1 Introduction and main results
In recent years, there has been an increasing amount of attention to problems involving nonlocal diffusion operators. These problems usually include the fractional Laplacian (-Δ)s (0<s<1) or fractional p-Laplacian operators (-Δp)s (p>1) with subcritical or critical nonlinearities and they arise in a quite natural way in many different applications, for instance in physical models, finances, fluid dynamics and image processing, etc.
The fractional Laplacian in ℝN is a nonlocal pseudo-differential operator defined by
(1.1)
(
-
Δ
)
α
2
u
(
x
)
=
C
n
,
α
lim
ε
→
0
∫
ℝ
N
∖
B
ε
(
x
)
u
(
x
)
-
u
(
z
)
|
x
-
z
|
N
+
α
𝑑
z
,
where α is any real number between 0 and 2. This operator is well defined in 𝒮(ℝN), the Schwartz space of rapidly decreasing C∞ functions in ℝN. One can extend this operator to a wider space of functions. Let
L
α
(
ℝ
N
)
=
{
u
:
ℝ
N
→
ℝ
,
∫
ℝ
N
|
u
(
x
)
|
(
1
+
|
x
|
)
N
+
α
𝑑
x
<
∞
}
.
Then it is easy to verify that for u∈Lα(ℝN)∩Cloc1,1(ℝN), the integral on the right-hand side of (1.1) is well defined. The non-locality of the fractional Laplacian makes it difficult to study. To circumvent this difficulty, Caffarelli and Silvestre introduced in their celebrated work [11] the extension method that reduced this nonlocal problem into a local one in higher dimensions. This fundamental extension method has been applied successfully to study equations involving the fractional Laplacian and many results have been obtained since then.
On the other hand, for the nonlocal operators such as fractional Laplacian and its nonlinear generalization, the fractional p-Laplacian, they can be regarded as an extension of the traditional local operators such as the Laplacian -Δ and p-Laplacian -Δp. The corresponding elliptic problems can be seen as an extension of many classical problems such as the well-known Brezis–Nirenberg problem. In 1983, Brezis and Nirenberg, in their celebrated paper [10], showed that the critical growth semi-linear problem
(1.2)
{
-
Δ
u
=
λ
u
+
u
|
u
|
2
*
-
2
in
Ω
,
u
=
0
on
∂
Ω
,
u
>
0
in
Ω
,
admits a classical solution provided that λ∈(0,λ1(2)) and N≥4, where λ1(2) is the first eigenvalue of -Δ with homogeneous Dirichlet boundary conditions and 2∗=2NN-2 is the critical Sobolev exponent. Furthermore, in dimension N=3, the same existence result holds provided that μ<λ<λ1(2), for a suitable μ>0 (if Ω is a ball, then μ=14λ1(2) is sharp). By Pohožaev identity, if λ∉(0,λ1(2)) and Ω is a star-sharped domain, then problem (1.2) admits no solution. Later on, Capozzi, Fortunato and Palmieri [12] established the existence of nontrivial solutions to the following problem for λ>0 when N≥4:
(1.3)
{
-
Δ
u
=
λ
u
+
u
|
u
|
2
*
-
2
in
Ω
,
u
=
0
on
∂
Ω
.
Subsequently, Ambrosetti and Struwe [3] used the dual approach to the problem (1.3), which allows a direct use of the celebrated Mountain-Pass Theorem and critical point theory of Ambrosetti and Rabinowitz [2, 39], and they presented a new and simpler proof for the existence of nontrivial solutions to (1.3). Furthermore, as an extension of problem (1.2), Azorero and Alonzo in [23] studied the existence of nontrivial solution for the problem
{
-
Δ
p
u
=
λ
u
|
u
|
p
-
2
+
u
|
u
|
p
*
-
2
in
Ω
,
u
=
0
on
∂
Ω
,
u
≥
0
in
Ω
,
where 0<λ<λ1(p) and N≥p2, where λ1(p) is the first eigenvalue of the p-Laplacian with the Dirichlet boundary condition. Then in 2015, Servadei and Valdinoci [40], among other things, generalized the well-known Brezis–Nirenberg problem to the critical fractional Laplacian:
(1.4)
{
(
-
Δ
)
s
u
=
λ
u
+
u
|
u
|
2
s
*
-
2
in
Ω
,
u
=
0
in
ℝ
N
∖
Ω
,
where s∈(0,1) and 2s*=2NN-2s. The authors showed that if λ1,s is the first eigenvalue of the nonlocal operator (-Δ)s with homogeneous Dirichlet boundary datum, then for any λ∈(0,λ1,s) there exists a nontrivial solution to (1.4) provided N≥4s. In [18], W. X. Chen and Zhu consider an indefinite fractional Laplacian problem. A corresponding Liouville-type theorem is established. Furthermore, they obtain a priori estimate for solutions in a bounded domain by blowing-up and rescaling. Substantially improved results for the indefinite fractional Laplacian problem have been further obtained by W. X. Chen, Li and Zhu [16].
Recently, Mosconi, Perera, Squassina and Yang [36] considered the Brezis–Nirenberg problem for the fractional p-Laplacian. More precisely, they investigated the problem
(1.5)
{
(
-
Δ
p
)
s
u
=
λ
u
|
u
|
p
-
2
+
u
|
u
|
p
s
*
-
2
in
Ω
,
u
=
0
in
ℝ
N
∖
Ω
,
where λ>0, s∈(0,1), p>1 and ps*=pNN-ps. Based on the results in [6] obtained by Brasco, Mosconi and Squassina and an abstract linking theorem based on the cohomological index proved in [43] by Yang and Perera, the authors obtained the existence of nontrivial solutions to problem (1.5).
On the other hand, as another form of the extension to the classical Brezis–Nirenberg problem, Alves, de Morais Filho and Souto in [1] considered a certain system of Laplacian equations with nonlinearities of critical or subcritical Sobolev growth and established the existence and nonexistence results of nontrivial solutions to such a system. Then de Morais Filho and Souto in [19] generalized the aforesaid results to a more general class of nonlinear systems of p-Laplacian operators:
(1.6)
{
-
Δ
p
u
=
Q
u
(
u
,
v
)
+
H
u
(
u
,
v
)
in
Ω
,
-
Δ
p
v
=
Q
v
(
u
,
v
)
+
H
v
(
u
,
v
)
in
Ω
,
u
=
v
=
0
on
∂
Ω
,
u
,
v
≥
0
,
u
,
v
≠
0
in
Ω
,
where p>1 and Ω is a bounded domain and Q and H satisfy certain homogeneity conditions. By using the concentration compactness principle and certain asymptotic estimates for minimizers of the Sobolev inequality, the authors obtained the existence of nontrivial weak solutions to problem (1.6).
Motivated by the aforementioned works, it is natural to ask whether system (1.6) has a nontrivial solution when the p-Laplacian is replaced by the fractional p-Laplacian. As far as we know, there is no related work in this direction so far. In this paper, we consider the following problem:
(1.7)
{
(
-
Δ
p
)
s
u
=
Q
u
(
u
,
v
)
+
H
u
(
u
,
v
)
in
Ω
,
(
-
Δ
p
)
s
v
=
Q
v
(
u
,
v
)
+
H
v
(
u
,
v
)
in
Ω
,
u
=
v
=
0
in
ℝ
N
∖
Ω
,
u
,
v
≥
0
,
u
,
v
≠
0
in
Ω
,
where Ω is a bounded domain in ℝN with Lipschitz boundary, s∈(0,1), 1<p<∞ and ps<N, the fractional p-Laplacian (-Δp)s is the nonlinear nonlocal operator defined on smooth functions by
(
-
Δ
p
)
s
u
(
x
)
=
2
lim
ε
→
0
∫
B
ε
(
x
)
c
|
u
(
x
)
-
u
(
y
)
|
p
-
2
(
u
(
x
)
-
u
(
y
)
)
|
x
-
y
|
N
+
p
s
𝑑
y
,
x
∈
ℝ
N
.
This definition is consistent, up to a normalization constant depending on N and s, with the usual definition of the linear fractional Laplacian operator (-Δ)s when p=2. It is worth noting that in recent years, there is a rapidly growing literature on the investigation for the fractional p-Laplacian operator. For instance, the fractional p-eigenvalue problems have been studied in Brasco, Parini and Squassina [8], Brasco and Parini [7], Franzina and Palatucci [22] and Lindgren and Lindqvist [32], etc. Regularity of solutions was obtained in Brasco and Lindren [5], Chen, Li and Qi [15], Di Castro, Kuusi and Palatucci [21, 20], Iannizzotto, Mosconi and Squassina [25], Kuusi, Mingione and Sire [26]. Monotonicity and symmetry of solutions was studied by Chen, Li and Wu [42, 14]. Existence via Morse theory was investigated in Iannizzotto, Liu, Perera and Squassina [24]. This operator has been studied extensively in recent years. We refer the reader to, for example, [17, 13, 37] and many references therein for details. In particular, Brasco, Mosconi and Squassina [6] obtained the optimal decay of extremal functions for the fractional Sobolev inequality, which plays an important role in studying the problem (1.7). Moreover, Marano and Mosconi extended the results to the fractional Hardy–Sobolev inequality. The authors proved the existence of optimizers for the fractional Hardy–Sobolev inequality and they studied the asymptotic properties of the optimizers. The reader is referred to [35] for details. For more recent progress on investigations of the Hardy–Sobolev-type inequalities, we refer the interested reader to [34] and the references therein.
In this paper, we consider problem (1.7) in the Banach space W=W0s,p(Ω)×W0s,p(Ω). The methods we use here are through the variational arguments and combining with the concentration compactness principle. More precisely, we introduce the nontrivial solution for problem (1.7) by finding a minimum point of the energy functional I:W→ℝ
(1.8)
I
(
u
,
v
)
=
1
p
∫
ℝ
2
N
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
+
1
p
∫
ℝ
2
N
|
v
(
x
)
-
v
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
-
∫
Ω
Q
(
u
,
v
)
𝑑
x
,
subjected to the constraint-manifold
ℋ
=
{
(
u
,
v
)
∈
W
:
∫
Ω
H
(
u
+
,
v
+
)
𝑑
x
=
1
}
.
Moreover, we say (u,v)∈W is a weak solution of problem (1.7) if
∫
ℝ
2
N
|
u
(
x
)
-
u
(
y
)
|
p
-
2
(
u
(
x
)
-
u
(
y
)
)
(
φ
(
x
)
-
φ
(
y
)
)
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
+
∫
ℝ
2
N
|
v
(
x
)
-
v
(
y
)
|
p
-
2
(
v
(
x
)
-
v
(
y
)
)
(
ψ
(
x
)
-
ψ
(
y
)
)
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
=
∫
Ω
[
φ
Q
u
(
u
,
v
)
+
ψ
Q
v
(
u
,
v
)
]
𝑑
x
+
η
∫
Ω
[
φ
H
u
(
u
,
v
)
+
ψ
H
v
(
u
,
v
)
]
𝑑
x
,
where η∈ℝ is a constant satisfying [I′(u,v)-ηJ′(u,v)](φ,ψ)=0 and J(u,v)=∫ΩH(u,v)𝑑x-1. Namely, η∈ℝ is a Lagrange multiplier satisfying the Euler–Lagrange equation corresponding to the functional I defined in (1.8) among functions in the constraint manifold ℋ. For further details of this variational framework, we refer the reader to Section 2.
Before we state our main results, we introduce several notations. In the sequel, we shall denote by λ1(p) the positive eigenvalue of the fractional p-Laplacian eigenvalue problem, subjected to Dirichlet boundary condition, and use its variational characterization
λ
1
=
λ
1
(
p
)
=
inf
u
∈
W
0
s
,
p
(
Ω
)
∖
{
0
}
∫
ℝ
2
N
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
∫
Ω
|
u
|
p
𝑑
x
.
Let us also denote
(1.9)
μ
1
p
=
min
{
Q
(
u
,
v
)
:
u
,
v
≥
0
,
u
p
+
v
P
=
1
}
and
(1.10)
μ
2
p
=
max
{
Q
(
u
,
v
)
:
u
,
v
≥
0
,
u
p
+
v
P
=
1
}
.
We are ready to state our first result of existence of nontrivial solutions to problem (1.7) when the system of the fractional p-Laplacian has homogeneous nonlinearities of subcritical Sobolev growth. This extends the result in the local case when s=1 in [19].
Theorem 1.1 (Subcritical Case).
Let Q be a C1(R+×R+,R) function such that
(1.11)
Q
(
λ
u
,
λ
v
)
=
λ
p
Q
(
u
,
v
)
for all
λ
>
0
,
u
,
v
≥
0
(
p
-homogenity)
and
(1.12)
Q
u
(
0
,
1
)
>
0
,
Q
v
(
1
,
0
)
>
0
.
Let H be another C1(R+×R+,R) function such that
H
(
λ
u
,
λ
v
)
=
λ
q
H
(
u
,
v
)
for all
λ
>
0
,
u
,
v
≥
0
,
p
<
q
<
p
s
*
,
(1.13)
H
(
u
,
v
)
≥
0
for all
u
,
v
≥
0
,
and
(1.14)
H
u
(
0
,
1
)
=
0
,
H
v
(
1
,
0
)
=
0
.
Then, for μ2<λ1, system (1.7) has a nontrivial solution.
Next is the main result of this paper. Namely, we will establish the following existence result of nontrivial solutions to the problem (1.7) when the system of the fractional p-Laplacian has homogeneous nonlinearities of critical Sobolev growth.
Theorem 1.2 (Critical Case).
Let Q be a C1(R+×R+,R) function defined as in Theorem 1.1 and let H be a C1(R+×R+,R) function satisfying (1.13) and (1.14). In addition, H is ps*-homogeneous, i.e.,
(1.15)
H
(
λ
u
,
λ
v
)
=
λ
p
s
*
H
(
u
,
v
)
for all
λ
>
0
,
u
,
v
≥
0
.
The 1-homogeneous function G defined by
G
(
s
p
s
*
,
t
p
s
*
)
=
H
(
s
,
t
)
for all
s
,
t
≥
0
,
is concave. Suppose also that N>p2s and 0<μ1≤μ2<λ1. Then system (1.7) has a nontrivial solution.
For the critical case, as ps* is the limiting fractional Sobolev exponent for the embedding W0s,p(Ω)↪Lps*(Ω), the lack of compactness arise new difficulties. In order to overcome these difficulties, we work with certain asymptotic estimates for minimizers obtained in [6] to establish a relationship for the fractional Sobolev-type constants which ensure the compactness occurs. On the other hand, we establish a concentration compactness principle associated with the system of the fractional p-Laplacian in bounded domain (see Lemma 5.1) which is of its independent interest. This principle is in the spirit to the work for a single fractional p-Laplacian equation [4] recently obtained by Bonder, Saintier and Silva, but nevertheless is more difficult to prove in our case of the system of the fractional p-Laplacians.
The paper is organized as follows: we give some preliminary results in Section 2. In Section 3, we consider the subcritical case and give the proof of Theorem 1.1. In Section 4, we will prove some technical lemmas and in the Section 5 we will prove a version of the concentration compactness lemmas associated with the system of the fractional p-Laplacian in bounded domain (Lemma 5.1). Finally, we complete the proof of Theorem 1.2.
2 Preliminary Results
Let Ω be a bounded domain in ℝN with Lipschitz boundary. In [17, 36, 37], Mosconi, Perera, Squassina and Chen et al. discussed the Dirichlet boundary value problem for the fractional p-Laplacian by means of variational methods. Since the problems they considered are in a bounded domain, they introduced the function space (W0s,p(Ω),[⋅]s,p). For the entire space, it is well known that the usual fractional Sobolev space is defined as
W
s
,
p
(
ℝ
N
)
=
{
u
∈
L
p
(
ℝ
N
)
:
[
u
]
s
,
p
<
∞
}
,
endowed with the norm
∥
u
∥
s
,
p
=
(
|
u
|
p
+
[
u
]
s
,
p
p
)
1
p
,
where |⋅|p is the norm in Lp(ℝN) and
[
u
]
s
,
p
=
(
∫
ℝ
2
N
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
)
1
p
is the Gagliardo seminorm of a measurable function u:ℝN→ℝ. In this paper, we continue to work in the closed subspace W0s,p(Ω) which is defined as
W
0
s
,
p
(
Ω
)
=
{
u
∈
W
s
,
p
(
ℝ
N
)
:
u
=
0
a.e. in
ℝ
N
∖
Ω
}
,
it is a uniformly convex Banach space for any p>1 and equivalently renormed by setting
(2.1)
∥
⋅
∥
W
0
s
,
p
(
Ω
)
=
[
⋅
]
s
,
p
=
(
∫
X
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
N
+
p
s
d
x
d
y
)
1
p
,
where X=ℝ2N∖(𝒞Ω×𝒞Ω) with 𝒞Ω=ℝN∖Ω. The second equality holds in (2.1) since u=0 in ℝN∖Ω and the integral in (2.1) can be reduced to over X. The embedding W0s,p(Ω)↪Lr(Ω) is continuous for r∈[1,ps*] and compact for r∈[1,ps*). It is worth noting that when s=1,p=N, the corresponding Sobolev space W01,N(Ω) is a borderline case for Sobolev embedding and the Sobolev inequality transforms into the well-known Moser-Trudinger inequality (see [38, 41]). In this regard, many authors have contributed to the study of nonlinear PDEs with nonlinearity of exponential growth and there is a vast literature in the subject. We only refer the reader to [28, 27, 30, 29] and many references therein.
In the following we shall find weak solutions of (1.7) in the space
W
=
W
0
s
,
p
(
Ω
)
×
W
0
s
,
p
(
Ω
)
,
endowed with the norm
∥
(
u
,
v
)
∥
p
=
[
u
]
s
,
p
p
+
[
v
]
s
,
p
p
=
∫
X
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
+
∫
X
|
v
(
x
)
-
v
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
.
Under the assumption of (1.11)–(1.12), we can extend the function Q to the whole plane as a 𝒞1-function as
Q
(
s
,
t
)
=
{
Q
(
s
,
t
)
,
s
,
t
≥
0
,
Q
(
0
,
t
)
+
Q
s
(
0
,
t
)
s
,
s
≤
0
≤
t
,
Q
(
s
,
0
)
+
Q
t
(
s
,
0
)
t
,
t
≤
0
≤
s
,
0
,
s
,
t
≤
0
.
It is particularly noteworthy that under the above extension, we have
(2.2)
Q
u
(
u
,
v
)
≥
0
for all
u
≤
0
,
v
∈
ℝ
,
Q
v
(
u
,
v
)
≥
0
for all
v
≤
0
,
u
∈
ℝ
.
(We note that the continuous differentiability of Q comes from Remark 2.1 (iii) and (iv) below.)
On the other hand, under assumption (1.14), by taking a similar argument as above, we can give a 𝒞1 extension of H to the whole plane as
H
(
u
,
v
)
=
H
(
u
+
,
v
+
)
for all
u
,
v
∈
ℝ
,
where u+:=max{u,0} and u-:=min{u,0}. Hence, in this paper, we shall find the solutions of system (1.7) as a minimum of the functional I:W→ℝ
I
(
u
,
v
)
=
1
p
(
∫
X
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
+
∫
X
|
v
(
x
)
-
v
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
)
-
∫
Ω
Q
(
u
,
v
)
𝑑
x
,
subjected to the constraint-manifold
ℋ
=
{
(
u
,
v
)
∈
W
:
∫
Ω
H
(
u
+
,
v
+
)
𝑑
x
=
1
}
.
Remark 2.1.
Since the 𝒞1-functions Q and H are homogeneous functions, for convenience of the reader we list several properties of homogeneous functions: Let F be a q (q≥1) homogeneous 𝒞1 function. Then:
There exists MF>0 such that
|
F
(
s
,
t
)
|
≤
M
F
(
|
s
|
q
+
|
t
|
q
)
for all
s
,
t
∈
ℝ
,
M
F
=
max
{
F
(
s
,
t
)
:
s
,
t
∈
ℝ
,
|
s
|
q
+
|
t
|
q
=
1
}
.
The maximum MF is attained for some (s0,t0)∈ℝ2 (both not vanishing), since
{
(
s
,
t
)
:
s
,
t
∈
ℝ
2
,
|
s
|
q
+
|
t
|
q
=
1
}
,
is a compact set and F continuous on it.
For all s,t∈ℝ,
(2.3)
s
F
s
(
s
,
t
)
+
t
F
t
(
s
,
t
)
=
q
F
(
s
,
t
)
.
∇
F
is a (q-1) homogeneous function and by (2.3), when q=1, we have: F is concave if and only if Fst(s,t)≥0.
Remark 2.2.
For the convenience of the reader, we list some explicit examples below for the homogeneous functions Q and H, which can provide us better understanding of the problem we consider in this paper. Let us denote
P
l
(
u
,
v
)
=
a
u
l
+
∑
α
i
+
β
i
=
l
b
i
u
α
i
v
β
i
+
c
v
l
,
where i∈ℐ (♯ℐ<∞), l>0, αi≥1, βi≥1, a,bi,c∈ℝ. With suitable bi and l, sums ∑∞ may be allowed. Then the following elementary functions and some others possible combinations of them, with convenient a,bi,c, satisfy hypotheses (1.11) and (1.12):
Q
1
(
u
,
v
)
=
P
p
(
u
,
v
)
,
Q
2
(
u
,
v
)
=
P
l
2
(
u
,
v
)
P
l
3
(
u
,
v
)
with
l
2
-
l
3
=
p
.
Moreover, H(u,v)=Pps∗(u,v) satisfies (1.15) and additional assumption given in Theorem 1.2. For more examples and properties of the homogeneous functions Q and H, we can see [19] for more details.
3 Proof of Theorem 1.1
This section is devoted to proving Theorem 1.1. We first introduce a lemma which will be used in our proof of Theorem 1.1.
Lemma 3.1 (Chen, Mosconi and Squassina [17]).
For any u∈W0s,p(Ω) it holds
〈
(
-
Δ
p
)
s
u
±
,
u
±
〉
≤
〈
(
-
Δ
p
)
s
u
,
u
±
〉
≤
〈
(
-
Δ
p
)
s
u
,
u
〉
,
with strict inequality as long as u is sign-changing.
Proof of Theorem 1.1.
Let {(un,vn)}∈ℋ be a minimizing sequence for RQ:=min(u,v)∈ℋI(u,v), i.e.,
(3.1)
R
Q
=
I
(
u
n
,
v
n
)
+
o
n
(
1
)
,
where on(1), from now on, is such that on(1)→0 as n→∞.
From (1.10) we get that
(3.2)
I
(
u
n
,
v
n
)
≥
1
p
min
{
1
,
(
1
-
μ
2
λ
1
)
}
(
∫
X
|
u
n
(
x
)
-
u
n
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
+
∫
X
|
v
n
(
x
)
-
v
n
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
)
.
Hence ∥(un,vn)∥ is bounded and (3.2) implies that I≥0 and I>0 on ℋ.
As we are in the subcritical case, there are subsequence {un}, {vn} in W0s,p(Ω) such that un⇀u,vn⇀v weakly in W0s,p(Ω) and these convergences also hold pointwisely on Ω and un⇀u,vn⇀v strongly in Lr(Ω) for r∈[1,ps*). Moreover, there is h∈Lr(Ω) such that |un(x)|,|vn(x)|≤h(x), a.e. in Ω.
Therefore, passing to the limit in (3.1) we get RQ=I(u,v) and (u,v)∈ℋ.
By the Lagrange multiplier theorem, if J(u,v)=∫ΩH(u,v)𝑑x-1, there exists η∈ℝ such that
(3.3)
[
I
′
(
u
,
v
)
-
η
J
′
(
u
,
v
)
]
⋅
(
φ
,
ψ
)
=
0
for all
(
φ
,
ψ
)
∈
ℋ
.
Here I′,J′ are the Fréchet derivatives, i.e., from (3.3), we have
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
p
-
2
(
u
(
x
)
-
u
(
y
)
)
(
φ
(
x
)
-
φ
(
y
)
)
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
+
∫
ℝ
N
∫
ℝ
N
|
v
(
x
)
-
v
(
y
)
|
p
-
2
(
v
(
x
)
-
v
(
y
)
)
(
ψ
(
x
)
-
ψ
(
y
)
)
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
=
∫
Ω
(
φ
Q
u
(
u
,
v
)
+
ψ
Q
v
(
u
,
v
)
)
𝑑
x
+
η
∫
Ω
(
φ
H
u
(
u
,
v
)
+
ψ
H
v
(
u
,
v
)
)
𝑑
x
for all (φ,ψ)∈W. Particularly, setting φ=u,ψ=v in the above equation, and using the homogeneity properties (2.3), we get pI(u,v)=qη, which implies that η=pqRQ is a positive number. In addition, it is easy to verify that (ρu,ρv), with ρ=(pqRQ)1p-q>0, is the desired solution. This solution, now denoted by (u,v), is nonnegative. In fact, replacing (φ,ψ) by (u-,v-), combining with Lemma 3.1, we have
0
≤
∥
(
u
-
,
v
-
)
∥
=
〈
(
-
Δ
p
)
s
u
-
,
u
-
〉
+
〈
(
-
Δ
p
)
s
v
-
,
v
-
〉
≤
〈
(
-
Δ
p
)
s
u
,
u
-
〉
+
〈
(
-
Δ
p
)
s
v
,
v
-
〉
=
∫
Ω
u
-
[
Q
u
(
u
,
v
)
+
H
u
(
u
,
v
)
]
𝑑
x
+
∫
Ω
v
-
[
Q
v
(
u
,
v
)
+
H
v
(
u
,
v
)
]
𝑑
x
=
∫
Ω
u
-
[
Q
u
(
u
-
,
v
)
+
H
u
(
u
,
v
)
]
𝑑
x
+
∫
Ω
v
-
[
Q
v
(
u
,
v
-
)
+
H
v
(
u
,
v
)
]
𝑑
x
.
Notice that u-,v-≤0, H(u,v)=H(u+,v+)≥0 for all u,v∈ℝ, then combining (2.2), we get that
∫
Ω
u
-
[
Q
u
(
u
-
,
v
)
+
H
u
(
u
,
v
)
]
𝑑
x
+
∫
Ω
v
-
[
Q
v
(
u
,
v
-
)
+
H
v
(
u
,
v
)
]
𝑑
x
≤
0
,
which implies that
u
-
=
v
-
=
0
.
By (1.7), (1.12) and (1.14), we see that u=0 if and only if v=0. Hence, u,v≥0,u,v≠0. Thus Theorem 1.1 is proved. ∎
4 Some Technical Lemmas
In this section, we establish several auxiliary estimates that will be used in the proof of Theorem 1.2. First, we recall the fractional Sobolev inequality. Let the symbol W0s,p(ℝN) denotes the homogeneous fractional Sobolev space
W
0
s
,
p
(
ℝ
N
)
:=
{
u
∈
L
p
s
*
(
ℝ
N
)
:
[
u
]
s
,
p
<
∞
}
,
endowed with the norm ∥⋅∥W0s,p(ℝN)=[⋅]s,p, and let
(4.1)
S
=
S
(
ℝ
N
)
=
inf
u
∈
W
0
s
,
p
(
ℝ
N
)
∖
{
0
}
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
(
∫
ℝ
N
|
u
|
p
s
*
𝑑
x
)
p
p
s
*
,
it is well known that S(ℝN) is achieved in ℝN. In particular, for p=2, the explicit formula for the minimizers for S has been given by Lieb in [31]. More precisely, the minimizers are of the form cU(|x-x0|ε), where
U
(
x
)
=
1
(
1
+
|
x
|
2
)
N
-
2
s
2
,
x
∈
ℝ
N
,
c
≠
0
, x0∈ℝN and ε>0. But for p≠2, it is not known the explicit formula for the minimizers of S(ℝN) till now. On the other hand, in [6], Brasco, Mosconi and Squassina investigated the existence and properties for the minimization problem (4.1). Recently, Marano, Mosconi extended the results in [6] and they obtained the existence and asymptotic properties for optimizers of the fractional Hardy–Sobolev inequality, see [35] and the references therein.
Now, we define
S
~
=
S
(
Ω
)
:=
inf
u
∈
W
0
s
,
p
(
Ω
)
∖
{
0
}
∫
ℝ
N
∫
ℝ
N
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
(
∫
Ω
|
u
|
p
s
*
𝑑
x
)
p
p
s
*
=
inf
u
∈
W
0
s
,
p
(
Ω
)
∖
{
0
}
∫
X
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
(
∫
Ω
|
u
|
p
s
*
𝑑
x
)
p
p
s
*
,
(4.2)
S
~
H
=
S
~
H
(
Ω
)
:=
inf
(
u
,
v
)
∈
W
∖
{
0
}
∫
X
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
+
∫
X
|
v
(
x
)
-
v
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
(
∫
Ω
H
(
u
+
,
v
+
)
𝑑
x
)
p
p
s
*
.
We next prove a relationship between S~ and S~H, which will be used later. To prove it, we introduce some definitions and recall some known results. Let (s0,t0) (s0,t0≥0) be the one defined in Remark 2.1 (ii) for the p-homogeneous function F(s,t)=H(s,t)pps*, and m-1=MF. Then we have
m
H
(
s
,
t
)
p
p
s
*
≤
|
s
|
p
+
|
t
|
p
for all
s
,
t
∈
ℝ
and
(4.3)
m
H
(
s
0
,
t
0
)
p
p
s
*
=
s
0
p
+
t
0
p
.
We now state a lemma which will be needed in the proof of Lemma 4.2 below.
Lemma 4.1 (de Morais Filho and Souto [19]).
Let H be a ps*-homogeneous continuous function satisfying: The 1-homogeneous function G, defined by
G
(
s
p
s
*
,
t
p
s
*
)
=
H
(
s
,
t
)
for all
s
,
t
≥
0
,
is concave. Then the Hölder-type inequality holds:
∫
Ω
H
(
u
,
v
)
𝑑
x
≤
H
(
∥
u
∥
L
p
s
*
(
Ω
)
,
∥
v
∥
L
p
s
*
(
Ω
)
)
for all
u
,
v
∈
L
p
s
*
(
Ω
)
u
,
v
≥
0
.
Lemma 4.2.
Let Ω be a domain (not necessarily bounded). If (4.3) holds, then
S
~
H
=
m
S
~
.
Moreover, if S~ is attained at ω0, then S~H is attained at (s0ω0,t0ω0) for all s0,t0>0 satisfying (4.3).
Proof.
Consider {ωn}⊆W0s,p(Ω) a minimizing sequence for S~ and the sequence {(s0,t0)ωn}. Substituting this sequence in quotient (4.2), using (4.3) and taking the limit, we obtain
S
~
H
≤
m
S
~
.
For the reverse inequality, we choose {(un,vn)} be a minimizing sequence for S~H. By the definition of S~ and Lemma 4.1, we have
∫
X
|
u
n
(
x
)
-
u
n
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
+
∫
X
|
v
n
(
x
)
-
v
n
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
(
∫
Ω
H
(
u
n
,
v
n
)
𝑑
x
)
p
p
s
*
≥
S
~
∥
u
n
∥
L
p
s
*
(
Ω
)
p
+
∥
v
n
∥
L
p
s
*
(
Ω
)
p
H
(
∥
u
n
∥
L
p
s
*
(
Ω
)
,
∥
v
n
∥
L
p
s
*
(
Ω
)
)
p
p
s
*
≥
m
S
~
.
Taking the limit, we obtain the remained inequality. The rest of the proof is straightforward computation and we omit it. ∎
We also need the following definitions:
R
λ
(
u
)
=
∫
X
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
-
λ
∫
Ω
|
u
|
p
𝑑
x
(
∫
Ω
|
u
|
p
s
*
𝑑
x
)
p
p
s
*
,
λ
>
0
,
R
~
Q
(
u
,
v
)
=
1
p
(
∫
X
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
+
∫
X
|
v
(
x
)
-
v
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
)
-
∫
Ω
Q
(
u
,
v
)
𝑑
x
(
∫
Ω
H
(
u
+
,
v
+
)
𝑑
x
)
p
p
s
*
,
R
λ
=
inf
{
R
λ
(
u
)
:
u
∈
W
0
s
,
p
(
Ω
)
and
∥
u
∥
L
p
s
*
(
Ω
)
=
1
}
,
R
~
Q
=
inf
{
R
~
Q
(
u
,
v
)
:
(
u
,
v
)
∈
W
and
∫
Ω
H
(
u
+
,
v
+
)
𝑑
x
=
1
}
.
In the following, we establish a relationship between R~Q and S~H which is crucial to prove Theorem 1.2. In order to establish the result, we need some estimates obtained in [6, 17]. From [6], we know that there exist a minimizer for S(ℝN), and for every minimizer U, there exist x0∈ℝN and a constant sign monotone function u:ℝ→ℝ such that U(x)=u(|x-x0|). Next we fix a radially nonnegative decreasing minimizer U=U(r) for S. Multiplying U by a positive constant if necessary, we may assume that
(
-
Δ
p
)
s
U
=
U
p
s
*
-
1
.
Lemma 4.3 (Brasco, Mosconi and Squassina [6]).
There exist c1,c2>0 and θ>1 such that for all r≥1,
c
1
r
N
-
p
s
p
-
1
≤
U
(
r
)
≤
c
2
r
N
-
p
s
p
-
1
,
U
(
θ
r
)
U
(
r
)
≤
1
2
for all
r
≥
1
.
For every δ≥ε>0. Let us set
m
ε
,
δ
=
U
ε
(
δ
)
U
ε
(
δ
)
-
U
ε
(
θ
δ
)
and
g
ε
,
δ
(
t
)
=
{
0
if
0
≤
t
≤
U
ε
(
θ
δ
)
,
m
ε
,
δ
p
(
t
-
U
ε
(
θ
δ
)
)
if
U
ε
(
θ
δ
)
≤
t
≤
U
ε
(
δ
)
,
t
+
U
ε
(
δ
)
(
m
ε
,
δ
p
-
1
-
1
)
if
t
≥
U
ε
(
δ
)
,
as well as
G
ε
,
δ
(
t
)
=
∫
0
t
g
ε
,
δ
′
(
τ
)
1
p
𝑑
τ
=
{
0
if
0
≤
t
≤
U
ε
(
θ
δ
)
,
m
ε
,
δ
(
t
-
U
ε
(
θ
δ
)
)
if
U
ε
(
θ
δ
)
≤
t
≤
U
ε
(
δ
)
,
t
if
t
≥
U
ε
(
δ
)
.
The functions gε,δ and Gε,δ are nondecreasing and absolutely continuous. Consider now the radially symmetric nonincreasing function
u
ε
,
δ
(
r
)
=
G
ε
,
δ
(
U
ε
(
r
)
)
,
which satisfies
u
ε
,
δ
(
r
)
=
{
U
ε
(
r
)
if
r
≤
δ
,
0
if
r
≥
θ
δ
.
Then uε,δ∈W0s,p(Ω), for any δ<θ-1δΩ (δΩ:=dist(0,∂Ω)).
Lemma 4.4 (Brasco, Mosconi and Squassina [6], Chen, Mosconi and Squassina [17]).
There exists C>0 such that, for any 0<2ε≤δ<θ-1δΩ, there holds
∫
ℝ
2
N
|
u
ε
,
δ
(
x
)
-
u
ε
,
δ
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
≤
S
N
p
s
+
C
(
ε
δ
)
N
-
p
s
p
-
1
and
∫
ℝ
N
u
ε
,
δ
(
x
)
p
s
*
𝑑
x
≥
S
N
p
s
-
C
(
ε
δ
)
N
p
-
1
.
Moreover, for any β>0, there exists Cβ such that
∫
ℝ
N
u
ε
,
δ
(
x
)
β
𝑑
x
≥
C
β
{
ε
N
-
N
-
p
s
p
β
|
log
ε
δ
|
if
β
=
p
s
*
p
′
,
ε
N
-
p
s
p
(
p
-
1
)
β
δ
N
-
N
-
p
s
p
-
1
β
if
β
<
p
s
*
p
′
,
ε
N
-
N
-
p
s
p
β
if
β
>
p
s
*
p
′
.
Lemma 4.5.
If μ1>0 and N>p2s, then
(4.4)
R
~
Q
<
S
~
H
p
.
Proof.
Without loss of generality we may assume that 0∈Ω. Let us consider the truncated function uε,δ, and we let uε=s0uε,δ,vε=t0uε,δ. Using (1.9) and (4.2), we have
1
p
∫
X
|
u
ε
(
x
)
-
u
ε
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
+
1
p
∫
X
|
v
ε
(
x
)
-
v
ε
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
-
∫
Ω
Q
(
u
ε
,
v
ε
)
𝑑
x
(
∫
Ω
H
(
u
ε
,
v
ε
)
𝑑
x
)
p
p
s
*
≤
1
p
∫
X
|
u
ε
,
δ
(
x
)
-
u
ε
,
δ
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
-
μ
1
p
∫
Ω
u
ε
,
δ
(
x
)
p
𝑑
x
m
-
1
(
∫
Ω
u
ε
,
δ
(
x
)
p
s
*
𝑑
x
)
p
p
s
*
=
m
p
R
μ
1
(
u
ε
,
δ
)
,
that is,
(4.5)
R
~
Q
(
u
ε
,
v
ε
)
≤
m
p
R
μ
1
(
u
ε
,
δ
)
.
On the other hand, for μ1>0 and N>p2s, combining Lemma 4.4, we have
R
μ
1
(
u
ε
,
δ
)
=
∫
X
|
u
ε
,
δ
(
x
)
-
u
ε
,
δ
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
-
μ
1
∫
Ω
u
ε
,
δ
(
x
)
p
𝑑
x
(
∫
Ω
u
ε
,
δ
(
x
)
p
s
*
𝑑
x
)
p
p
s
*
=
∫
ℝ
2
N
|
u
ε
,
δ
(
x
)
-
u
ε
,
δ
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
-
μ
1
∫
Ω
u
ε
,
δ
(
x
)
p
𝑑
x
(
∫
Ω
u
ε
,
δ
(
x
)
p
s
*
𝑑
x
)
p
p
s
*
≤
S
N
p
s
+
C
(
ε
δ
)
N
-
p
s
p
-
1
-
C
p
μ
1
ε
N
-
N
-
p
s
p
p
[
S
N
p
s
-
C
(
ε
δ
)
N
p
-
1
]
N
-
p
s
N
=
[
S
N
p
s
+
C
(
ε
δ
)
N
-
p
s
p
-
1
-
C
p
μ
1
ε
p
s
]
[
S
N
p
s
-
C
(
ε
δ
)
N
p
-
1
]
p
s
N
S
N
p
s
-
C
(
ε
δ
)
N
p
-
1
≤
[
S
N
p
s
+
C
(
ε
δ
)
N
-
p
s
p
-
1
-
C
p
μ
1
ε
p
s
]
S
S
N
p
s
-
C
(
ε
δ
)
N
p
-
1
.
Since N>p2s, we have min{N-psp-1,Np-1}>ps, which implies that for small ε, we have
S
N
p
s
+
C
(
ε
δ
)
N
-
p
s
p
-
1
-
C
p
μ
1
ε
p
s
<
S
N
p
s
-
C
(
ε
δ
)
N
p
-
1
.
Combining Lemma 4.2 and (4.5), we have that R~Q(uε,vε)<S~Hp for a small ε, and hence (4.4) holds. ∎
5 A Concentration Compactness Principle Associated with the Fractional p-Laplacian System
In the next section we shall prove Theorem 1.2 using the fact that if R~Q<S~Hp, a minimizing sequence for R~Q indeed converges, i.e., compactness occurs for R~Q<S~Hp. This will be possible due to a concentration compactness principle associated with the fractional p-Laplacian system that we are to prove. The method is based on the seminal paper of P. L. Lions [33], and here we provide a more general proof which is inspired by the results recently obtained by Bonder, Saintier and Silva in [4].
First, we define the fractional (s,p)-gradient of a function u∈W0s,p(ℝN) as
|
D
s
u
(
x
)
|
p
=
∫
ℝ
N
|
u
(
x
+
h
)
-
u
(
x
)
|
p
|
h
|
N
+
p
s
𝑑
h
.
In the following, we shall use the notation |Dsu| denotes the fractional gradient of a function u, and the (s,p)-gradient is well defined a.e. in ℝN and |Dsu|∈Lp(ℝN).
Lemma 5.1.
Let Ω be a bounded domain of RN with Lipschitz boundary and let {(un,vn)}⊆W0s,p(Ω)×W0s,p(Ω) be a sequence such that un⇀u, vn⇀v in W0s,p(Ω). Let us assume
|
D
s
u
n
|
p
⇀
μ
,
|
D
s
v
n
|
p
⇀
ν
,
H
(
u
n
,
v
n
)
⇀
σ
,
in the sense of measure, where μ,ν and σ are bounded nonnegative measure on RN. Then there exist at most a countable set J, a family of distinct points {xj}j∈J⊆Ω¯ and {μj}j∈J,{νj}j∈J and {σj}j∈J⊆(0,∞) such that
(5.1)
σ
=
H
(
u
,
v
)
+
∑
j
∈
J
σ
j
δ
x
j
,
(5.2)
μ
≥
|
D
s
u
|
p
+
∑
j
∈
J
μ
j
δ
x
j
,
(5.3)
ν
≥
|
D
s
v
|
p
+
∑
j
∈
J
ν
j
δ
x
j
.
(Here δx indicates the Dirac mass at x.) Moreover, the following relations holds:
S
~
H
⋅
(
σ
j
)
p
p
s
*
≤
μ
j
+
ν
j
.
Before proving Lemma 5.1, we need the following lemmas.
Lemma 5.2.
Given ε>0, there exists Cε such that
|
H
(
s
+
a
,
t
+
b
)
-
H
(
s
,
t
)
|
≤
ε
(
|
s
|
p
s
*
+
|
t
|
p
s
*
)
+
C
ε
(
|
a
|
p
s
*
+
|
b
|
p
s
*
)
.
Proof.
Using the mean value theorem,
(5.4)
|
H
(
s
+
a
,
t
+
b
)
-
H
(
s
,
t
)
|
=
|
∇
H
(
s
+
θ
a
,
t
+
θ
b
)
⋅
(
a
,
b
)
|
,
for some θ∈(0,1). Using the homogeneity properties (Remark 2.1), and the inequality
|
A
+
θ
B
|
p
s
*
-
1
≤
C
p
s
*
-
1
(
|
A
|
p
s
*
-
1
+
|
B
|
p
s
*
-
1
)
for all
A
,
B
∈
ℝ
,
in (5.4), we obtain
|
H
(
s
+
a
,
t
+
b
)
-
H
(
s
,
t
)
|
≤
C
(
|
s
|
p
s
*
-
1
|
a
|
+
|
s
|
p
s
*
-
1
|
b
|
+
|
t
|
p
s
*
-
1
|
a
|
+
|
t
|
p
s
*
-
1
|
b
|
+
|
a
|
p
s
*
+
|
b
|
p
s
*
+
|
a
|
p
s
*
-
1
|
b
|
+
|
b
|
p
s
*
-
1
|
a
|
)
.
Applying Young’s inequality to the last inequality, we get our desired result. ∎
Lemma 5.3.
Let Ω⊆RN and suppose that σ is a measure on Ω, un(x)→u(x),vn(x)→v(x) a.e. on Ω and ∥un∥Lps*(Ω,dσ),∥vn∥Lps*(Ω,dσ) are bounded sequence. Then we have
(5.5)
∫
Ω
H
(
u
n
,
v
n
)
𝑑
σ
-
∫
Ω
H
(
u
n
-
u
,
v
n
-
v
)
𝑑
σ
=
∫
Ω
H
(
u
,
v
)
𝑑
σ
+
o
n
(
1
)
.
Proof.
The idea of this proof was borrowed from [9, 19]. Let us fix some ε>0 and let Cε be as in Lemma 5.2. Let us define the function
g
n
=
|
H
(
u
n
,
v
n
)
-
H
(
u
n
-
u
,
v
n
-
v
)
-
H
(
u
,
v
)
|
.
Notice that
g
n
≤
H
(
u
,
v
)
+
ε
(
|
u
n
-
u
|
p
s
*
+
|
v
n
-
v
|
p
s
*
)
+
C
ε
(
|
u
|
p
s
*
+
|
v
|
p
s
*
)
,
where we have used Lemma 5.2 with s=un-u, t=vn-v, u=a and v=b. Define
w
n
,
ε
=
g
n
-
ε
(
|
u
n
-
u
|
p
s
*
+
|
v
n
-
v
|
p
s
*
)
,
then Remark 2.1 (i) yields
|
w
n
,
ε
|
=
|
g
n
-
ε
(
|
u
n
-
u
|
p
s
*
+
|
v
n
-
v
|
p
s
*
)
|
≤
(
M
H
+
C
ε
)
(
|
u
|
p
s
*
+
|
v
|
p
s
*
)
∈
L
1
(
Ω
,
d
σ
)
.
Since un→u,vn→v a.e. x∈Ω, we have wn,ε(x)→0 a.e. in Ω as n→∞. The Lebesgue dominated convergence theorem implies that
lim
n
→
∞
∫
Ω
w
n
,
ε
𝑑
σ
=
0
.
Therefore, since gn=wn,ε+ε(|un-u|ps*+|vn-v|ps*) and ∥un∥Lps*(Ω,dσ),∥vn∥Lps*(Ω,dσ) are bounded sequences, we have
lim sup
n
→
∞
∫
Ω
g
n
𝑑
σ
≤
ε
lim sup
n
→
∞
∫
Ω
(
|
u
n
-
u
|
p
s
*
+
|
v
n
-
v
|
p
s
*
)
𝑑
σ
≤
C
ε
.
Since ε>0 is arbitrary, we conclude that (5.5) holds. ∎
In the following, we give two properties of the nonlocal (s,p)-gradient. The first one is a scaling property and the second one is a decay estimate for the nonlocal gradient of a function with compact support. Via these two techniques, we can give the proof of Lemma 5.1. Now, for u∈W0s,p(ℝN), we define ur,x0(x)=u(x-x0r), where r>0 and x0∈ℝN. By a straightforward computation, we have
(5.6)
|
D
s
u
r
,
x
0
(
x
)
|
p
=
1
r
p
s
|
D
s
u
(
x
-
x
0
r
)
|
p
.
Lemma 5.4 (Bonder, Saintier and Silva [4]).
Let v∈W1,∞(RN) be such that supp(v)⊂B1(0). Then there exists a constant C>0 depending on N,s,p and ∥v∥1,∞ such that
|
D
s
v
(
x
)
|
p
≤
C
min
{
1
,
|
x
|
-
(
N
+
p
s
)
}
.
Combining (5.6) and Lemma 5.4, we obtain the following:
Lemma 5.5.
Let ϕ∈W1,∞(RN) be such that supp(ϕ)⊂B1(0), and given r>0 and x0∈RN, we define
ϕ
r
,
x
0
=
ϕ
(
x
-
x
0
r
)
.
Then
|
D
s
ϕ
r
,
x
0
(
x
)
|
p
≤
C
min
{
r
-
p
s
,
r
N
|
x
-
x
0
|
-
(
N
+
p
s
)
}
,
where C>0 depends on N,s,p and ∥ϕ∥1,∞.
Lemma 5.6 (Bonder, Saintier and Silva [4]).
Let 0<s<1<p be such that ps<N and let p≤q<ps*. Let ω∈L∞(RN) be such that there exists α>0 and C>0 such that
0
≤
ω
(
x
)
≤
C
|
x
|
-
α
.
Then if α>sq-Nq-pp, W0s,p(RN)⊂⊂Lq(ωdx;RN). That is, for any bounded sequence {un}n∈N⊂W0s,p(RN), there exist a subsequence {unj}j∈N⊂{un}n∈N and a function u∈W0s,p(RN) such that unj⇀u weakly in W0s,p(RN) and
∫
ℝ
N
|
u
n
j
-
u
(
x
)
|
q
ω
(
x
)
𝑑
x
→
0
as
j
→
∞
.
For the detailed proofs of Lemma 5.4, Lemma 5.5 and Lemma 5.6, we refer to [4] and the references therein. Furthermore, it is worth noting that in the case p=q, we need α>ps in Lemma 5.6. So if ϕ∈W1,∞(ℝN) has compact support, then ω=|Dsϕ|p verifies the hypotheses of Lemma 5.6 with p=q.
Proof of Lemma 5.1.
The idea of proving (5.1) is passing to the limit in the following identity resulting from Lemma 5.3:
(5.7)
∫
Ω
|
ϕ
|
p
s
*
H
(
u
n
,
v
n
)
𝑑
x
=
∫
Ω
|
ϕ
|
p
s
*
H
(
u
,
v
)
𝑑
x
+
∫
Ω
|
ϕ
|
p
s
*
H
(
u
n
-
u
,
v
n
-
v
)
𝑑
x
+
o
n
(
1
)
.
Thus we consider the sequence {(u~n,v~n)}⊆W, where u~n=un-u,v~n=vn-v. In this particular case, from the assumption of Lemma 5.1, we know that u~n→0,v~n→0 a.e. on ℝN and in Llocq(ℝN) (1≤q<ps*). Moreover, we assume that |Dsu~n|p⇀u~, |Dsv~n|p⇀v~ and H(u~n,v~n)⇀σ~ in the sense of measure. Firstly, we show that the measures μ~,ν~ and σ~ verify a reverse Hölder inequality. In fact, given ϕ∈C0∞(ℝN), we will prove that
(5.8)
S
~
H
(
∫
ℝ
N
|
ϕ
|
p
s
*
𝑑
σ
~
)
p
p
s
*
≤
∫
ℝ
N
|
ϕ
|
p
(
d
μ
~
+
d
ν
~
)
.
From (4.2), we have
(5.9)
[
∫
ℝ
N
|
ϕ
|
p
s
*
H
(
u
n
-
u
,
v
n
-
v
)
𝑑
x
]
p
p
s
*
=
[
∫
Ω
|
ϕ
|
p
s
*
H
(
u
n
-
u
,
v
n
-
v
)
𝑑
x
]
p
p
s
*
=
[
∫
Ω
|
ϕ
|
p
s
*
H
(
u
~
n
,
v
~
n
)
𝑑
x
]
p
p
s
*
≤
S
~
H
-
1
[
∫
X
|
ϕ
(
x
)
u
~
n
(
x
)
-
ϕ
(
y
)
u
~
n
(
y
)
|
p
|
x
-
y
|
N
+
p
s
d
x
d
y
+
∫
X
|
ϕ
(
x
)
v
~
n
(
x
)
-
ϕ
(
y
)
v
~
n
(
y
)
|
p
|
x
-
y
|
N
+
p
s
d
x
d
y
]
=
S
~
H
-
1
[
∫
ℝ
2
N
|
ϕ
(
x
)
u
~
n
(
x
)
-
ϕ
(
y
)
u
~
n
(
y
)
|
p
|
x
-
y
|
N
+
p
s
d
x
d
y
+
∫
ℝ
2
N
|
ϕ
(
x
)
v
~
n
(
x
)
-
ϕ
(
y
)
v
~
n
(
y
)
|
p
|
x
-
y
|
N
+
p
s
d
x
d
y
]
=
S
~
H
-
1
[
∫
ℝ
N
|
D
s
(
ϕ
u
~
n
)
|
p
𝑑
x
+
∫
ℝ
N
|
D
s
(
ϕ
v
~
n
)
|
p
𝑑
x
]
.
For the right-hand side of (5.9), we consider the term ∫ℝN|Ds(ϕu~n)|p𝑑x. By using Minkowski’s inequality, we have
∥
D
s
(
ϕ
u
~
n
)
∥
p
=
(
∫
ℝ
N
|
D
s
(
ϕ
u
~
n
)
|
p
𝑑
x
)
1
p
≤
(
∫
ℝ
N
∫
ℝ
N
|
ϕ
(
y
)
(
u
~
n
(
x
)
-
u
~
n
(
y
)
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
)
1
p
+
(
∫
ℝ
N
∫
ℝ
N
|
u
~
n
(
x
)
(
ϕ
(
x
)
-
ϕ
(
y
)
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
)
1
p
=
(
∫
ℝ
N
∫
ℝ
N
|
ϕ
(
x
)
|
p
|
u
~
n
(
x
+
h
)
-
u
~
n
(
x
)
|
p
|
h
|
N
+
p
s
𝑑
h
𝑑
x
)
1
p
+
(
∫
ℝ
N
∫
ℝ
N
|
u
~
n
(
x
+
h
)
|
p
|
ϕ
(
x
+
h
)
-
ϕ
(
x
)
|
p
|
h
|
N
+
p
s
𝑑
h
𝑑
x
)
1
p
.
By a simple change of variable, we have
∫
ℝ
N
∫
ℝ
N
|
u
~
n
(
x
+
h
)
|
p
|
ϕ
(
x
+
h
)
-
ϕ
(
x
)
|
p
|
h
|
N
+
p
s
𝑑
h
𝑑
x
=
∫
ℝ
N
∫
ℝ
N
|
u
~
n
(
y
)
|
p
|
ϕ
(
y
)
-
ϕ
(
y
+
h
^
)
|
p
|
h
^
|
N
+
p
s
𝑑
h
^
𝑑
y
=
∫
ℝ
N
|
u
~
n
(
y
)
|
p
|
D
s
ϕ
(
y
)
|
p
𝑑
y
.
Hence, we get
∥
D
s
(
u
~
n
ϕ
)
∥
p
≤
(
∫
ℝ
N
|
ϕ
(
x
)
|
p
|
D
s
u
~
n
(
x
)
|
p
𝑑
x
)
1
p
+
(
∫
ℝ
N
|
u
~
n
(
x
)
|
p
|
D
s
ϕ
(
x
)
|
p
𝑑
x
)
1
p
.
Now, from Lemma 5.4, we know that the weight ω(x)=|Dsϕ(x)|p satisfies the hypotheses of Lemma 5.6, and hence u~n→0 strongly in Lp(ω). Therefore,
lim sup
n
→
∞
∥
D
s
(
ϕ
u
~
n
)
∥
p
≤
(
∫
ℝ
N
|
ϕ
|
p
𝑑
μ
~
)
1
p
.
Similarly, we have
lim sup
n
→
∞
∥
D
s
(
ϕ
v
~
n
)
∥
p
≤
(
∫
ℝ
N
|
ϕ
|
p
𝑑
ν
~
)
1
p
.
Using the above facts and passing to the limit in (5.9), we get that
S
~
H
(
∫
ℝ
N
|
ϕ
|
p
s
*
𝑑
σ
~
)
p
p
s
*
≤
∫
ℝ
N
|
ϕ
|
p
(
d
μ
~
+
d
ν
~
)
.
This concludes the proof of the reverse Hölder inequality (5.8). Since un=0 and vn=0 in Ωc, it is clearly that supp(σ~)⊆Ω¯. Then, from [33, Lemma 1.2], there exists a countable set J, points {xj}j∈J⊆Ω¯ and {σj}j∈J⊆(0,∞) such that
(5.10)
σ
~
=
∑
j
∈
J
σ
j
δ
x
j
.
From (5.7) we have σ=H(u,v)+σ~ and combining (5.10) we get that
σ
=
H
(
u
,
v
)
+
∑
j
∈
J
σ
j
δ
x
j
.
Now, to prove the relation between the weights σj and μj,νj, we take ϕ∈C0∞(ℝN), with 0≤ϕ≤1, ϕ(0)=1 and suppϕ=B(0,1). For given ε>0 and xj∈Ω¯, we consider the rescaled functions
ϕ
ε
,
x
i
(
x
)
=
ϕ
(
x
-
x
j
ε
)
.
Arguing as in the proof of inequality (5.9) again, we have
(5.11)
S
~
H
[
∫
ℝ
N
|
ϕ
ε
,
x
j
|
p
s
*
H
(
u
n
,
v
n
)
𝑑
x
]
p
p
s
*
≤
∫
ℝ
N
|
D
s
(
ϕ
ε
,
x
j
u
n
)
|
p
𝑑
x
+
∫
ℝ
N
|
D
s
(
ϕ
ε
,
x
j
v
n
)
|
p
𝑑
x
.
As before, we have
∥
D
s
(
ϕ
ε
,
x
j
u
n
)
∥
p
≤
(
∫
ℝ
N
|
ϕ
ε
,
x
j
(
x
)
|
p
|
D
s
u
n
(
x
)
|
p
𝑑
x
)
1
p
+
(
∫
ℝ
N
|
u
n
(
x
)
|
p
|
D
s
ϕ
ε
,
x
j
(
x
)
|
p
𝑑
x
)
1
p
.
Recall that from Lemma 5.5 we obtain that
(5.12)
|
D
s
ϕ
ε
,
x
j
(
x
)
|
p
≤
C
min
{
ε
-
s
p
,
ε
N
|
x
-
x
j
|
-
(
N
+
p
s
)
}
.
Now, (5.12) implies that |Dsϕε,xj|p satisfies the hypotheses of Lemma 5.6. Moreover, since un⇀u in W0s,p(Ω), we have that
(5.13)
lim
n
→
∞
∫
ℝ
N
|
u
n
(
x
)
|
p
|
D
s
ϕ
ε
,
x
j
(
x
)
|
p
𝑑
x
=
∫
ℝ
N
|
u
(
x
)
|
p
|
D
s
ϕ
ε
,
x
j
(
x
)
|
p
𝑑
x
.
Now we check that
(5.14)
∫
ℝ
N
|
u
(
x
)
|
p
|
D
s
ϕ
ε
,
x
j
(
x
)
|
p
𝑑
x
→
0
as
ε
→
0
.
Utilizing (5.12) again, we get
∫
ℝ
N
|
u
(
x
)
|
p
|
D
s
ϕ
ε
,
x
j
(
x
)
|
p
𝑑
x
≤
C
(
ε
-
p
s
∫
|
x
-
x
j
|
<
ε
|
u
|
p
𝑑
x
+
ε
N
∫
|
x
-
x
j
|
≥
ε
|
u
|
p
|
x
-
x
j
|
N
+
p
s
𝑑
x
)
=
C
(
I
+
I
I
)
.
The first one is the easiest one,
I
≲
ε
-
p
s
(
∫
|
x
-
x
j
|
<
ε
|
u
|
p
s
*
𝑑
x
)
p
p
s
*
⋅
|
B
ε
|
p
s
N
≲
(
∫
|
x
-
x
j
|
<
ε
|
u
|
p
s
*
𝑑
x
)
p
p
s
*
.
Since u∈Lps*(ℝN), thus I goes to zero as ε→0.
For the second term we proceed as follows:
I
I
=
∑
k
=
0
∞
ε
N
∫
2
k
ε
≤
|
x
-
x
j
|
≤
2
k
+
1
ε
|
u
|
p
|
x
-
x
j
|
N
+
p
s
𝑑
x
≤
∑
k
=
0
∞
1
2
k
(
N
+
p
s
)
ε
p
s
∫
|
x
-
x
j
|
≤
2
k
+
1
ε
|
u
|
p
𝑑
x
≤
∑
k
=
0
∞
1
2
k
(
N
+
p
s
)
ε
p
s
(
∫
|
x
-
x
j
|
≤
2
k
+
1
ε
|
u
|
p
s
*
𝑑
x
)
p
p
s
*
|
B
2
k
+
1
ε
|
p
s
N
=
c
∑
k
=
0
∞
1
2
k
N
(
∫
|
x
-
x
j
|
≤
2
k
+
1
ε
|
u
|
p
s
*
𝑑
x
)
p
p
s
*
,
where c depends only on N,s,p. Now, given δ>0, take k0∈ℕ such that c∑k=k0+1∞2-Nk<δ. So
I
I
≤
∥
u
∥
p
s
*
p
δ
+
c
∑
k
=
0
k
0
1
2
N
k
(
∫
|
x
-
x
j
|
<
2
k
0
+
1
ε
|
u
|
p
s
*
𝑑
x
)
p
p
s
*
=
∥
u
∥
p
s
*
p
δ
+
c
(
s
,
p
,
N
,
k
0
)
(
∫
|
x
-
x
j
|
<
2
k
0
+
1
ε
|
u
|
p
s
*
𝑑
x
)
p
p
s
*
.
Therefore, we obtain that lim supε→0II≤δ∥u∥ps*p, for any δ>0. This concluded the proof of (5.14).
Since |Dsun|p⇀μ, combined with (5.12)–(5.14), yields
lim
n
→
∞
∥
D
s
(
ϕ
ε
,
x
j
u
n
)
∥
p
≤
(
∫
ℝ
N
|
ϕ
ε
,
x
j
(
x
)
|
p
𝑑
μ
)
1
p
+
(
∫
ℝ
N
|
u
|
p
|
D
s
ϕ
ε
,
x
j
(
x
)
|
p
𝑑
x
)
1
p
→
μ
j
1
p
as
ε
→
0
.
Thus,
lim
n
→
∞
∫
ℝ
N
|
D
s
(
ϕ
ε
,
x
j
u
n
)
|
p
𝑑
x
≤
μ
j
as
ε
→
0
.
Similarly, we have
lim
n
→
∞
∫
ℝ
N
|
D
s
(
ϕ
ε
,
x
j
v
n
)
|
p
𝑑
x
≤
ν
j
as
ε
→
0
.
For the left-hand side of (5.11), (5.1) implies that
lim
n
→
∞
∫
ℝ
N
|
ϕ
ε
,
x
j
|
p
s
*
H
(
u
n
,
v
n
)
𝑑
x
=
∫
ℝ
N
|
ϕ
ε
,
x
j
|
p
s
*
H
(
u
,
v
)
𝑑
x
+
σ
j
as
ε
→
0
.
Furthermore, it is easy to verify that
∫
ℝ
N
|
ϕ
ε
,
x
j
|
p
s
*
H
(
u
,
v
)
𝑑
x
→
0
as
ε
→
0
.
Using the above fact and passing to the limit in (5.11), we obtain that
S
~
H
σ
j
p
p
s
*
≤
μ
j
+
ν
j
for all
j
∈
J
.
Finally, we have
μ
≥
∑
j
∈
J
μ
j
δ
x
j
,
ν
≥
∑
j
∈
J
ν
j
δ
x
j
.
By the Fatou Lemma, we obtain that μ≥|Dsu|p and ν≥|Dsv|p. Since |Dsu|p and |Dsv|p are orthogonal to ∑j∈Jμjδxj and ∑j∈Jνjδxj, respectively, it follows that (5.2) and (5.3) hold. ∎
6 Proof of Theorem 1.2
Following the subcritical case (Theorem 1.1), it only remains to show that the infimum
R
~
Q
:=
min
(
u
,
v
)
∈
ℋ
I
(
u
,
v
)
is attained. Let {(un,vn)}⊂ℋ be a minimizing sequence for R~Q, i.e.,
1
p
[
∫
ℝ
2
N
|
u
n
(
x
)
-
u
n
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
+
∫
ℝ
2
N
|
v
n
(
x
)
-
v
n
(
y
)
|
p
|
x
-
y
|
N
+
p
s
𝑑
x
𝑑
y
]
-
∫
Ω
Q
(
u
n
,
v
n
)
𝑑
x
=
1
p
(
∫
ℝ
N
|
D
s
u
n
|
p
𝑑
x
+
∫
ℝ
N
|
D
s
v
n
|
p
𝑑
x
)
-
∫
Ω
Q
(
u
n
,
v
n
)
𝑑
x
=
R
~
Q
+
o
n
(
1
)
.
As in the proof of Theorem 1.1, ∥(un,vn)∥ is bounded. Let us suppose that un⇀u,vn⇀v in W0s,p(Ω) and assume the assumptions of Lemma 4.2. Moreover, since H nonnegative, by the above convergences we have 0≤∫ΩH(u,v)𝑑x≤1. On the other hand, by Lemma 5.1, we have
(6.1)
I
(
u
,
v
)
=
1
p
[
∫
ℝ
N
|
D
s
u
|
p
𝑑
x
+
∫
ℝ
N
|
D
s
v
|
p
𝑑
x
]
-
∫
Ω
Q
(
u
,
v
)
𝑑
x
=
1
p
[
∫
ℝ
N
|
D
s
u
n
|
p
d
x
+
∫
ℝ
N
|
D
s
v
n
|
p
d
x
]
-
1
p
[
∫
ℝ
N
|
D
s
u
n
|
p
d
x
-
∫
ℝ
N
|
D
s
u
|
p
d
x
+
∫
ℝ
N
|
D
s
v
n
|
p
d
x
-
∫
ℝ
N
|
D
s
v
|
p
d
x
]
-
∫
Ω
Q
(
u
n
,
v
n
)
d
x
≤
R
~
Q
-
1
p
[
∫
ℝ
N
𝑑
μ
-
∫
ℝ
N
|
D
s
u
|
p
𝑑
x
+
∫
ℝ
N
𝑑
ν
-
∫
ℝ
N
|
D
s
v
|
p
𝑑
x
]
+
o
n
(
1
)
≤
R
~
Q
-
1
p
(
∑
j
∈
J
μ
j
+
∑
j
∈
J
ν
j
)
+
o
n
(
1
)
≤
R
~
Q
-
S
~
H
p
∑
j
∈
J
σ
j
p
p
s
*
.
In addition, by (5.1), we have
1
=
∫
Ω
𝑑
σ
=
∫
Ω
H
(
u
,
v
)
𝑑
x
+
∑
j
∈
J
σ
j
.
Case I.
We have σj=0 for all j∈J. In this case, we get that I(u,v)≤R~Q and ∫ΩH(u,v)𝑑x=1, i.e., (u,v)∈ℋ is the wanted minimum point.
Case II.
Suppose on the contrary, that there is at least one σj>0 for some j∈J. By Lemma 4.5, R~Q<S~Hp and hence (6.1) yields
(6.2)
I
(
u
,
v
)
<
R
~
Q
-
R
~
Q
∑
j
∈
J
σ
j
p
p
s
*
=
R
~
Q
(
1
-
∑
j
∈
J
σ
j
p
p
s
*
)
.
Since 0<σj<1, the following holds:
1
-
∑
j
∈
J
σ
j
p
p
s
*
≤
(
1
-
∑
j
∈
J
σ
j
)
p
p
s
*
=
(
∫
Ω
H
(
u
,
v
)
𝑑
x
)
p
p
s
*
.
The last inequality in (6.2) gives
(6.3)
I
(
u
,
v
)
<
R
~
Q
(
∫
Ω
H
(
u
,
v
)
𝑑
x
)
p
p
s
*
.
Since I(u,v)≥0, we have A=∫ΩH(u,v)𝑑x>0. Therefore (u,v)≠(0,0). Taking u~=uA1/ps* and v~=vA1/ps*, inequality (6.3) is reduced to
I
(
u
~
,
v
~
)
=
1
A
p
p
s
*
I
(
u
,
v
)
<
R
~
Q
(
1
A
∫
Ω
H
(
u
,
v
)
𝑑
x
)
p
p
s
*
=
R
~
Q
,
which contradicts the definition of R~Q. Hence, σj=0 for all j∈J and the proof of Theorem 1.2 is now completed.
Acknowledgements
The authors wish to thank the referee for his many useful comments which have improved the exposition of the paper.
References
[1] C. O. Alves, D. C. de Morais Filho and M. A. S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal. 42 (2000), no. 5, 771–787. 10.1016/S0362-546X(99)00121-2Suche in Google Scholar
[2] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381. 10.1016/0022-1236(73)90051-7Suche in Google Scholar
[3] A. Ambrosetti and M. Struwe, A note on the problem -Δu=λu+u|u|2∗-2, Manuscripta Math. 54 (1986), no. 4, 373–379. 10.1007/BF01168482Suche in Google Scholar
[4] J. F. Bonder, N. Saintier and A. Silva, The concentration-compactness principle for fractional order Sobolev spaces in unbounded domains and applications to the generalized fractional Brezis–Nirenberg problem, NoDEA Nonlinear Differential Equations Appl. 25 (2018), no. 6, Paper No. 52. 10.1007/s00030-018-0543-5Suche in Google Scholar
[5] L. Brasco and E. Lindgren, Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case, Adv. Math. 304 (2017), 300–354. 10.1016/j.aim.2016.03.039Suche in Google Scholar
[6] L. Brasco, S. Mosconi and M. Squassina, Optimal decay of extremals for the fractional Sobolev inequality, Calc. Var. Partial Differential Equations 55 (2016), no. 2, Article No. 23. 10.1007/s00526-016-0958-ySuche in Google Scholar
[7] L. Brasco and E. Parini, The second eigenvalue of the fractional p-Laplacian, Adv. Calc. Var. 9 (2016), no. 4, 323–355. 10.1515/acv-2015-0007Suche in Google Scholar
[8] L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional p-Laplacian, Discrete Contin. Dyn. Syst. 36 (2016), no. 4, 1813–1845. 10.3934/dcds.2016.36.1813Suche in Google Scholar
[9] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486–490. 10.1007/978-3-642-55925-9_42Suche in Google Scholar
[10] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477. 10.1002/cpa.3160360405Suche in Google Scholar
[11] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7–9, 1245–1260. 10.1080/03605300600987306Suche in Google Scholar
[12] A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 6, 463–470. 10.1016/s0294-1449(16)30395-xSuche in Google Scholar
[13] W. Chen and Y. Gui, Multiple solutions for a fractional p-Kirchhoff problem with Hardy nonlinearity, Nonlinear Anal. 188 (2019), 316–338. 10.1016/j.na.2019.06.009Suche in Google Scholar
[14] W. Chen and C. Li, Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math. 335 (2018), 735–758. 10.1016/j.aim.2018.07.016Suche in Google Scholar
[15] W. Chen, C. Li and S. Qi, A Hopf lemma and regularity for fractional p-Laplacian, preprint (2018), https://arxiv.org/abs/1805.02776v2. 10.3934/dcds.2020034Suche in Google Scholar
[16] W. Chen, C. Li and J. Zhu, Fractional equations with indefinite nonlinearities, Discrete Contin. Dyn. Syst. 39 (2019), no. 3, 1257–1268. 10.3934/dcds.2019054Suche in Google Scholar
[17] W. Chen, S. Mosconi and M. Squassina, Nonlocal problems with critical Hardy nonlinearity, J. Funct. Anal. 275 (2018), no. 11, 3065–3114. 10.1016/j.jfa.2018.02.020Suche in Google Scholar
[18] W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations 260 (2016), no. 5, 4758–4785. 10.1016/j.jde.2015.11.029Suche in Google Scholar
[19] D. C. de Morais Filho and M. A. S. Souto, Systems of p-Laplacean equations involving homogeneous nonlinearities with critical Sobolev exponent degrees, Comm. Partial Differential Equations 24 (1999), no. 7–8, 1537–1553. 10.1080/03605309908821473Suche in Google Scholar
[20] A. Di Castro, T. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal. 267 (2014), no. 6, 1807–1836.10.1016/j.jfa.2014.05.023Suche in Google Scholar
[21] A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 5, 1279–1299. 10.1016/j.anihpc.2015.04.003Suche in Google Scholar
[22] G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Math. Univ. Parma (N.S.) 5 (2014), no. 2, 373–386. Suche in Google Scholar
[23] J. P. García Azorero and I. Peral Alonso, Existence and nonuniqueness for the p-Laplacian: Nonlinear eigenvalues, Comm. Partial Differential Equations 12 (1987), no. 12, 1389–1430. 10.1080/03605308708820534Suche in Google Scholar
[24] A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var. 9 (2016), no. 2, 101–125. 10.1515/acv-2014-0024Suche in Google Scholar
[25] A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional p-Laplacian, Rev. Mat. Iberoam. 32 (2016), no. 4, 1353–1392. 10.4171/RMI/921Suche in Google Scholar
[26] T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys. 337 (2015), no. 3, 1317–1368. 10.1007/s00220-015-2356-2Suche in Google Scholar
[27] N. Lam and G. Lu, Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in ℝN, J. Funct. Anal. 262 (2012), no. 3, 1132–1165. 10.1016/j.jfa.2011.10.012Suche in Google Scholar
[28] N. Lam and G. Lu, Sharp Moser–Trudinger inequality on the Heisenberg group at the critical case and applications, Adv. Math. 231 (2012), no. 6, 3259–3287. 10.1016/j.aim.2012.09.004Suche in Google Scholar
[29] N. Lam and G. Lu, N-Laplacian equations in ℝN with subcritical and critical growth without the Ambrosetti–Rabinowitz condition, Adv. Nonlinear Stud. 13 (2013), no. 2, 289–308. 10.1515/ans-2013-0203Suche in Google Scholar
[30] N. Lam and G. Lu, Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti-Rabinowitz condition, J. Geom. Anal. 24 (2014), no. 1, 118–143. 10.1007/s12220-012-9330-4Suche in Google Scholar
[31] E. H. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349–374. 10.1007/978-3-642-55925-9_43Suche in Google Scholar
[32] E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations 49 (2014), no. 1–2, 795–826. 10.1007/s00526-013-0600-1Suche in Google Scholar
[33] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201. 10.4171/RMI/6Suche in Google Scholar
[34] G. Lu and Q. Yang, Paneitz operators on hyperbolic spaces and high order Hardy–Sobolev–Maz’ya inequalities on half spaces, Amer. J. Math. 141 (2019), no. 6, 1777–1816. 10.1353/ajm.2019.0047Suche in Google Scholar
[35] S. A. Marano and S. J. N. Mosconi, Asymptotics for optimizers of the fractional Hardy–Sobolev inequality, Commun. Contemp. Math. 21 (2019), no. 5, Article ID 1850028. 10.1142/S0219199718500281Suche in Google Scholar
[36] S. Mosconi, K. Perera, M. Squassina and Y. Yang, The Brezis-Nirenberg problem for the fractional p-Laplacian, Calc. Var. Partial Differential Equations 55 (2016), no. 4, Article ID 105. 10.1007/s00526-016-1035-2Suche in Google Scholar
[37] S. Mosconi and M. Squassina, Nonlocal problems at nearly critical growth, Nonlinear Anal. 136 (2016), 84–101. 10.1016/j.na.2016.02.012Suche in Google Scholar
[38] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71), 1077–1092. 10.1512/iumj.1971.20.20101Suche in Google Scholar
[39] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, American Mathematical Society, Providence, 1986. 10.1090/cbms/065Suche in Google Scholar
[40] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc. 367 (2015), no. 1, 67–102. 10.1090/S0002-9947-2014-05884-4Suche in Google Scholar
[41] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483. 10.1512/iumj.1968.17.17028Suche in Google Scholar
[42] L. Wu and W. Chen, The sliding methods for the fractional p-Laplacian, Adv. Math. 361 (2020), Article ID 106933. 10.1016/j.aim.2019.106933Suche in Google Scholar
[43] Y. Yang and K. Perera, N-Laplacian problems with critical Trudinger–Moser nonlinearities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), no. 4, 1123–1138. 10.2422/2036-2145.201406_004Suche in Google Scholar
