 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 , and moreover, For the fractional Sobolev inequality, consider the Hilbert space (ℝ ) defined as Gagliardo seminorm where 0 < < 1 , 2 * := 2/( - 2) is the fractional Sobolev critical exponent and the norm (1) := inf 1,2 (ℝ ) ̸ =0 |∇| 2 d (∫ || 2 * d) 2/2 * (2) , 0 () := (( - 2)) (-2)/4 ( 2 + - 0 2 ) (-2)/2 , (3) , 0 2 d =∫ , 0 2* d = /2 , (4) := inf 1,2 (ℝ ) ̸ =0 |∇| 2 d (∫ (|| 2 * () /|| )d) 2/2 * () , (5) () := (( - )( - 2)) (-2)/2(2-) (-2)/2 ( 2- + || 2- ) -(-2)/(2-) , (6) 2 d=∫ 2*() || d = ( ) (-)/(2-) . (7) (ℝ ) :={ 2 * (ℝ ) : () - () - (/2)+ 2 (ℝ 2 )}, (8) 2 := , 2 2 () - () 2 - +2 dd Hindawi Advances in Mathematical Physics Volume 2019, Article ID 6595961, 5 pages https://doi.org/10.1155/2019/6595961

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Page 1: Research Article - Hindawi Publishing CorporationAnnals of the Superior Normal School of Pisa—Science Class, vol. 18, no. 3, pp. 917–949, 2018. [2] H. Cheikh Ali, “Hardy–Sobolev

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

Page 2: Research Article - Hindawi Publishing CorporationAnnals of the Superior Normal School of Pisa—Science Class, vol. 18, no. 3, pp. 917–949, 2018. [2] H. Cheikh Ali, “Hardy–Sobolev

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�.

Page 3: Research Article - Hindawi Publishing CorporationAnnals of the Superior Normal School of Pisa—Science Class, vol. 18, no. 3, pp. 917–949, 2018. [2] H. Cheikh Ali, “Hardy–Sobolev

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�,

Page 4: Research Article - Hindawi Publishing CorporationAnnals of the Superior Normal School of Pisa—Science Class, vol. 18, no. 3, pp. 917–949, 2018. [2] H. Cheikh Ali, “Hardy–Sobolev

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�� ≤ ��,� + �,

Page 5: Research Article - Hindawi Publishing CorporationAnnals of the Superior Normal School of Pisa—Science Class, vol. 18, no. 3, pp. 917–949, 2018. [2] H. Cheikh Ali, “Hardy–Sobolev

5Advances in Mathematical Physics

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[20] Z. Guo, M. Melgaard, and W. Zou, “Schrödinger equations with magnetic fields and Hardy–Sobolev critical exponents,” Electronic Journal of Differential Equations, vol. 199, pp. 1–18, 2017.

[21] P. d’Avenia and M. Squassina, “Ground states for fractional magnetic operators,” ESAIM: Control, Optimisation and Calculus of Variations, vol. 24, no. 1, pp. 1–24, 2018.

[22] Z. Guo and M. Melgaard, “Fractional magnetic Sobolev inequalities with two variables,” Mathematical Inequalities & Applications, vol. 22, no. 2, pp. 703–718, 1998.

Page 6: Research Article - Hindawi Publishing CorporationAnnals of the Superior Normal School of Pisa—Science Class, vol. 18, no. 3, pp. 917–949, 2018. [2] H. Cheikh Ali, “Hardy–Sobolev

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