Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 11, pp. 1–17.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

EXISTENCE OF SOLUTIONS FOR CRITICAL FRACTIONAL

p-LAPLACIAN EQUATIONS WITH INDEFINITE WEIGHTS

NA CUI, HONG-RUI SUN

Abstract. This article concerns the critical fractional p-Laplacian equation

with indefinite weights

(−∆p)su = λg(x)|u|p−2u+ h(x)|u|p
∗
s−2u in RN ,

where 0 < s < 1 < p <∞, N > sp and p∗s = Np/(N−sp), the weight functions

g may be indefinite, and h changes sign. Specifically, based on the results

of asymptotic estimates for an extremal in the fractional Sobolev inequality
and the discrete spectrum of fractional p-Laplacian operator, we establish an

existence criterion for a nontrivial solution to this problem.

1. Introduction

The purpose of this article is to study the existence of nontrivial solutions for
the critical fractional p-Laplacian equation

(−∆p)
su = λg(x)|u|p−2u+ h(x)|u|p

∗
s−2u in RN , (1.1)

where 0 < s < 1 < p <∞, N > sp, p∗s = Np/(N −sp) and (−∆p)
s is the fractional

p-Laplacian operator which is defined as

(−∆p)
su(x) = 2 lim

ε→0+

∫
RN\Bε(x)

|u(x)− u(y)|p−2(u(x)− u(y))

|x− y|N+sp
dy, ∀x ∈ RN ,

the weight functions g may be indefinite and h changes sign.
In recent years, there has been an increasing attention to nonlocal diffusion

problems, in particular to the ones driven by the fractional Laplacian operator [15].
One of the reasons for this comes from the fact that this operator naturally arises
in several physical phenomena like flames propagation [9] and anomalous diffusion
[22], in geophysical fluid dynamics [11], or in mathematical finance [13] and so on. It
also provides a simple model to describe certain jump Lévy processes in probability
theory [3].

For fractional Laplacian problems on bounded domains, an approach was pro-
posed by Caffarelli and Silvestre in [10], which allows to transform a nonlocal prob-
lem into a local problem via the Dirichlet-Neumann map. By employing the har-
monic extension, a large number of existence results have been obtained for Dirichlet
problems involving the fractional Laplacian in bounded domain. In particular, to

2010 Mathematics Subject Classification. 35R11, 35J92, 35B33.

Key words and phrases. Fractional p-Laplacian; critical exponent; indefinite weight.
c©2021 Texas State University.

Submitted April 1, 2020. Published March 5, 2021.

1



2 N. CUI, H.-R. SUN EJDE-2021/11

deal with the critical nonlinearity, we refer to [4, 5, 12] and the references therein.
Furthermore, Servadei and Valdinoci [25] studied the Brezis-Nirenberg problem

(−∆p)
su = λ|u|p−2u+ |u|p

∗
s−2u in Ω,

u = 0 in RN\Ω,
and proved the existence of solutions for the above problem with p = 2. Mosconi
et al. [23] deduced the existence of nontrivial solution for the above problem un-
der different conditions of N , s, p and λ, by utilizing the asymptotic estimates
for the minimizers obtained in [7] and an abstract linking theorem based on the
cohomological index [29]. It is worth noting that these results were established for
a bounded domain.

There are a lot of novel researches and interesting results of fractional Laplacian
in the whole space, see [1, 2, 6, 8, 16, 18, 20, 24, 26, 30] and the references therein.
For instance, Dipierro et al. [16] considered the existence of a positive solution for
the problem

(−∆)su = εf(x)uq + u2∗s−1 in RN ,
where 0 ≤ q < 2∗s − 1, ε > 0 is a small parameter and f is a continuous and
compactly supported function. Due to the lack of regularity of the associated
energy functional, the case 0 < q < 1 is particularly difficult. Moreover, in [8],
Bucur and Medina also investigated the above problem when 1 ≤ q < 2∗s − 1 by
using different methods from [16]. Bonder et al. [6] dealt with the following critical
equation involving the fractional p-Laplacian

(−∆p)
su = λf(x)|u|q−2u+K(x)|u|p

∗
s−2u in RN ,

where p ≤ q < p∗s, 0 ≤ f ∈ L1
loc(RN ) is such that the embedding Ds,p(RN ) ↪→

Lq(fdx;RN ) is compact and K is nonnegative, bounded and has a limit at ∞. By
the concentration-compactness principle and mountain pass theorem, they obtained
some existence results for two cases: q = p and p < q < p∗s, respectively. Recently,
Ambrosio et al. [1] investigated the existence and concentration of positive solutions
for the p-fractional Schrödinger equation

εsp(−∆p)
su+ V (x)|u|p−2u = f(x) + γ|u|p

∗
s−2u in RN ,

where ε is a small parameter, γ ∈ {0, 1}, V is a continuous positive potential having
a local minimum and f is a superlinear continuous function with subcritical growth.
The main results were deduced via penalization techniques and suitable variational
arguments. However, it is worth emphasizing that, we can find that weight functions
were all assumed to be positive in these above mentioned literatures.

Motivated by the above analysis, we study the existence of nontrivial solution
for problem (1.1), which is closely related to the principle eigenvalue of the problem

(−∆p)
su = λg(x)|u|p−2u in RN . (1.2)

In [14], we established the existence of a sequence of eigenvalues which converges to
infinity for problem (1.2), and the principle eigenvalue is simple. The corresponding
eigenfunction may be positive in RN (see Lemma 2.1 below), which is a key step to
prove the (PS)c condition. In this article we extend the results in [17] for critical
classical Laplacian problem to the fractional setting.

It is worth mentioning that, as far as we know, there is no result for fractional
p-Laplacian in RN with indefinite weights. This problem will present us with two
main difficulties: firstly, the lack of an explicit formula of the extremal for the



EJDE-2021/11 CRITICAL FRACTIONAL p-LAPLACIAN EQUATION 3

Sobolev embedding Ds,p(RN ) ↪→ Lp
∗
s (RN ). For p = 2, we know that the extremal

is of the explicit form cU( |x−x0|
ε ) with

U(x) =
1

(1 + |x|p′)
N−sp
p

, x ∈ RN ,

where p′ = p/(p− 1) is the Hölder conjugate of p, c 6= 0, x0 ∈ RN and ε > 0, which
was firstly proved by Lieb in [19] and can be devoted to proving the existence
of solution. Although it has been conjectured that this extremal has a similar
explicit form for the general case, to the best of our knowledge, it is still open
until now. However, we can verify an existence of nontrivial solution by utilizing
the asymptotic estimates for the extremal, which was obtained by Brasco et al.
[7]. Secondly, the nonlinearities contain some indefinite weights, which give us
further difficulty. Indeed, the indefinite weight case is more complicated than that
of the positive one (see proof of Lemma 3.1), when we consider the convergence
in the corresponding topology space, although we can define some spaces, their
topology (or norm) are naturally related to indefinite weight functions, to give some
technical assistance. Concerning the whole space, it causes the lack of the compact
embedding, we will overcome the difficulty by the new concentration-compactness
principle obtained in Bonder et al. [6] recently.

In this article, we assume that the weight functions g and h satisfy the following

conditions: let g = g+ − g− with g+, g− ≥ 0, g+ ∈ L∞(RN ) ∩ L
N
sp (RN ), g− ∈

L∞(RN ), and

(A1) h ∈ L∞(RN ) and h+ 6≡ 0;

(A2) there exist constants ρ > 0 and θ > 1 such that h(x) = h(0) + o(|x|
N
p ) for

x ∈ B(0, ρθ);
(A3) h(0) = ‖h‖∞ and h(x) > 0 for x ∈ B(0, ρθ);
(A4) g(x) ≥ g0 > 0 in B(0, ρθ).

Next, we state the main result of this article.

Theorem 1.1. Assume that (A1)–(A4) hold, then for any λ ∈ (0, λ+
1 ), problem

(1.1) admits at least a nontrivial solution.

The remainder of this paper is organized as follows. In Section 2, we collect
some notation and known results, and we prove some preliminaries. In Section 3,
by using the concentration-compactness principle and mountain pass theorem, we
prove Theorem 1.1.

2. Preliminaries

We recall that for any 0 < s < 1, the fractional Sobolev space is defined by

Ds,p(RN ) =
{
u ∈ Lp

∗
s (RN ) : [u]s,p <∞

}
,

where the term

[u]s,p =
(∫∫

R2N

|u(x)− u(y)|p

|x− y|N+sp
dx dy

)1/p

is the so-called Gagliardo seminorm of u. With the induced norm ‖u‖Ds,p(RN ) =

[u]s,p, the space Ds,p(RN ) is a uniformly convex Banach space, and there exists a
positive constant Cp∗s such that

‖u‖p∗s ≤ Cp∗s [u]s,p for u ∈ Ds,p(RN ) .



4 N. CUI, H.-R. SUN EJDE-2021/11

Let the best Sobolev constant in this inequality be

S = inf
u∈Ds,p(RN )\{0}

[u]ps,p
‖u‖pp∗s

. (2.1)

Next, we define the fractional (s, p)-gradient of a function u ∈ Ds,p(RN ) as (see [6])

|Dsu(x)|p =

∫
RN

|u(x+ t)− u(x)|p

|t|N+sp
dt.

Observe that this (s, p)-gradient is well defined in RN and |Dsu(x)| ∈ Lp(RN ).
We introduce a weight function

ω(x) =
1

(1 + |x|)sp
, x ∈ RN ,

and define w(x) = max{g−(x), ω(x)}. Let X be the completion of C∞0 (RN ) with
respect to the norm

‖u‖X =
(

[u]ps,p +

∫
RN

w|u|p dx
)1/p

,

then X is a uniformly convex Banach space [14, Lemma 2.1]. Obviously, note that
the embedding X ↪→ Ds,p(RN ) is continuous, and since Ds,p(RN ) ↪→ Lσloc(RN ) is
compact for σ ∈ [1, p∗s), we deduce that the embedding X ↪→ Lσloc(RN ) is compact
for σ ∈ [1, p∗s).

Weak solutions of (1.1) coincide with critical points of the C1-functional Jλ :
X → R,

Jλ(u) =
1

p
[u]ps,p −

λ

p

∫
RN

g|u|p dx− 1

p∗s

∫
RN

h|u|p
∗
s dx, u ∈ X. (2.2)

Lemma 2.1 ([14, Theorem 1.1]). Suppose that g+ ∈ L
N
sp (RN ) ∩ L∞(RN ), g− ∈

L∞(RN ) and g+, g− ≥ 0. Then there exists a simple eigenvalue λ+
1 > 0 such that

the eigenvalue problem (1.2) has a positive eigenfunction u+
1 ∈ X associated with

λ+
1 . Moreover, for any λ ∈ (0, λ+

1 ], we have∫∫
R2N

|u(x)− u(y)|p

|x− y|N+sp
dx dy − λ

∫
RN

g|u|p dx ≥ 0, u ∈ X.

The following lemma provides the concentration-compactness principle for frac-
tional Laplacian operator in unbounded domains.

Lemma 2.2 ([6, Theorem 1.1]). Let {uk}k∈N ⊂ Ds,p(RN ) be a weakly convergent
sequence with weak limit u. Then there exist two bounded measures µ and ν, an at
most enumerable set of indices I, xi ∈ RN , and positive real numbers µi, νi, i ∈ I,
such that the following convergence hold weakly∗ in the sense of measures,

|Dsuk|p dx ⇀ µ ≥ |Dsu|p dx+
∑
i∈I

µiδxi ,

|uk|p
∗
s dx ⇀ ν = |u|p

∗
s dx+

∑
i∈I

νiδxi ,

Sν
p/p∗s
i ≤ µi, for all i ∈ I.

Moreover, if we define

µ∞ = lim
R→∞

lim sup
k→∞

∫
|x|>R

|Dsuk|p dx, ν∞ = lim
R→∞

lim sup
k→∞

∫
|x|>R

|uk|p
∗
s dx,



EJDE-2021/11 CRITICAL FRACTIONAL p-LAPLACIAN EQUATION 5

then

lim sup
k→∞

∫
RN
|Dsuk|p dx = µ(RN ) + µ∞,

lim sup
k→∞

∫
RN
|uk|p

∗
s dx = ν(RN ) + ν∞,

Sν
p/p∗s∞ ≤ µ∞.

Now, we estimate the decay of this nonlocal gradient and a compactness conclu-
sion with weights.

Lemma 2.3 ([6, Coroll. 2.3]). Let φ ∈W 1,∞(RN ) be such that supp(φ) ⊂ B(0, 1),
given r > 0 and x0 ∈ RN , we define φr,x0

(x) = φ(x−x0

r ). Then

|Dsφr,x0
(x)|p ≤ C min

{
r−sp, rN |x− x0|−(N+sp)

}
,

where C > 0 depends on N, s, p and ‖φ‖W 1,∞(RN ).

Lemma 2.4 ([6, Lemma 2.4]). If a ∈ L∞(RN ), there exist α > 0 and C > 0 such
that

0 ≤ a(x) ≤ C|x|−α.
Then, if α > sp, the embedding Ds,p(RN ) ↪→ Lp(adx;RN ) is compact. That is, for
any bounded sequence {uk}k∈N ⊂ Ds,p(RN ), there exists a subsequence {ukj}j∈N ⊂
{uk}k∈N and a function u ∈ Ds,p(RN ) such that∫

RN
a|ukj − u|p dx→ 0 as j →∞.

To verify the existence of solution for (1.1), we first need to show some prelimi-
nary results.

Lemma 2.5. Assume that un ⇀ u in Ds,p(RN ). Then for f ∈ L∞(RN ), there
exists a subsequence of {un}, still denoted by {un}, such that for any ϕ ∈ C∞0 (RN ),

(i)
∫
RN f |un|

p∗s−2unϕdx→
∫
RN f |u|

p∗s−2uϕdx;

(ii)
∫
RN f |un|

p−2unϕdx→
∫
RN f |u|

p−2uϕdx.

Proof. (i) Since un ⇀ u in Ds,p(RN ) and the embedding Ds,p(RN ) ↪→ Lσloc(RN ) is
compact for σ ∈ [1, p∗s), we can assume that

un → u a.e. in RN ,
so f |un|p

∗
s−2un → f |u|p∗s−2u a.e. in RN . Since {un} is bounded in Lp

∗
s (RN ), it

follows from f ∈ L∞(RN ) that {f |un|p
∗
s−2un} is bounded in L

p∗s
p∗s−1 (RN ). Thus,

we can get that f |un|p
∗
s−2un ⇀ f |u|p∗s−2u in L

p∗s
p∗s−1 (RN ) (see [27]). For any ϕ ∈

C∞0 (RN ), we obtain∫
RN

f |un|p
∗
s−2unϕdx→

∫
RN

f |u|p
∗
s−2uϕdx.

(ii) It can be proved by a similar method as (i), so we omit it here. �

Lemma 2.6. Suppose that un ⇀ u in Ds,p(RN ), then there exists a subsequence
of {un}, denoted still by {un}, such that

|un(x)− un(y)|p−2(un(x)− un(y))

|x− y|
N+sp
p′

⇀
|u(x)− u(y)|p−2(u(x)− u(y))

|x− y|
N+sp
p′



6 N. CUI, H.-R. SUN EJDE-2021/11

in Lp
′
(RN × RN ), where p′ = p

p−1 is the Hölder conjugate of p.

Proof. For simplicity, we define

ξn(x, y) =
|un(x)− un(y)|p−2(un(x)− un(y))

|x− y|
N+sp
p′

.

Since {un} is bounded in Ds,p(RN ), we obtain that {ξn} is bounded in Lp
′
(RN ×

RN ), hence there exists ξ ∈ Lp′(RN × RN ) such that ξn ⇀ ξ in Lp
′
(RN × RN ) up

to a subsequence.
In addition, since un ⇀ u inDs,p(RN ) and the embeddingDs,p(RN ) ↪→ Lσloc(RN )

is compact for σ ∈ [1, p∗s), we can assume that un → u a.e. in RN . Then it follows

that ξn(x, y)→ |u(x)−u(y)|p−2(u(x)−u(y))

|x−y|
N+sp
p′

a.e. in RN × RN . Combining this with the

boundedness of {ξn}, we obtain that

ξn(x, y) ⇀
|u(x)− u(y)|p−2(u(x)− u(y))

|x− y|
N+sp
p′

in Lp
′
(RN × RN ).

Thus

ξ(x, y) =
|u(x)− u(y)|p−2(u(x)− u(y))

|x− y|
N+sp
p′

.

The proof is complete. �

Lemma 2.7. Let {un} ⊂ Ds,p(RN ) be a bounded sequence such that Jλ(un) → c
and J ′λ(un) → 0 as n → ∞, and let µi, νi be as in Lemma 2.2. Then we have the
following estimates:

νi ≥ S
N
sph(xi)

− N
sp , µi ≥ S

N
sph(xi)

1− N
sp if h(xi) > 0,

νi = µi = 0 if h(xi) = 0.

Proof. The proof closely follows the technique of [6, Lemma 3.6]. We first show that
the set I in Lemma 2.2 is finite. Fix a concentration point xi, let φ ∈ C∞0 (RN , [0, 1])
be such that

φ(x) =

{
1, if |x| ≤ 1,

0, if |x| ≥ 2,

and let φδ(x) = φ(x−xiδ ) for δ > 0. According to J ′λ(un)→ 0 as n→∞, taking the
test function ϕ = unφδ, then we have

lim
n→∞

(
〈(−∆p)

sun, unφδ〉 − λ
∫
RN
|un|pφδ dx−

∫
RN

h|un|p
∗
sφδ dx

)
= lim
n→∞

〈J ′λ(un), unφδ〉 = 0.

Thus, it follows from Lemma 2.2, g, h ∈ L∞(RN ) and un → u in Lploc(RN ) that

lim
n→∞

〈(−∆p)
sun, unφδ〉 = λ

∫
RN

g|u|pφδ dx+

∫
RN

hφδdν. (2.3)

Next, we verify that

lim
δ→0

lim
n→∞

〈(−∆p)
sun, unφδ〉 = µi. (2.4)

We can write

〈(−∆p)
sun, unφδ〉



EJDE-2021/11 CRITICAL FRACTIONAL p-LAPLACIAN EQUATION 7

=

∫∫
R2N

|un(x)− un(y)|p−2(un(x)− un(y))(un(x)φδ(x)− un(y)φδ(y))

|x− y|N+sp
dx dy

=

∫∫
R2N

|un(x)− un(y)|pφδ(x)

|x− y|N+sp
dx dy

+

∫∫
R2N

|un(x)− un(y)|p−2(un(x)− un(y))(φδ(x)− φδ(y))un(y)

|x− y|N+sp
dx dy

= I1 + I2.

Clearly, I1 =
∫
RN |D

sun|pφδ dx, then by using un ⇀ u in Ds,p(RN ) and Lemma
2.2, we have

lim
n→∞

I1 =

∫
RN

φδdµ.

Since µ−
∑
i∈I µiδxi has no atoms and φδ → 0 as δ → 0 for any x 6= xi, i ∈ I, we

conclude that limδ→0 limn→∞ I1 = µi. Now,we estimate I2 using Hölder inequality
we have

I2 ≤
∫
RN

(|Dsun|p)
p−1
p (|Dsφδ|p)1/p|un|dy

≤ ‖Dsun‖p−1
p

(∫
RN
|un|p|Dsφδ|p dx

)1/p

≤ c
(∫

RN
|un|p|Dsφδ|p dx

)1/p

.

Then by Lemmas 2.3 and 2.4, we know that

lim sup
n→∞

I2 ≤ c
(∫

RN
|u|p|Dsφδ|p dx

)1/p

. (2.5)

Furthermore, we check that limδ→0

∫
RN |u|

p|Dsφδ|p dx = 0. In fact, thanks to
Lemma 2.3 and Hölder inequality, we deduce that∫

RN
|u|p|Dsφδ|p dx

≤ C
(
δ−sp

∫
|x|<δ

|u|p dx+ δN
∫
|x|≥δ

|u|p

|x|N+sp
dx
)

≤ Cδ−sp
(∫
|x|<δ

|u|p
∗
s dx

)p/p∗s
|Bδ|

sp
N + CδN

∞∑
k=0

∫
2kδ≤|x|≤2k+1δ

|u|p

|x|N+sp
dx

≤ c1
(∫
|x|<δ

|u|p
∗
s dx

)p/p∗s
+ C

∞∑
k=0

2−k(N+sp)δ−sp
(∫
|x|≤2k+1δ

|u|p
∗
s dx

)p/p∗s
|B2k+1δ|

sp
N

= c1

(∫
|x|<δ

|u|p
∗
s dx

)p/p∗s
+ c2

∞∑
k=0

2−kN
(∫
|x|≤2k+1δ

|u|p
∗
s dx

)p/p∗s
.



8 N. CUI, H.-R. SUN EJDE-2021/11

Since u ∈ Lp
∗
s (RN ), then it is easy to see that limδ→0

∫
|x|<δ |u|

p∗s dx = 0. Given

ε > 0, taking k0 ∈ N such that c2
∑∞
k=k0+1 2−kN < ε, we obtain

c2

∞∑
k=0

2−kN
(∫
|x|≤2k+1δ

|u|p
∗
s dx

)p/p∗s
≤ ε‖u‖pp∗s + c2

k0∑
k=0

2−kN
(∫
|x|≤2k0+1δ

|u|p
∗
s dx

)p/p∗s
.

Then,

lim sup
δ→0

c2

∞∑
k=0

2−kN
(∫
|x|≤2k+1δ

|u|p
∗
s dx

)p/p∗s
≤ ε‖u‖pp∗s .

Hence,

lim
δ→0

∫
RN
|u|p|Dsφδ|p dx = 0,

and from (2.5) it follows that limδ→0 limn→∞ I2 = 0. So, the limit equality (2.4)
holds. On the other hand, by applying the property of the function φδ, we know
that

lim
δ→0

∫
RN

g|u|pφδ dx = 0 and lim
δ→0

∫
RN

hφδdν = h(xi)νi.

Then, in view of (2.3), we have h(xi)νi = µi for any i ∈ I, which implies that
h(xi) ≥ 0. So, it follows from Lemma 2.2 that

νi ≥ S
N
sph(xi)

− N
sp , µi ≥ S

N
sph(xi)

1− N
sp if h(xi) > 0,

νi = µi = 0 if h(xi) = 0.

This completes the proof. �

3. Main results

In this section, we prove the existence of solutions for critical fractional p-
Laplacian with indefinite weights of problem (1.1). By using the concentration-
compactness principle and mountain pass theorem, we prove Theorem 1.1. Firstly,
we prove that the functional Jλ satisfies the (PS)c condition for small energy levels.

Lemma 3.1. For any λ ∈ (0, λ+
1 ), the functional Jλ satisfies the (PS)c condition

for all c < s
N S

N
sp ‖h‖1−

N
sp

∞ .

Proof. Let {un} be a (PS)c sequence; that is,

Jλ(un)→ c and J ′λ(un)→ 0 as n→∞. (3.1)

First of all, we show that the sequence {un} is bounded in X. If this is not true, we
may suppose that, up to a subsequence, still denoted by {un} such that ‖un‖X →∞
as n→∞. For n ∈ N, let vn = un

‖un‖X , then we can assume that there exists v ∈ X
such that

vn ⇀ v in X, vn → v in Lploc(RN ), vn → v a.e. in RN . (3.2)

Obviously, by (3.1), we have

c+ o(1)‖un‖X = Jλ(un)− 1

p∗s
〈J ′λ(un), un〉 =

s

N

(
[un]ps,p − λ

∫
RN

g|un|p dx
)
.



EJDE-2021/11 CRITICAL FRACTIONAL p-LAPLACIAN EQUATION 9

Multiplying both sides by ‖un‖−pX and letting n→∞, we deduce that

[vn]ps,p − λ
∫
RN

g|vn|p dx→ 0. (3.3)

Since the embedding X ↪→ Ds,p(RN ) ↪→ Lp
∗
s (RN ) is continuous and ‖vn‖X = 1,

we know that {vn} is bounded in Lp
∗
s (RN ), then {|vn|p} is bounded in L

p∗s
p (RN ).

Combining this with (3.2), we have |vn|p ⇀ |v|p in L
p∗s
p (RN ). So, in view of

g+ ∈ L
N
sp (RN ), we obtain∫

RN
g+|vn|p dx→

∫
RN

g+|v|p dx. (3.4)

For λ ∈ (0, λ+
1 ), by (3.2)-(3.4), we have

0 ≤ [v]ps,p − λ
∫
RN

g|v|p dx

= [v]ps,p − λ
∫
RN

(g+ − g−)|v|p dx

≤ lim inf
n→∞

(
[vn]ps,p − λ

∫
RN

g+|vn|p dx+ λ

∫
RN

g−|vn|p dx
)

= 0.

(3.5)

Thus, the variational characterization of the principle eigenvalue λ+
1 (see Lemma

2.1) implies that v ≡ 0. Then from (3.4) and (3.5) it follows that∫
RN

g+|vn|p dx→ 0, lim
n→∞

[vn]ps,p = 0, lim
n→∞

∫
RN

g−|vn|p dx = 0.

By the Hardy-type inequality [21, Theorem 2], we obtain

0 ≤
∫
RN

ω|vn|p dx ≤ CN,s,p[vn]ps,p,

we then deduce that limn→∞ ‖vn‖X = 0, which contradicts that ‖vn‖X = 1. There-
fore, {un} is bounded in X.

Next, we verify that un → u in X. Since {un} is bounded in X, we can assume
that there exists u ∈ X such that un ⇀ u in X. Then applying Lemmas 2.5 and
2.6 and g, h ∈ L∞(RN ), for any ϕ ∈ C∞0 (RN ), we have∫∫

R2N

|un(x)− un(y)|p−2(un(x)− un(y))(ϕ(x)− ϕ(y))

|x− y|N+sp
dx dy

→
∫∫

R2N

|u(x)− u(y)|p−2(u(x)− u(y))(ϕ(x)− ϕ(y))

|x− y|N+sp
dx dy,∫

RN
g|un|p−2unϕdx→

∫
RN

g|u|p−2uϕdx,∫
RN

h|un|p
∗
s−2unϕdx→

∫
RN

h|u|p
∗
s−2uϕdx.

Hence, in view of 〈J ′λ(un), ϕ〉 → 0, we obtain∫∫
R2N

|u(x)− u(y)|p−2(u(x)− u(y))(ϕ(x)− ϕ(y))

|x− y|N+sp
dx dy

− λ
∫
RN

g|u|p−2uϕdx−
∫
RN

h|u|p
∗
s−2uϕdx = 0,



10 N. CUI, H.-R. SUN EJDE-2021/11

that is, 〈J ′λ(u), ϕ〉 = 0.
From the proof of Lemma 2.7, we know that for any i ∈ I,

h(xi)νi = µi, (3.6)

νi = 0 if h(xi) = 0, νi ≥ S
N
sph(xi)

− N
sp if h(xi) > 0. (3.7)

Suppose that νi 6= 0 for some i ∈ I. Becasue un ⇀ u in X, we may obtain that
un → u a.e. in RN . On the other hand, we can get that {|un|p} is bounded in

L
p∗s
p (RN ), since the embedding X ↪→ Lp

∗
s (RN ) is continuous. Then, it follows from

g+ ∈ L
N
sp (RN ) that ∫

RN
g+|un|p dx→

∫
RN

g+|u|p dx. (3.8)

Thus, by (3.1), Lemma 2.2 and (3.8), we deduce that

c+ o(1)‖un‖X = Jλ(un)− 1

p∗s
〈J ′λ(un), un〉

=
s

N

(
[un]ps,p − λ

∫
RN

g|un|p dx
)

=
s

N

(
[un]ps,p − λ

∫
RN

g+|un|p dx+ λ

∫
RN

g−|un|p dx
)

≥ s

N

(
[u]ps,p − λ

∫
RN

g|u|p dx+
∑
i∈I

µi

)
+ o(1).

Furthermore, combining J ′λ(u) = 0, (3.6) with (3.7), we obtain

c+ o(1)‖un‖X ≥
s

N

∫
RN

h|u|p
∗
s dx+

s

N

∑
i∈I

h(xi)νi + o(1)

≥ s

N

∫
RN

h|u|p
∗
s dx+

s

N
S
N
sp

∑
i∈I

h(xi)
1− N

sp + o(1).

In view of c < s
N S

N
sp ‖h‖1−

N
sp

∞ , we obtain that
∫
RN h|u|

p∗s dx < 0. However, by the

fact J ′λ(u) = 0 and λ ∈ (0, λ+
1 ), we have

∫
RN

h|u|p
∗
s dx = [u]ps,p − λ

∫
RN

g|u|p dx ≥ 0,

which is a contraction. Thus νi = µi = 0 for any i ∈ I, then it follows from Lemma
2.2 that ∫

RN
|un|p

∗
s dx→

∫
RN
|u|p

∗
s dx.



EJDE-2021/11 CRITICAL FRACTIONAL p-LAPLACIAN EQUATION 11

This together with the weak convergence of un ⇀ u in Lp
∗
s (RN ) imply that un → u

in Lp
∗
s (RN ). By un ⇀ u in X and (3.1), we derive that

〈J ′λ(un)− J ′λ(u), un − u〉

=

∫∫
R2N

( |un(x)− un(y)|p−2(un(x)− un(y))

|x− y|N+sp

− |u(x)− u(y)|p−2(u(x)− u(y))

|x− y|N+sp

)
(un(x)− un(y)− u(x) + u(y)) dx dy

− λ
∫
RN

g(|un|p−2un − |u|p−2u)(un − u)dx

−
∫
RN

h(|un|p
∗
s−2un − |u|p

∗
s−2u)(un − u)dx→ 0.

(3.9)

Since un ⇀ u in X, un → u in Lp
∗
s (RN ), g+ ∈ L

N
sp (RN ) and h ∈ L∞(RN ), we have

the convergence ∫
RN

g+(|un|p−2un − |u|p−2u)(un − u)dx→ 0,∫
RN

h(|un|p
∗
s−2un − |u|p

∗
s−2u)(un − u)dx→ 0.

(3.10)

Then, combining (3.9) with (3.10), we obtain∫∫
R2N

( |un(x)− un(y)|p−2(un(x)− un(y))

|x− y|N+sp
− |u(x)− u(y)|p−2(u(x)− u(y))

|x− y|N+sp

)
× (un(x)− un(y)− u(x) + u(y)) dx dy

+ λ

∫
RN

g−(|un|p−2un − |u|p−2u)(un − u)dx→ 0.

(3.11)
Using Hölder inequality, we deduce that∫∫

R2N

( |un(x)− un(y)|p−2(un(x)− un(y))

|x− y|N+sp
− |u(x)− u(y)|p−2(u(x)− u(y))

|x− y|N+sp

)
× (un(x)− un(y)− u(x) + u(y)) dx dy

≥ [un]ps,p + [u]ps,p − [un]p−1
s,p [u]s,p − [u]p−1

s,p [un]s,p

=
(
[un]p−1

s,p − [u]p−1
s,p

)
([un]s,p − [u]s,p)

and similarly, we have∫
RN

g−(|un|p−2un − |u|p−2u)(un − u)dx

≥
((∫

RN
g−|un|p dx

) p−1
p −

(∫
RN

g−|u|p dx
) p−1

p
)

×
((∫

RN
g−|un|p dx

)1/p

−
(∫

RN
g−|u|p dx

)1/p)
.

Let fb(t) = (tb−1 − βb−1)(t− β) for t ∈ R+, b > 1, we know that fb(t) ≥ 0. Hence,
it follows from (3.11) that(

[un]p−1
s,p − [u]p−1

s,p

)
([un]s,p − [u]s,p)→ 0,



12 N. CUI, H.-R. SUN EJDE-2021/11((∫
RN

g−|un|p dx
) p−1

p −
(∫

RN
g−|u|p dx

) p−1
p
)

×
((∫

RN
g−|un|p dx

)1/p

−
(∫

RN
g−|u|p dx

)1/p)
→ 0.

The function fb also has the following property: if fb(tn)→ 0, then tn → β. Thus,

[un]ps,p → [u]ps,p and

∫
RN

g−|un|p dx→
∫
RN

g−|u|p dx as n→∞.

By the Hardy-type inequality, we derive that
∫
RN ω|un|

p dx→
∫
RN ω|u|

p dx as n→
∞. So, ‖un‖X → ‖u‖X , this together with the weak convergence of un ⇀ u in X
implies that un → u in X. �

To verify that Jλ has the geometric structure required by the mountain pass
theorem, we need to introduce the following results.

Lemma 3.2 ([7, Proposition 3.1]). Let 1 < p < ∞, s ∈ (0, 1), N > sp and let S
be as in (2.1). Then

(i) there exists a minimizer for S;
(ii) for every minimizer U ∈ Ds,p(RN ), there exist x0 ∈ RN and a constant

sign monotone function u : R→ R such that U(x) = u(|x− x0|);
(iii) for every minimizer U , there exists λU > 0 such that∫∫

R2N

|U(x)− U(y)|p−2(U(x)− U(y))(ϕ(x)− ϕ(y))

|x− y|N+sp
dx dy

= λU

∫
RN
|U |p

∗
s−2Uϕdx,

for ϕ ∈ Ds,p(RN ).

In the following, we fix a radially symmetric nonnegative decreasing minimizer
U = U(r) for S. Multiplying U by a positive constant if necessary, we may assume
that U is a radial solution of

(−∆p)
sU = |U |p

∗
s−2U.

Taking the test function U and applying (2.1) yield

[U ]ps,p = ‖U‖p
∗
s
p∗s

= S
N
sp . (3.12)

For ε > 0, the function

Uε(|x|) = ε−
N−sp
p U

( |x|
ε

)
is also a minimizer for S satisfying (3.12).

Lemma 3.3 ([7, Corollary 3.7]). There exist constants C1, C2 > 0 and θ > 1 such
that for all r ≥ 1,

C1

r
N−sp
p−1

≤ U(r) ≤ C2

r
N−sp
p−1

and
U(θr)

U(r)
≤ 1

2
.

Now, we give some auxiliary functions and estimate their norms. In what follows,
θ is the constant in Lemma 3.3 that depends only on N , s and p. For ε, ρ > 0 and
θ > 1, let us set

mε,ρ =
Uε(ρ)

Uε(ρ)− Uε(ρθ)
.



EJDE-2021/11 CRITICAL FRACTIONAL p-LAPLACIAN EQUATION 13

Moreover, let us define

gε,ρ(t) =


0, if 0 ≤ t ≤ Uε(ρθ),
mp
ε,ρ(t− Uε(ρθ)), if Uε(ρθ) < t ≤ Uε(ρ),

t+ Uε(ρ)(mp−1
ε,ρ − 1), if t > Uε(ρ),

and

Gε,ρ(t) =

∫ t

0

g′ε,ρ(τ)1/pdτ =


0, if 0 ≤ t ≤ Uε(ρθ),
mε,ρ(t− Uε(ρθ)), if Uε(ρθ) < t ≤ Uε(ρ),

t, if t > Uε(ρ).

The functions gε,ρ and Gε,ρ are nondecreasing and absolutely continuous. Consider
the radially symmetric non-increasing function

uε,ρ(r) = Gε,ρ(Uε(r)),

which satisfies

uε,ρ(r) =

{
Uε(r), if r ≤ ρ,
0, if r ≥ ρθ.

Then, we have the following estimates for uε,ρ.

Lemma 3.4 ([23, Lemma 2.7]). There exists a constant C = C(N, p, s) > 0 such
that, for any 0 < ε ≤ ρ

2 , it holds

[uε,ρ]
p
s,p ≤ S

N
sp + C

( ε
ρ

)N−sp
p−1 , (3.13)

‖uε,ρ‖pp ≥

{
1
C ε

sp log(ρε ), if N = sp2,
1
C ε

sp, if N > sp2,
(3.14)

‖uε,ρ‖
p∗s
p∗s
≥ S

N
sp − C

( ε
ρ

) N
p−1 . (3.15)

Now we define the function

vε,ρ(x) =
uε,ρ(x)

‖uε,ρ(x)‖p∗s
.

Lemma 3.5. There exist ε, ρ, t0 > 0 such that for λ ∈ (0, λ+
1 ), Jλ(t0vε,ρ) < 0 and

sup
t≥0

Jλ(tvε,ρ) <
s

N
S
N
sp ‖h‖1−

N
sp

∞ .

Proof. For any λ ∈ (0, λ+
1 ), by (2.1), we have

Jλ(u) ≥ 1

p

(
1− λ

λ1

)
[u]ps,p −

1

p∗s
‖h‖∞S−

p∗s
p [u]

p∗s
s,p,

so we can obtain that Jλ(u) ≥ c, when ‖u‖X is sufficiently small. Furthermore, it
is easy from (2.2) to see that limt→∞ Jλ(tvε,ρ) = −∞, hence Jλ(tvε,ρ) attains its
maximum at some tε ∈ (0,∞) with ψ′λ(tε) = 0, where

ψλ(t) = Jλ(tvε,ρ) =
tp

p
[vε,ρ]

p
s,p −

tp

p
λ

∫
RN

g|vε,ρ|p dx−
tp
∗
s

p∗s

∫
RN

h|vε,ρ|p
∗
s dx.



14 N. CUI, H.-R. SUN EJDE-2021/11

Then, we obtain

0 = ψ′λ(tε) = tp−1
ε

(
[vε,ρ]

p
s,p − λ

∫
RN

g|vε,ρ|p dx
)
− tp

∗
s−1
ε

∫
RN

h|vε,ρ|p
∗
s dx,

moreover, combining (A2), (A3) and (A4), we deduce that

t
p∗s−p
ε =

[vε,ρ]
p
s,p − λ

∫
RN g|vε,ρ|

p dx∫
RN h|vε,ρ|p

∗
s dx

≤
[vε,ρ]

p
s,p

h(0)‖vε,ρ‖
p∗s
p∗s

. (3.16)

Clearly, we have

Jλ(tεvε,ρ) = sup
t≥0

Jλ(tvε,ρ) = I1 − I2, (3.17)

where

I1 =
tpε
p

[vε,ρ]
p
s,p −

t
p∗s
ε

p∗s
h(0)

∫
RN
|vε,ρ|p

∗
s dx,

I2 =
tpε
p
λ

∫
RN

g|vε,ρ|p dx−
t
p∗s
ε

p∗s

∫
RN

(h(0)− h)|vε,ρ|p
∗
s dx.

In view of (3.13) and (3.15), we obtain

[vε,ρ]
p
s,p =

[uε,ρ]
p
s,p

‖uε,ρ(x)‖pp∗s
≤ S +O

( ε
ρ

)N−sp
p−1 .

For positive numbers a and b, the maximum of ~(t) = a t
p

p − b t
p∗s
p∗s

for t ≥ 0 is

attained at t = (ab )
N−sp
sp2 , then, by the assumption (A3) and above inequality, we

can deduce that

I1 ≤
1

p

( [vε,ρ]
p
s,p

h(0)
∫
RN |vε,ρ|p

∗
s dx

)N−sp
sp

[vε,ρ]
p
s,p

− h(0)

p∗s

( [vε,ρ]
p
s,p

h(0)
∫
RN |vε,ρ|p

∗
s dx

) N
sp

∫
RN
|vε,ρ|p

∗
s dx

=
s

N
‖h‖1−

N
sp

∞

(
[vε,ρ]

p
s,p

) N
sp
(∫

RN
|vε,ρ|p

∗
s dx

)1− N
sp

≤ s

N
S
N
sp ‖h‖1−

N
sp

∞ .

Without loss of generality, we can assume that {tε} is bounded. In fact, since {tε} is
bounded from blew, otherwise one concludes easily from (3.17) that Jλ(tεvε,ρ)→ 0
as ε → 0. In addition, according to (3.16), we know that {tε} is bounded from
above for ε > 0 small. Next, we estimate the Lκ-norm of uε,ρ, for κ ∈ [1,∞),
Lemma 3.3 yields∫

RN
|uε,ρ(x)|κ dx ≥

∫
Bρ(0)

|uε,ρ(x)|κ dx =

∫
Bρ(0)

|Uε(x)|κ dx

= ε−
(N−sp)κ

p

∫
Bρ(0)

∣∣U(x
ε

)∣∣κ dx
≥ Cκ1 ε

Np−(N−sp)κ
p

∫ ρ/ε

1

r−
N−sp
p−1 κ+N−1dr .



EJDE-2021/11 CRITICAL FRACTIONAL p-LAPLACIAN EQUATION 15

Then ∫
RN
|uε,ρ(x)|κ dx ≥ cκ


εN−

N−sp
p κ, if κ > N(p−1)

N−sp ,

εN−
N−sp
p κ| log ρ

ε |, if κ = N(p−1)
N−sp ,

ε
N−sp
p(p−1)

κρN−
N−sp
p−1 κ, if κ < N(p−1)

N−sp .

Thus, by assumption (A4), we have

∫
RN

g|vε,ρ|p dx ≥ cp


g0ε

sp, if N > sp2,

g0ε
sp| log ρ

ε |, if N = sp2,

g0ε
N−sp
p−1 ρN−

N−sp
p−1 p, if N < sp2.

Similarly, in view of assumption (A2), we obtain∫
RN

(h(0)− h)|vε,ρ|p
∗
s dx =

1

‖uε,ρ‖
p∗s
p∗s

∫
RN

(h(0)− h)|uε,ρ|p
∗
s dx

≥ ε
N
p

∫ ρ/ε

1

r−
N

p(p−1)
−1dr.

Since N
p ≥ sp if N ≥ sp2 and N

p > N−sp
p−1 and N < sp2, then I2 can be dominated

by
∫
RN g|vε,ρ|

p dx. Thus, we conclude that

Jλ(tεvε,ρ) ≤


s
N S

N
sp ‖h‖1−

N
sp

∞ −K1ε
sp, if N > sp2,

s
N S

N
sp ‖h‖1−

N
sp

∞ −K1ε
sp| log ρ

ε |, if N = sp2,

s
N S

N
sp ‖h‖1−

N
sp

∞ −K1ε
N−sp
p−1 ρN−

N−sp
p−1 p, if N < sp2,

then

Jλ(tεvε,ρ) <
s

N
S
N
sp ‖h‖1−

N
sp

∞

if ε > 0 is sufficiently small. This completes the proof. �

Proof of Theorem 1.1. For any λ ∈ (0, λ+
1 ), by (2.1), we have

Jλ(u) ≥ 1

p

(
1− λ

λ1

)
[u]ps,p −

1

p∗s
‖h‖∞S−

p∗s
p [u]

p∗s
s,p,

then it follows that Jλ(u) ≥ c > 0, when ‖u‖X is sufficiently small. And since
Jλ(0) = 0, 0 is a local minimum of Jλ. In addition, noting that

Jλ(tvε,ρ) =
tp

p

(
[vε,ρ]

p
s,p − λ

∫
RN

g|vε,ρ|p dx
)
− tp

∗
s

p∗s

∫
RN

h|vε,ρ|p
∗
s dx→ −∞

as t→ +∞, fix t1 > 0 so large that Jλ(t1vε,ρ) < 0. Now, we construct the set

Γ = {γ ∈ C([0, 1], X) : γ(0) = 0, γ(1) = t1vε,ρ},

and let

c = inf
γ∈Γ

sup
t∈[0,1]

Jλ(γ(t)).

In view of Lemma 3.5, it is easy to see that

c <
s

N
S
N
sp ‖h‖1−

N
sp

∞ ,



16 N. CUI, H.-R. SUN EJDE-2021/11

and hence Jλ satisfies the (PS)c condition by Lemma 3.1. Then, c is a critical level
of Jλ via the mountain pass theorem [28], and we obtain that problem (1.1) has a
nontrivial solution. �

Acknowledgement. This work was supported by the NSFC (Grant No 11671181).

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Na Cui

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000,
China

Email address: cuin17@lzu.edu.cn

Hong-Rui Sun (corresponding author)
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000,

China
Email address: hrsun@lzu.edu.cn


	1. Introduction
	2. Preliminaries
	3. Main results
	Acknowledgement

	References