The minimizing problem involving p-Laplacian and Hardy–Littlewood–Sobolev upper critical exponent

The minimizing problem involving p-Laplacian and Hardy–Littlewood–Sobolev upper critical exponent

Yu Su

and Haibo Chen

School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, P.R. China Received 3 April 2018, appeared 31 August 2018

Communicated by Patrizia Pucci Abstract. In this paper, we study the minimizing problem

S_p,1,α,µ:= inf

u∈W^1,p(R^N)\{0}

R

R^N|∇u|^pdx−µR

R^N |u|^p

|x|^pdx

R

R^N

R

R^N

|u(x)|^p∗α|u(y)|^p∗α

|x−y|^α dxdy ^p

2·p∗ α

,

where N>3,p∈(1,N),µ∈0, ^N−_p^pp

,α∈(0,N)andp^∗_α= ^p₂ ^2N−α_N−pis the Hardy–

Littlewood–Sobolev upper critical exponent. Firstly, by using refinement of the Hardy–

Littlewood–Sobolev inequality, we prove that Sp,1,α,µ is achieved in R^N by a radially symmetric, nonincreasing and nonnegative function. Secondly, we give a estimation of extremal function.

Keywords: refinement of Hardy–Littlewood–Sobolev inequality, Hardy–Littlewood–

Sobolev upper critical exponent, p-Laplacian, minimizing.

2010 Mathematics Subject Classification: 35J50, 35J60.

1 Introduction

In this paper, we consider the minimizing problem:

S_p,1,α,µ:= inf

u∈W^1,p(_R^N)\{0}

R

R^N|∇u|^pdx−µR

R^N |u|^p

|x|^pdx R

R^N

R

R^N

|u(x)|^p∗α|u(y)|^p∗α

|x−y|^α dxdy_2·^p_p∗ α

, (P)

where N > ^{3, p} ∈ (1,N), µ ∈ 0, ^N−_p^pp

, α ∈ (0,N) and p^∗_α = ^p₂ ^2N_N−⁻_p^α is the Hardy–

Littlewood–Sobolev upper critical exponent.

The paper was motivated by some papers appeared in recent years. For p = 2, problem (P) is closely related to the nonlinear Choquard equation as follows:

−_∆u+V(x)u= (|x|^α∗ |u|^q)|u|^q−2u, inR^N, (1.1)

BCorresponding author. Emails: yizai52@qq.com (Y. Su), math_chb@163.com (H. Chen)

where α ∈ (0,N) and ^2N_N^−α 6 ^q 6 ^2N_N−⁻₂^{α. For q} = 2 and α = 1, the equation (1.1) goes back to the description of the quantum theory of a polaron at rest by Pekar in 1954 [19] and the modeling of an electron trapped in its own hole in 1976 in the work of Choquard, as a certain approximation to Hartree–Fock theory of one-component plasma [20]. For q = ^2N_N−⁻₂¹ andα=1, by using the Green function, it is obvious that equation (1.1)can be regarded as a generalized version of Schrödinger–Newton system:

(−_∆u+V(x)u=|u|^NN+−12φ, inR^N,

−_∆φ=|u|^2NN−−21, inR^N.

The existence and qualitative properties of solutions of Choquard type equations (1.1) have been widely studied in the last decades (see [16]). Moroz and Van Schaftingen [15] considered equation (1.1) with lower critical exponent ^2N_N^−α if the potential 1−V(x)should not decay to zero at infinity faster than the inverse of |x|². In [1], the authors studied the equation (1.1) with critical growth in the sense of Trudinger–Moser inequality and studied the existence and concentration of the ground states. In 2018, Gao and Yang [11] firstly investigated the following critical Choquard equation:

−_∆u= _Z

R^N

|u|^2∗α

|x−y|^αdy

|u|^2∗α−2u+λu, inΩ, (1.2) where Ω is a bounded domain of R^N with Lipschitz boundary, N > ^{3, α} ∈ (0,N), λ > 0 and 2^∗_α = ^2N_N−⁻₂^α. By using variational methods, they established the existence, multiplicity and nonexistence of nontrivial solutions to equation (1.2). In 2017, Mukherjee and Sreenadh [17]

considered the following fractional Choquard equation:

(−_∆)^su= _Z

R^N

|u|^2∗α,s

|x−y|^αdy

|u|^2∗α,s−2u+λu, inΩ, (1.3) where Ω is a bounded domain of R^N with C^1,1 boundary, s ∈ (0, 1), N > ^{2s, α} ∈ (0,N), λ > 0 and 2^∗_α,s = ^2N_N−⁻_2s^α is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. By using variational methods, they established the existence, multiplicity and nonexistence of nontrivial solutions to equation (1.3). For details and recent works, we refer to [2,6,7,12,23–26,30] and the references therein.

For p 6= 2, in 2017, Pucci, Xiang and Zhang [22] studied the Schrödinger–Choquard–

Kirchhoff equations involving the fractional p-Laplacian as follows:

(a+bkuk_s^p(θ−1))[(−_∆)^s_pu+V(x)|u|^p−2u] =λf(x,u) +

_Z

R^N

|u|^p∗α,s

|x−y|^αdy

|u|^p∗α,s−2u inR^N,

(1.4)

wherekuk_s= R

R^NR

R^N |u(x)−u(y)|^p

|x−y|^N+ps dxdy+R

R^NV(x)|u|^pdx

,a,b∈ _R⁺₀ witha+b>0, λ>0 is a parameter,s∈(_{0, 1})_,N> ps,θ ∈_1,N−^N_ps,α∈(_0,N)_,p^∗_α,s= ^p(2N−α)

2(N−sp) is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality, and f : R^N → _R is a Carathéodory function,V:R^N →_R⁺is a potential function. By using variational methods, they established the existence of nontrivial nonnegative solution to equation (1.4).

There is an open problem in [22]. We define the best constant:

S_p,s,α,µ := inf

u∈W^s,p(_R^N)\{0}

R

R^N

R

R^N |u(x)−u(y)|^p

|x−y|^N+ps dxdy−µR

R^N |u|^p

|x|^psdx

R

R^N

R

R^N

|u(x)|^p∗α,s|u(y)|^p∗α,s

|x−y|^α dxdy ^p

2·p∗ α,s

, (1.5)

where N > ^{3, p} ∈ (1,N), s ∈ (0, 1], α ∈ (0,N) and µ ∈ [0,C_N,s,p), C_N,s,p is defined in [9, Theorem 1.1]. And p^∗_α,s = ^p(2N−α)

2(N−sp) is the critical exponent in the sense of Hardy–Littlewood–

Sobolev inequality.

Open problem: Is the best constantSp,s,α,µachieved?

(Result 1) For p = 2, s = 1, µ = 0 andα ∈ (0,N), Gao and Yang [11] showed that S_2,1,α,0 is achieved inR^N by the extremal function:

w_σ(x) =C₁σ⁻

N−2

2 w(x), w(x) = ^b1

(b₁²+|x−a₁|²)^N−22, whereC₁ >0 is a fixed constant,a₁∈_R^N andb₁∈ (0,∞).

(Result 2) For p = 2, s ∈ (0, 1), µ = 0 and α ∈ (0,N), Mukherjee and Sreenadh [17] proved that S2,s,α,0 is achieved inR^N by the extremal function:

w_σ(x) =C₂σ⁻

N−2s

2 w(x), w(x) = ^b2

(b₂²+|x−a₂|²)^N−22s, whereC₂ >0 is a fixed constant,a₂∈_R^N andb₂∈ (0,∞).

(Result 3) For p= 2, s ∈(0, 1), µ∈ ^h0, 4^sΓ2(N

+2s

4 )

Γ²(^N−₄^2s)

andα∈ (0,N), Yang and Wu [34] showed that S2,s,α,µ is achieved inR^N.

For Open problem, we study the case ofp∈ (_1,N)_,s=_1,µ∈0, ^N−_p^pp

andα∈(_0,N)_. By using the refinement of Sobolev inequality in [18, Theorem 2], we show that S_p,1,α,µ is achieved inR^N (see Theorem1.1).

For the case p 6= 2, one expects that the minimizers of Sp,s,α,µ have a form similar to the functionωσ. However, it is not known the explicit formula of the extremal function. We give the estimation of extremal function (see Theorem1.2and Theorem1.3).

The first main result of this paper reads as follows.

Theorem 1.1. Let N> ^{3, p}∈ (1,N),α∈(0,N)andµ∈0, ^N−_p^pp

. Then S_p,1,α,µ is achieved in R^N by a radially symmetric, nonincreasing and nonnegative function.

The second main result of this paper reads as follows. For p = 2 and s ∈ (0, 1), by using fractional Coulomb–Sobolev space and endpoint refined Sobolev inequality in [4], we give a estimation of extremal function.

Theorem 1.2. Let N>^{3, p}= 2,α∈ (0,N), s∈ (0, 1)andµ∈[0, ¯µ). Any nonnegative minimizer u of S2,s,α,µis radially symmetric and nonincreasing, and it satisfies for x6=0that

C₄

µ¯

¯ µ−µ

S2,s,α,µ

⁽_2N^N−_(N^α)(₊^N_2s⁻₋^2s)

α) N ω_N−1

^N_2N^−2s 1

|x|^N−22s >^u(x), whereω_N−1is the area of the unit sphere inR^N.

The third main result of this paper reads as follows. For p 6= 2 and s = 1, we give a estimation of extremal function.

Theorem 1.3. Let N >^{3, p} ∈ (1,N),α ∈ (0,N)andµ∈ [0, ˜µ). Any nonnegative minimizer u of S_p,1,α,µ is radially symmetric and nonincreasing, and it satisfies for x6=0that

2^αN² ω²_N−1

! ¹

2·p∗

α 1

|x|^N−pp >^u(x), whereω_N−1is the area of the unit sphere inR^N.

2 Preliminaries

The Sobolev spaceW^1,p(_R^N)is the completion ofC₀^∞(_R^N)with respect to the norm kuk_W^p =

Z

R^N|∇u|^pdx.

Fors∈ (0, 1)and p∈(1,N), the fractional Sobolev spaceW^s,p(_R^N)is defined by W^s,p(_R^N):=

u∈ L^{NN p−sp}(_R^N)

Z

R^N

Z

R^N

|u(x)−u(y)|^p

|x−y|^{N+ps dxdy}<_∞

. Fors∈ (0, 1)and p∈(1,N), we introduce the Hardy inequalities:

¯ µ

Z

R^N

|u|²

|x|^2sdx6

Z

R^N

Z

R^N

|u(x)−u(y)|²

|x−y|^{N+2s dxdy, for anyu}∈W^s,2(_R^N)and ¯µ=4^sΓ²(^N+₄^2s) Γ²(^N−₄^2s)^, and

˜ µ

Z

R^N

|u|^p

|x|^pdx 6

Z

R^N|∇u|^pdx, for anyu∈W^1,p(_R^N)and ˜µ=

N−p p

p

. The fractional Coulomb–Sobolev space [4] is defined by

E^{s,α,2∗α,s}(_R^N) = _Z

R^N

Z

R^N

|u(x)−u(y)|²

|x−y|^{N+2s dxdy}< _∞ and

Z

R^N

Z

R^N

|u(x)|^2∗α,s|u(y)|^2∗α,s

|x−y|^{α dxdy}<_∞

.

(2.1)

We endow the spaceE^{s,α,2∗α,s}(_R^N)with the norm kuk²_E,α =

Z

R^N

Z

R^N

|u(x)−u(y)|²

|x−y|^{N+2s dxdy}+ _Z

R^N

Z

R^N

|u(x)|^2∗α,s|u(y)|^2∗α,s

|x−y|^{α dxdy} ₂∗¹

α,s . (2.2) We could define the best constant:

S_p,1,0,µ := inf

u∈W^1,p(_R^N)\{0}

kuk_W^p −µR

R^N |u|^p

|x|^pdx (R

R^N|u|^p∗dx)

p

p∗ , (2.3)

whereS_p,1,0,µ is attained inR^N (see [8]).

Lemma 2.1 (Hardy–Littlewood–Sobolev inequality, [14]). Let t,r > 1 and 0 < µ < N with

1

t +¹_r+ _N^µ =2, f ∈ L^t(_R^N)and h∈ L^r(_R^N). There exists a sharp constant C₂ >0, independent of f,g such that

Z

R^N

Z

R^N

|f(x)||h(y)|

|x−y|^{µ dxdy}6^C2kfk_tkhk_r.

A measurable functionu:R^N →_Rbelongs to the Morrey spacekuk_Lr,v(_R^N)withr∈ [_1,∞) andv ∈(0,N]if and only if

kuk^r_Lr,v(RN)= sup

R>0,x∈_R^N

R^v−N Z

B(x,R)

|u(y)|^rdy<_∞.

Lemma 2.2([18]). For any1 < p < N, let p^∗ = _N^{N p}_−p. There exists C₃ > ₀ such that forθ andϑ satisfying _p^p∗ 6^θ <1,16^ϑ< p^∗ = _N^{N p}_−p, we have

_Z

R^N|u|^p∗dx _p¹_∗

6^C3kuk^θ_Wkuk^1−θ

L^ϑ,ϑ

(N−p) p (_R^N)

, for any u∈W^1,p(_R^N).

Lemma 2.3 (Endpoint refined Sobolev inequality, [4, Theorem 1.2]). Let α ∈ (0,N)and s ∈ (0, 1). Then there exists a constant C₄ >0such that the inequality

kuk

L^N2N−2s(_R^N)6^C4

Z

R^N

Z

R^N

|u(x)−u(y)|²

|x−y|^{N+2s dxdy}

(N−α)(N−2s) 2N(N+2s−α)

×

Z

R^N

Z

R^N

|u(x)|^{2NN−−2sα}|u(y)|^{2NN−−2sα}

|x−y|^{α dxdy}

!_N^s₍⁽_N^N₊⁻_2s^2s₋⁾

α)

, holds for all u∈ E^{s,α,2∗α,s}(_R^N).

3 The proof of Theorem 1.1

We show the refinement of the Hardy–Littlewood–Sobolev inequality. This inequality plays a key role in the proof of Theorem1.1.

Lemma 3.1. For any1< p <N andα∈(0,N), there exists C₅ >0such that forθandϑsatisfying

p

p^∗ 6^θ <1,16^ϑ< p^∗ = _N^{N p}_−p, we have _Z

R^N

Z

R^N

|u(x)|^p∗α|u(y)|^p∗α

|x−y|^{α dxdy} _p¹∗

α 6^C5kuk^2θ_Wkuk^2(1−θ)

L^ϑ,

ϑ(N−p) p (_R^N)

, for any u∈W^1,p(_R^N).

Proof. By using Lemma2.2, we have _Z

R^N|u|^p∗dx _p¹_∗

6^C3kuk^θ_Wkuk^1−θ

L^ϑ,ϑ

(N−p) p (_R^N)

. (3.1)

By the Hardy–Littlewood–Sobolev inequality and (3.1), we obtain _Z

R^N

Z

R^N

|u(x)|^p∗α|u(y)|^p∗α

|x−y|^{α dxdy} _p¹∗

α 6^C

1 p∗

α

2 kuk²

L^p∗(_R^N) 6^C

1 p∗

α

2 C²₃kuk_W^2θkuk^2(1−θ)

L^ϑ,ϑ

(N−p) p (_R^N)

. In [18], there is a misprint, the authors point out it by themselves. The right one is

L^p∗(_R^N),→ L^r,rN−pp(_R^N) (3.2) for any p∈(_1,N)_andr ∈[_1,p^∗). This embedding plays a key role in the proof of Theorem1.1.

Proof of Theorem1.1.

Step 1.Suppose now 06^µ< µ˜ = ^N−_p^pp. Applying Lemma3.1withϑ= p, we have _Z

R^N

Z

R^N

|u(x)|^p∗α|u(y)|^p∗α

|x−y|^{α dxdy} _p¹∗

α 6^C

kuk_W^p −µ Z

R^N

|u|^p

|x|^{pdx 2θ}_p

kuk^2(1−θ)

L^p,N−p(_R^N), (3.3) foru∈W^1,p(_R^N). Let{u_n}be a minimizing sequence ofS_p,1,α,µ, that is

kunk^p_W−µ Z

R^N

|u_n|^p

|x|^{p dx} →Sp,1,α,µ, asn→_∞, and

Z

R^N

Z

R^N

|u_n(x)|^p∗α|u_n(y)|^p∗α

|x−y|^{α dxdy} =1.

Inequality (3.3) enables us to findC>0 independent ofn such that

kunk_Lp,N−p(_R^N)>^C>0. (3.4) We have the chain of inclusions

W^1,p(_R^N),→ L^p∗(_R^N),→ L^p,N−p(_R^N), (3.5) which implies that

ku_nk_Lp,N−p(_R^N)6^{C. (3.6)} Applying (3.4) and (3.6), there exists C>0 such that

0<C6ku_nk_Lp,N−p(_R^N)6^C−1.

Combining the definition of Morrey space and above inequalities, for alln ∈ N, we get the existence ofλ_n>0 andx_n ∈_R^N such that

1 λ_n^p

Z

B(xn,λn)

|u_n(y)|^pdy>ku_nk_L^p_p,N−p(RN)− ^C

2n >^C˜ >0, for some new positive constant ˜Cthat does not depend on n.

Letvn(x) =λ

N−p p

n un(λnx). Notice that, by using the scaling invariance, we have kvnk^p_W−µ

Z

R^N

|vn|^p

|x|^{p dx}→S_p,1,α,µ, asn→_∞, and

Z

R^N

Z

R^N

|v_n(x)|^p∗α|v_n(y)|^p∗α

|x−y|^{α dxdy}=1.

Then

Z

B(^xn

λn,1)

|v_n(y)|^pdy= ¹ λ_n^p

Z

B(xn,λn)

|u_n(y)|^pdy>^C˜ >0.

We can also show thatv_n is bounded inW^1,p(_R^N). Hence, we may assume

v_n *v inW^1,p(_R^N), v_n→v a.e. inR^N, v_n→vin L^q_loc(_R^N) for allq∈[p,p^∗). We claim that {^xn

λn} is uniformly bounded in n. Indeed, for any 0 < β < p, by Hölder’s inequality, we observe that

0<C^˜ 6

Z

B(^xn_λn,1)

|vn|^pdy=

Z

B(^xn_λn,1)

|y|

pβ p(N−β)

N−p |v_n|^p

|y|

pβ p(N−β)

N−p

dy

6

Z

B(_λ^xn

n,1)

|y|^{β(pN−−βp)}dy

!1−^N_N⁻₋^p_β

 Z

B(_λ^xn

n,1)

|vn|^{p(NN−−pβ)}

|y|^{β dy}





N−p N−β

.

By the rearrangement inequality, see [14, Theorem 3.4], we have Z

B(^xn_λn,1)

|y|^β

(N−p) p−β dy6

Z

B(0,1)

|y|^β

(N−p)

p−β dy6^C.

Therefore,

0<C6

Z

B(^xn_λn,1)

|vn|^p

(N−β) N−p

|y|^{β dy. (3.7)}

Now, suppose on the contrary, that _λ^xn_n → _∞ asn → ∞. Then, for anyy ∈ B _λ^xn_n, 1

, we have

|y|>|^xn

λn| −1 fornlarge. Thus, Z

B(^xn

λn,1)

|v_n|^{p(NN−−pβ)}

|y|^{β dy}6 ¹ (|^xn

λn| −1)^β

Z

B(^xn

λn,1)

|v_n|^{p(NN−−pβ)}dy

6 B(^xn

λn, 1)

β N

(|^xn

λn| −1)^β

Z

B(^xn

λn,1)

|v_n|^{NN p−p}dy

!^N_N^−β

6 B(^x_λⁿ

n, 1)

β N

(|^xn

λn| −₁)^β ·kvnk

N−β N

W

S

N−β N−p

p,1,0,0

6 (|^{xn C}

λn| −₁)^β →0 asn→_∞, which contradicts (3.7). Hence,{^xn

λn}is bounded, and there existsR>0 such that Z

B(0,R)

|v_n(y)|^pdy>

Z

B(_λn^xn,1)

|v_n(y)|^pdy>^C˜ >_0.

Since the embeddingW^1,p(_R^N),→L^q_loc(_R^N)q∈[p,p^∗)is compact, we deduce that Z

B(0,R)

|v(y)|^pdy>^C˜ >_0, which means v6≡_0.

Step 2.Set

h(t) =t²

·p∗

pα, t>⁰(1< p< N). Sincep ∈(1,N)andα∈(0,N), we get

2·p^∗_α

p = ^2N−α

N−p >1 and N+p−α>0.

We know that

h⁰⁰(t) = (2N−α)(N+p−α) (N−p)^{2 t}

2p−α N−p >^0,

which implies thath(t)is a convex function. By usingh(0) =0 andl∈[0, 1], we know h(lt) =h(lt+ (1−l)·0)6^lh(t) + (1−l)h(0) =lh(t). (3.8) For anyt₁,t₂∈[_0,∞), applying last inequality withl= _t^t1

1+t₂ andl= _t^t2

1+t₂, we get h(t₁) +h(t2) =h

(t₁+t2) ^t1 t₁+t₂

+h

(t₁+t2) ^t2 t₁+t₂

6 ^t1

t₁+t₂h(t₁+t₂) + ^t2

t₁+t₂h(t₁+t₂) (by(3.8))

=h(t₁+t2).

(3.9)

Now, we claim thatvn →vstrongly inW^1,p(_R^N). Set K(u,v) =

Z

R^N|∇u|^p−2∇u∇vdx−µ Z

R^N

|u|^p−2uv

|x|^{p dx.}

Since{vn}is a minimizing sequence,

nlim→_∞K(v_n,v_n) =S_p,1,α,µ.

By using Brézis–Lieb type lemma [5] and [22, Theorem 2.3], we know K(v,v) + lim

n→_∞K(vn−v,vn−v) = lim

n→_∞K(vn,vn) +o(1) =Sp,1,α,µ+o(1), (3.10) and

Z

R^N

Z

R^N

|vn(x)|^p∗α|vn(y)|^p∗α

|x−y|^{α dxdy}−

Z

R^N

Z

R^N

|vn(x)−v(x)|^p∗α|vn(y)−v(y)|^p∗α

|x−y|^{α dxdy}

=

Z

R^N

Z

R^N

|v(x)|^p∗α|v(y)|^p∗α

|x−y|^{α dxdy}+o(1),

(3.11)

whereo(1)denotes a quantity that tends to zero asn→_∞. Therefore, 1= lim

n→_∞ Z

R^N

Z

R^N

|v_n(x)|^p∗α|v_n(y)|^p∗α

|x−y|^{α dxdy}

= _lim

n→_∞ Z

R^N

Z

R^N

|v_n(x)−v(x)|^p∗α|v_n(y)−v(y)|^p∗α

|x−y|^{α dxdy} +

Z

R^N

Z

R^N

|v(x)|^p∗α|v(y)|^p∗α

|x−y|^{α dxdy} 6^S−

2·p∗ pα

p,1,α,µ

nlim→_∞K(vn−v,vn−v)

2·p∗ pα

+S⁻

2·p∗ pα

p,1,α,µ(K(v,v))

2·p∗ pα

6^S−

2·p∗ pα

p,1,α,µ

nlim→_∞K(v_n−v,v_n−v) +K(v,v)

2·p∗ pα

(_by(_3.9)) 6¹ (by(3.10)).

Therefore, all the inequalities above have to be equalities. We know that

nlim→_∞K(vn−v,vn−v)

2·p∗ pα

+ (K(v,v))

2·p∗

pα =lim

n→_∞K(vn−v,vn−v) +K(v,v)

2·p∗ pα

. (3.12) We show that lim_n→_∞K(v_n−v,v_n−v) =0. Combining (3.9) and (3.12), we know that

either lim

n→_∞K(vn−v,vn−v) =0 or K(v,v) =0.

Sincev6≡0, soK(v,v)6=0. Therefore,

nlim→_∞K(v_n−v,v_n−v) =_{0. (3.13)} This implies that v_n →vstrongly inW^1,p(_R^N). Moreover, we get

nlim→_∞ Z

R^N

Z

R^N

|v_n(x)−v(x)|^p∗α|v_n(y)−v(y)|^p∗α

|x−y|^{α dxdy}=0. (3.14)

Step 3. Sincev6≡0, putting (3.13) into (3.10), and inserting (3.14) into (3.11), we know

nlim→_∞

kvnk_W^p −µ Z

R^N

|vn|^p

|x|^{p dx}

→S_p,1,α,µ= kvk_W^p −µ Z

R^N

|v|^p

|x|^pdx, and

Z

R^N

Z

R^N

|v(x)|^p∗α|v(y)|^p∗α

|x−y|^{α dxdy}=1.

Thenvis an extremal.

In addition,|v| ∈W^1,p(_R^N)and|∇|v||= |∇v|a.e. inR^N, therefore,|v|is also an extremal, and then there exist non–negative extremals.

Let ¯v>0 be an extremal. Denote by ¯v∗ the symmetric–decreasing rearrangement of ¯v(See [14, Section 3]). From [21] it follows that

Z

R^N|∇v¯∗|^pdx6

Z

R^N|∇v¯|^pdx. (3.15)

According to the simplest rearrangement inequality in [14, Theorem 3.4], we get Z

R^N

|v¯|^p

|x|^pdx6

Z

R^N

|v¯∗|^p

|x|^{p dx. (3.16)}

By using Riesz’s rearrangement inequality in [14, Theorem 3.7], we have Z

R^N

Z

R^N

|v¯(x)|^p∗α|v¯(y)|^p∗α

|x−y|^{α dxdy}6

Z

R^N

Z

R^N

|v¯∗(x)|^p∗α|v¯∗(y)|^p∗α

|x−y|^{α dxdy. (3.17)} Combining (3.15), (3.16) and (3.17), and the fact thatµ>0, we get that ¯v∗ is also an extremal, and then there exist radially symmetric and nonincreasing extremal.

4 Proof of Theorem 1.2

For p = 2 and s ∈ (0, 1), we give a estimation of extremal function u(x). The proof of Theorem 1.2 is based on the Coulomb–Sobolev space E^{s,α,2∗α,s}(_R^N) and the endpoint refined Sobolev inequality in Lemma2.3.

Proof of Theorem1.2. We show some properties of radially symmetric, nonincreasing and non- negative functionu(x). Let ¯µ = 4^sΓ2(N

+2s

4 )

Γ²(^N−₄^2s). By the definition of extremal u (see the proof of Theorem1.1), we know

Z

R^N

Z

R^N

|u(x)−u(y)|²

|x−y|^{N+2s dxdy}−µ Z

R^N

|u|²

|x|^2dx=S_2,s,α,µ, (4.1)

and

Z

R^N

Z

R^N

|u(x)|^2∗α,s|u(y)|^2∗α,s

|x−y|^{α dxdy}=1. (4.2)

Applying (4.1), (4.2) and the definition of Coulomb–Sobolev space E^{s,α,2∗α,s}(_R^N), we get u ∈ E^{s,α,2∗α,s}(_R^N).

By using (4.1), (4.2),u∈ E^{s,α,2∗α,s}(_R^N)and Lemma2.3, we have kuk

L^N2N−2s(_R^N)6^C4

_Z

R^N

Z

R^N

|u(x)−u(y)|²

|x−y|^{N+2s dxdy}

(N−α)(N−2s) 2N(N+2s−α)

×

Z

R^N

Z

R^N

|u(x)|^{2NN−−2sα}|u(y)|^{2NN−−2sα}

|x−y|^{α dxdy}

!_N^s₍⁽_N^N₊⁻_2s^2s₋⁾

α)

=C_{4 Z}

R^N

Z

R^N

|u(x)−u(y)|²

|x−y|^{N+2s dxdy}

(N−α)(N−2s) 2N(N+2s−α)

6^C4

µ¯

¯ µ−µ

S_2,s,α,µ

⁽_2N^N−_(N^α)(₊^N_2s⁻₋^2s)

α)

.

(4.3)

For any 0< R<_∞andB(R)_:= B(_0,R)⊂_R^N, we obtain C₄

µ¯

¯ µ−µ

S2,s,α,µ

⁽_2N^N−_(N^α)(₊^N_2s⁻₋^2s)

α)

>

Z

R^N|u(x)|^N2N−2sdx ^N_2N^−2s

>

_Z

B(R)

|u(x)|^N2N−2sdx ^N_2N^−2s

>|u(R)|ω

N−2s 2N

N−1

_{Z R}

0 ρ^N−1dρ ^N_2N^−2s

=|u(R)|^ωN−1 N

^N_2N^−2s R^N−22s, which implies

C₄

µ¯

¯ µ−µ

S_2,s,α,µ

⁽_2N^N−_(N^α)(₊^N_2s⁻₋^2s)

α) N ω_N−1

^N_2N^−2s 1

|x|^N−22s >|u(x)|_.