Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 23, pp. 1–27.

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

EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR

FRACTIONAL LAPLACE PROBLEMS WITH

CRITICAL GROWTH

YAJING ZHANG, QIAOQIN LI, LU PANG

Abstract. We prove the existence of multiple positive solutions of fractional
Laplace problems with critical growth, we consider the concave power case or

the convex power case. We establish the relationship between the number of

the local maximum points of the coefficient function of the critical nonlinearity
and the number of the positive solutions of the equation.

1. Introduction

Considerable attention has been devoted to fractional and non-local operators of
elliptic type in recent years, both for their interesting theoretical structure and
in view of concrete applications, like flame spropagation, chemical reactions of
liquids, population dynamics, geophysical fluid dynamics, and American option,
see [3, 12, 13, 17, 30, 31] and the references therein.

In this article we consider the critical problem involving the fractional Laplacian

(−∆)su = λuq−1 +Q(x)up−1 in Ω,

u > 0 in Ω,

u = 0 in RN \ Ω,

(1.1)

where s ∈ (0, 1) is fixed and (−∆)s is the fractional Laplace operator, Ω ⊂ RN (N >
2s) is a smooth bounded domain, 1 < q < p = 2∗s := 2N

N−2s , λ > 0, and Q ∈ C(Ω̄)
is a positive function.

The fractional Laplace operator (−∆)s (up to normalization factors) is defined
by

−(−∆)su(x) =

∫
RN

(
u(x+ y) + u(x− y)− 2u(x)

)
K(y)dy, x ∈ RN ,

where K(x) = |x|−(N+2s), x ∈ RN . We will denote by Hs(RN ) the usual fractional
Sobolev space endowed with the so-called Gagliardo norm

‖u‖Hs(RN ) = ‖u‖L2(RN ) +
(∫

R2N

|u(x)− u(y)|2K(x− y) dx dy
)1/2

,

2010 Mathematics Subject Classification. 49J35, 35A15, 35S15.
Key words and phrases. Positive solutions; fractional Laplace problems; critical growth;

variational method; Nehari manifold.
c©2021 Texas State University.

Submitted June 9, 2020. Published March 31, 2021.

1



2 Y. ZHANG, Q. LI, L. PANG EJDE-2021/23

while X0 is the function space defined as

X0 = {u ∈ Hs(RN ) : u = 0 a.e. in RN \ Ω}.

We refer to [18, 23, 24] for a general definition of X0 and its properties. The
embedding X0 ↪→ Lq(Ω) is continuous for any q ∈ [1, 2∗s] and compact for any
q ∈ [1, 2∗s). The space X0 is endowed with the norm

‖u‖X0 =
(∫

R2N

|u(x)− u(y)|2K(x− y) dx dy
)1/2

.

By [23, Lemma 5.1] we have C2
0 (Ω) ⊂ X0. Thus X0 is non-empty. Note that

(X0, ‖ · ‖X0) is a Hilbert space with scalar product

(u, v)X0 =

∫
R2N

(u(x)− u(y))(v(x)− v(y))K(x− y) dx dy.

Problems similar to (1.1) have been also studied in the local setting by several
authors. In particular, Brezis and Nirenberg[9] studied the equation

−∆u = |u|2
∗−2u+ f(x, u),

where f(x, u) is a lower order perturbation of |u|2∗−2u in the sense that f(x, t)/t2
∗ →

0 as t→ +∞, and 2∗ = 2N
N−2 . A typical example to which their results apply is

−∆u = λuq−1 + u2∗−1 in Ω,

u > 0 in Ω,

u = 0 on ∂Ω,

(1.2)

where λ > 0 is a parameter and 2 < q < 2∗. When N ≥ 4, problem (1.2) has a
positive solution for every λ > 0. When N = 3 and 4 < q < 6, problem (1.2) has a
positive solution. When N = 3 and 2 < q ≤ 4, it is only for large values of λ that
problem (1.2) has a positive solution. The case q = 2 in (1.2) is also studied by
them. Ambrosetti et al. [1] investigated the following problem with concave-convex
power nonlinearities,

−∆u = λuq−1 + up−1 in Ω,

u > 0 in Ω,

u = 0 on ∂Ω,

(1.3)

where 1 < q < 2 < p ≤ 2∗. They proved that there exists λ0 > 0 such that
(1.3) admits at least two positive solutions for λ ∈ (0, λ0), one positive solution for
λ = λ0, and no positive solution for λ > λ0. After the work[1], several papers have
been devoted to problem (1.3), see for example [2, 7, 9, 10, 16].

Now, we focus our attention on critical nonlocal fractional problems. It is worth
noting here that problem (1.1) with λ = 0 and Q ≡ 1 has no positive solution
whenever Ω is a star-shaped domain, see [15, 21]. This fact motivates the pertur-
bation terms λuq−1 since we are interested in the existence of positive solutions
of (1.1). Servadei and Valdinoci[25, 26] studied problem (1.1), with q = 2 and
Q ≡ 1, and obtained Brezis-Nirenberg type results. When Q ≡ 1, Barrios et al.
[6] studied problem (1.1) and showed the existence and multiplicity of solutions to
problem (1.1). Note that one can also define a fractional power of the Laplacian
using spectral decomposition. The similar problem with (1.1) but for this spectral
fractional Laplacian has been treated in [5, 11].



EJDE-2021/23 FRACTIONAL LAPLACE PROBLEMS WITH CRITICAL GROWTH 3

Taking into account that we are looking for positive solutions, we consider the
energy functional associated with (1.1),

Iλ(u) =
1

2

∫
R2N

|u(x)− u(y)|2K(x− y) dx dy − λ

q

∫
Ω

(u+)qdx

− 1

p

∫
Ω

Q(x)(u+)pdx,

(1.4)

where u+ = max{u, 0} denotes the positive part of u. By the Maximum Princi-
ple(Proposition 2.2.8 in [27]), it is easy to check that critical points of I are the
positive solutions of (1.1).

We assume that Q satisfies the following hypotheses.

(H1) Q ∈ C(Ω̄) is a positive function;
(H2) there exist m local maximum points a1, a2, . . . , am ∈ Ω of Q such that

Q(ai) = max
x∈Ω̄

Q(x) = 1 for 1 ≤ i ≤ m,

Q(x)−Q(ai) = o(|x− ai|σ)

as x→ ai uniformly in i, where σ := N−2s
2 ;

(H2’) there exist m local maximum points a1, a2, . . . , am ∈ Ω of Q such that

Q(ai) = max
x∈Ω̄

Q(x) = 1 for 1 ≤ i ≤ m,

Q(x)−Q(ai) = o(|x− ai|σ)

as x→ ai uniformly in i, for some σ := N − (N−2s)q
2 ;

(H3) there exists ρ0 > 0 such that

Bρ0(ai) ∩Bρ0(aj) = ∅ for i 6= j and 1 ≤ i, j ≤ m,

and ∪mi=1Bρ0(ai) ⊂ Ω, where Bρ0(ai) = {x ∈ RN : |x− ai| < ρ0}.
We now summarize the main results of the paper. Note that we are facing two
cases of |u|q−2u in problem (1.1), the concave case: 1 < q < 2, and the convex case:
2 < q < 2∗s. Firstly, in Section 2 we look at the problem (1.1) in the concave case
and prove the following result.

Theorem 1.1. Assume that 1 < q < 2 and Q satisfies (H1)–(H3). There exists
a positive number Λ∗ such that for λ ∈ (0,Λ∗), problem (1.1) has at least m + 1
positive solutions.

The convex case is treated in Section 3. While the existence result for problem
(1.1) is given in the next theorem.

Theorem 1.2. Assume 2 < q < 2∗s, N ≥ 4, and Q satisfies (H1), (H2’), (H3).
Then there exists a positive number Λ∗ such that for λ ∈ (0,Λ∗), problem (1.1) has
at least m positive solutions.

We prove Theorem 1.1 and Theorem 1.2 by variational methods. We construct
m compact Palais-Smale sequences which are localized in correspondence of m local
maximum points of Q in Ω. Thus, we could prove multiplicity of positive solutions
of (1.1). This paper is organized as follows. In Section 2 we study problem (1.1)
in the case of the exponent 1 < q < 2. In Section 3 we we study problem (1.1) in
the case of the exponent 2 < q < 2∗s.



4 Y. ZHANG, Q. LI, L. PANG EJDE-2021/23

2. Critical and concave case 1 < q < 2

This section is devoted to the study of problem (1.1) when the exponent satisfies
1 < q < 2. Firstly, we prove the existence of a ground state solution of (1.1).
By a ground state solution, we mean a solution w ∈ X0 such that Iλ(w) ≤ Iλ(v)
for every nontrivial solution v of (1.1). Next, we establish the existence of m + 1
positive solution of (1.1).

2.1. Preliminaries and Nehari manifold. Note that Iλ is unbounded below.
We restrict Iλ to a suitable set in order to get rid of this problem. We define the
Nehari manifold

Nλ = {u ∈ X0 \ {0} : 〈I ′λ(u), u〉 = 0}

=
{
u ∈ X0 \ {0} : ‖u‖2X0

= λ

∫
Ω

(u+)qdx+

∫
Ω

Q(x)(u+)pdx
}
.

Obviously, the Nehari manifold contains all the nontrivial critical points of Iλ.

Lemma 2.1. The functional Iλ is coercive and bounded from below on Nλ.

Proof. For every u ∈ Nλ, we have

Iλ(u) =
(1

2
− 1

p

)
‖u‖2X0

− λ
(1

q
− 1

p

) ∫
Ω

(u+)qdx

≥ s

N
‖u‖2X0

− λ
(1

q
− 1

p

)
|Ω|

p−q
p |u|qp

≥ s

N
‖u‖2X0

− λ
(1

q
− 1

p

)
|Ω|

p−q
p S−q/2s ‖u‖qX0

,

(2.1)

consequently, Iλ is coercive and bounded from below on Nλ since 1 < q < 2. �

We define ψλ(u) = 〈I ′λ(u), u〉. Then for u ∈ Nλ, we have

〈ψ′λ(u), u〉 = 2‖u‖2X0
− λq

∫
Ω

(u+)qdx− p
∫

Ω

Q(x)(u+)pdx (2.2)

= (2− q)‖u‖2X0
− (p− q)

∫
Ω

Q(x)(u+)pdx (2.3)

= λ(p− q)
∫

Ω

(u+)qdx− (p− 2)‖u‖2X0
. (2.4)

Adopting a method similar to that used in [29], we split Nλ into three parts:

N+
λ = {u ∈ Nλ : 〈ψ′λ(u), u〉 > 0};
N 0
λ = {u ∈ Nλ : 〈ψ′λ(u), u〉 = 0};
N−λ = {u ∈ Nλ : 〈ψ′λ(u), u〉 < 0}.

In our context, the Sobolev constant is

Ss = inf
u∈Hs(RN )\{0}

∫
R2N (u(x)− u(y))2K(x− y) dx dy( ∫

RN |u(x)|pdx
)2/p . (2.5)

Set

Λ :=
p− 2

p− q
(2− q
p− q

) 2−q
p−2 |Ω|−

p−q
p S

p−q
p−2
s . (2.6)

Lemma 2.2. If λ ∈ (0,Λ), then N 0
λ = ∅.



EJDE-2021/23 FRACTIONAL LAPLACE PROBLEMS WITH CRITICAL GROWTH 5

Proof. Arguing by contradiction and assume that there exists λ0 ∈ (0,Λ) such that
N 0
λ 6= ∅. For u ∈ N 0

λ , by (2.3), we have

(2− q)‖u‖2X0
= (p− q)

∫
Ω

Q(x)(u+)pdx ≤ (p− q)S−p/2s ‖u‖pX0
.

Consequently,

‖u‖X0
≥
(2− q
p− q

S
p
2
s

) 1
p−2

. (2.7)

Similarly, by (2.4), we have

‖u‖X0 ≤
(
λ0
p− q
p− 2

|Ω|
p−q
p S−q/2s

) 1
2−q

. (2.8)

Combing (2.7) and (2.8), we have

λ0 ≥
p− 2

p− q
(2− q
p− q

) 2−q
p−2 |Ω|−

p−q
p S

p−q
p−2
s = Λ.

We have a contradiction. �

Set

X+
0 = X0 \ {u ∈ X0 : u+(x) = 0 a.e. in Ω}.

Lemma 2.3. For λ ∈ (0,Λ) and u ∈ X+
0 , there exist unique positive numbers t+(u)

and t−(u) such that t+(u)u ∈ N+
λ , t

−(u)u ∈ N−λ , and

Iλ(t+(u)u) = inf
t∈[0,tmax]

Iλ(tu), Iλ(t−(u)u) = sup
t∈[tmax,+∞)

Iλ(tu), (2.9)

where

tmax =
[ (2− q)‖u‖2X0

(p− q)
∫

Ω
Q(x)(u+)pdx

] 1
p−2 .

Proof. Set γ(t) = t2−q‖u‖2X0
− tp−q

∫
Ω
Q(x)(u+)pdx and ϕ(t) = Iλ(tu) for t ≥ 0.

Clearly, tu ∈ Nλ if and only if γ(t) = λ
∫

Ω
(u+)qdx. Moreover,

γ′(t) = (2− q)t1−q‖u‖2X0
− (p− q)tp−q−1

∫
Ω

Q(u+)pdx, (2.10)

and so it is easy to see that, if tu ∈ Nλ, then tu ∈ N+
λ (or N−λ ) if and only

if γ′(t) > 0 (or < 0). By (2.10), γ(t) has a unique critical point at t = tmax,
and γ is strictly increasing on (0, tmax) and strictly decreasing on (tmax,+∞) with
limt→+∞ γ(t) = −∞. By (2.5) and λ ∈ (0,Λ), we have

γ(tmax) =
p− 2

p− q
(2− q
p− q

) 2−q
p−2

‖u‖
2(p−q)
p−2

X0(∫
Ω
Q(x)(u+)pdx

) 2−q
p−2

≥ p− 2

p− q

(2− q
p− q

Q−1
M

) 2−q
p−2

S
p(2−q)
2(p−2)
s ‖u‖qX0

> λS−q/2s |Ω|
p−q
p ‖u‖qX0

≥ λ
(∫

Ω

(u+)pdx
) q
p |Ω|

p−q
p

≥ λ
∫

Ω

(u+)qdx.



6 Y. ZHANG, Q. LI, L. PANG EJDE-2021/23

Thus, we have unique t+(u) with 0 < t+(u) < tmax < t−(u) such that

γ(t+(u)) =

∫
Ω

(u+)qdx = γ(t−(u))

and γ′(t+) > 0 > γ′(t−). Equivalently, t+(u)u ∈ N+
λ and t−(u)u ∈ N−λ . Since

ϕ′(t) = tq−1
[
γ(t)− λ

∫
Ω

(u+)qdx
]
,

we derive that ϕ is decreasing on the intervals (0, t+(u)) and (t−(u),+∞), and
increasing on the interval (t+(u), t−(u)). Then we obtain (2.9). �

Applying Lemma 2.1 and Lemma 2.2, we write Nλ = N+
λ ∪N

−
λ and define

αλ = inf
u∈Nλ

Iλ(u), α+
λ = inf

u∈N+
λ

Iλ(u), α−λ = inf
u∈N−

λ

Iλ(u).

Lemma 2.4. (i) If λ ∈ (0,Λ), then αλ ≤ α+
λ < 0;

(ii) if λ ∈ (0, q2Λ), then α−λ ≥ d0, where

d0 =
(2− q
p− q

) q
p−2S

pq
2(p−2)
s

[ s
N

(2− q
p− q

) 2−q
p−2S

p(2−q)
2(p−2)
s − λ

(1

q
− 1

p

)
|Ω|

p−q
p S−q/2s

]
> 0.

Proof. (i) By Lemma 2.2, Nλ = N+
λ ∪ N

−
λ . Let w0 ∈ X+

0 , by Lemma 2.7, there

exists t(w0) > 0 such that t(w0)w0 ∈ N+
λ . By (2.3), we have

Iλ(t(w0)w0) =
(1

2
− 1

q

)
t2(w0)‖w0‖2X0

+
(1

q
− 1

p

)
tp(w0)

∫
Ω

Q(x)(w+
0 )pdx

< −2− q
q

(1

2
− 1

p

)
t2(w0)‖w0‖2X0

< 0.

(2.11)

(ii) For u ∈ N−λ , by (2.3) and (2.5), we have

2− q
p− q

‖u‖2X0
<

∫
Ω

Q(x)(u+)pdx ≤ S−p/2s ‖u‖pX0
,

which implies

‖u‖X0
>
(2− q
p− q

) 1
p−2S

p
2(p−2)
s . (2.12)

Consequently,

Iλ(u) =
(1

2
− 1

p

)
‖u‖2X0

− λ
(1

q
− 1

p

) ∫
Ω

(u+)qdx

≥ s

N
‖u‖2X0

− λ
(1

q
− 1

p

)
|Ω|

p−q
p S−q/2s ‖u‖qX0

= ‖u‖qX0

[ s
N
‖u‖2−qX0

− λ
(1

q
− 1

p

)
|Ω|

p−q
p S−q/2s

]
> d0

for λ ∈ (0, q2Λ). �

As a consequence of Lemma 2.2 we have the following result.

Lemma 2.5. For each u ∈ Nλ, there exist ε > 0 and a differentiable function
ξ : Bε(0) ⊂ X0 → (0,+∞) such that

ξ(0) = 1, ξ(w)(u− w) ∈ Nλ for w ∈ Bε(0),

〈ξ′(0), w〉 =
2(u,w)X0 − λq

∫
Ω

(u+)q−1wdx− p
∫

Ω
Q(u+)pdx

(2− q)‖u‖2X0
− (p− q)

∫
Ω
Q(x)(u+)pdx

(2.13)



EJDE-2021/23 FRACTIONAL LAPLACE PROBLEMS WITH CRITICAL GROWTH 7

for all w ∈ X0.

Proof. We define F : R×X0 → R as

F (ξ, w) = 〈I ′λ(ξ · (u− w)), ξ · (u− w)〉

= ξ2‖u− w‖2X0
− λξq

∫
Ω

[(u− w)+]qdx− ξp
∫

Ω

Q(x)[(u− w)+]pdx.

Then F (1, 0) = 0, and by Lemma 2.2, we have

∂F

∂ξ

∣∣∣
(1,0)

= 〈ψ′λ(u), u〉 6= 0

We can apply the implicit function theorem at the point (1, 0) and obtain the
result. �

2.2. Existence of a ground state solution. We follow the idea in [29] to show
the existence of a (PS)αλ sequence and a (PS)α−

λ
sequence in X0 for Iλ.

Lemma 2.6. (i) For λ ∈ (0,Λ), there exists a (PS)αλ sequence {un} ⊂ Nλ
for Iλ;

(ii) For λ ∈ (0, q2Λ), there exists a (PS)α−
λ

sequence {un} ⊂ N− for Iλ.

Proof. We only prove (i). (ii) has a similar proof. Applying Ekeland’s variational
principle[14] to the minimization problem αλ = infu∈Nλ Iλ(u) we have a minimizing
sequence {un} ⊂ Nλ with the following properties:

Iλ(un) < αλ +
1

n
, (2.14)

Iλ(w) ≥ Iλ(un)− 1

n
‖w − un‖X0

, ∀w ∈ Nλ. (2.15)

By taking n large, from (2.14) and (2.11), we have

Iλ(un) =
(1

2
− 1

p

)
‖un‖2X0

− λ
(1

q
− 1

p

) ∫
Ω

(u+
n )qdx

< αλ +
1

n

< −2− q
q

(1

2
− 1

p

)
t2(w0)‖w0‖2X0

(2.16)

for some w0 ∈ X+
0 , which implies that

|Ω|
p−q
p S−q/2s ‖un‖qX0

≥
∫

Ω

(u+
n )qdx >

(2− q)(p2 − 1)

λ(p− q)
t2(w0)‖w0‖2X0

. (2.17)

By (2.16) and (2.17), we have

L1 < ‖un‖X0 < L2, (2.18)

where

L1 =
(
|Ω|−

p−q
p Sq/2s

(2− q)(p2 − 1)

λ(p− q)
t2(w0)‖w0‖2X0

)1/q

,

L2 =
(
λ

p− q
(p2 − 1)q

|Ω|
p−q
p S−q/2s

) 1
2−q

.

Now we show that I ′λ(un)→ 0 as n→∞.



8 Y. ZHANG, Q. LI, L. PANG EJDE-2021/23

Let v ∈ X0 with ‖v‖X0
= 1. Applying Lemma 2.5 with u = un and w = ρv, ρ > 0

small, we obtain ξn(ρv) such that wρ := ξn(ρv)(un−ρv) ∈ Nλ. By (2.15), we deduce

〈I ′λ(un), wρ − un〉+ o(‖wρ − un‖X0
) = Iλ(wρ)− Iλ(un) ≥ − 1

n
‖wρ − un‖X0

.

Therefore,

〈I ′λ(un),−ρv〉+ [ξn(ρv)− 1]〈I ′λ(un), un− ρv〉 ≥ −
1

n
‖wρ− un‖X0 + o(‖wρ− un‖X0).

Dividing by ρ we have

〈I ′λ(un), v〉

≤ ξn(ρv)− 1

ρ
〈I ′λ(un), un − ρv〉+

1

nρ
‖wρ − un‖X0 +

o(‖wρ − un‖X0
)

ρ

= [1− ξn(ρv)]〈I ′λ(un), v〉+
1

nρ
‖wρ − un‖X0 +

o(‖wρ − un‖X0)

ρ
.

(2.19)

Clearly,

‖wρ − un‖X0
≤ |ξn(ρv)− 1| · ‖un‖X0

+ ρ|ξn(ρv)|,

lim
ρ→0

|ξn(ρv)− 1|
ρ

≤ ‖ξ′n(0)‖X∗
0
.

Consequently, passing to the limit as ρ → 0 in (2.19), we find a constant C > 0
independent of ρ such that

〈I ′λ(un), v〉 ≤ C

n
(1 + ‖ξ′n(0)‖X∗

0
).

The will be complte once we show that ‖ξ′n(0)‖X∗
0

is uniformly bounded in n. From
(2.13) and (2.18) we obtain

〈ξ′n(0), v〉 ≤ C1

|(2− q)‖un‖2X0
− (p− q)

∫
Ω
Q(x)(u+

n )pdx|

for some suitable constant C1 > 0. We only need to show that |(2 − q)‖un‖2X0
−

(p−q)
∫

Ω
Q(u+

n )pdx| is bounded away from zero. Arguing by contradiction, assume
that for a subsequence, which we still call {un}, we have

(2− q)‖un‖2X0
− (p− q)

∫
Ω

Q(x)(u+
n )pdx = o(1). (2.20)

By (2.5), we have

(2− q)‖un‖2X0
= (p− q)

∫
Ω

Q(x)(u+
n )pdx+ o(1) ≤ (p− q)S−p/2s ‖u0‖pX0

+ o(1).

Since ‖un‖X0
is bounded away from zero by (2.18), we obtain

‖un‖X0
≥
(2− q
p− q

S
p
2
s

) 1
p−2

+ o(1). (2.21)

In addition (2.20), and the fact that un ∈ Nλ give

λ

∫
Ω

(u+
n )qdx = ‖un‖2X0

−
∫

Ω

Q(x)(u+
n )pdx =

p− 2

p− q
‖un‖2X0

+ o(1),

‖un‖X0
≤
[
λ
p− q
p− 2

|Ω|
p−q
p S−q/2s

] 1
2−q + o(1).

(2.22)



EJDE-2021/23 FRACTIONAL LAPLACE PROBLEMS WITH CRITICAL GROWTH 9

Combing (2.21) and (2.22), we have

λ ≥ Λ + o(1),

which is clearly impossible. We obtain 〈I ′λ(un), v〉 ≤ C
n . This completes the proof

of (i). �

Proposition 2.7. Assume that 1 < q < 2 and Q satisfies (H1)–(H3). Then (1.1)
has at least one positive ground state solution if λ ∈ (0,Λ).

Proof. By Lemma 2.6 (i), there exists a minimizing sequence {un} ⊂ Nλ for Iλ
such that

Iλ(un)→ αλ, I ′λ(un)→ 0 (2.23)

as n → ∞. Since Iλ is coercive on Nλ by Lemma 2.1, we obtain that ‖un‖X0
is

bounded. Going if necessary to a subsequence, we can assume that

un ⇀ u in X0,

un → u in Lr(Ω) for r ∈ [1, p),

un → u a.e. in Ω.

From (2.23) We obtain that 〈I ′λ(u), w〉 = 0,∀w ∈ X0, i.e. u is a solution of (1.1). In
particular, u ∈ Nλ. By the Maximum Principle [27, Proposition 2.2.8], u is strictly
positive in Ω. By the definition of αλ and weak lower semicontinuity of the norm,
we have

αλ ≤ Iλ(u) =
(1

2
− 1

p

)
‖u‖2X0

− λ
(1

q
− 1

p

) ∫
Ω

(u+)qdx

≤ lim inf
n→∞

[ s
N
‖un‖2X0

− λ
(1

q
− 1

p

) ∫
Ω

(u+
n )qdx

]
≤ lim inf

n→∞
Iλ(un) = αλ.

It follows that Iλ(u) = αλ and un → u strongly in X0.
We claim that u ∈ N+

λ . Assume by the contradiction that u ∈ N−λ . By Lemma

2.3, there exist positive numbers t+(u) < tmax < t−(u) = 1 such that t+(u)u ∈ N+
λ

and t−λ (u)u ∈ N−λ , and

Iλ(t+(u)u) < Iλ(t−(u)u) = Iλ(u) = αλ,

which is impossible. Hence, u ∈ N+
λ and Iλ(u) = αλ = α+

λ . �

2.3. Proof of Theorem 1.1. In this section, we prove that (1.1) admits m positive
solutions. First of all, we show that Iλ satisfies the (PS)β condition in X0 for
β < β∗, where

β∗ :=
s

N
S
N
2s
s + αλ.

Lemma 2.8. Iλ satisfies the (PS)β condition in X0 for β < β∗.

Proof. Let {un} be a (PS)β sequence for Iλ such that

Iλ(un)→ β and I ′λ(un)→ 0. (2.24)

Then, for n large enough, we have

β + 1 + ‖un‖X0
≥ Iλ(un)− 1

p
〈I ′λ(un), un〉



10 Y. ZHANG, Q. LI, L. PANG EJDE-2021/23

=
s

N
‖un‖2X0

− λ
(1

q
− 1

p

) ∫
Ω

(u+
n )qdx

≥ s

N
‖u‖2X0

− λ
(1

q
− 1

p

)
|Ω|

p−q
p S−q/2s ‖u‖qX0

.

It follows that ‖un‖X0
is bounded. Going if necessary to a subsequence, we can

assume that
un ⇀ u0 in X0,

un → u0 in Lr(Ω) for r ∈ [1, p),

un → u0 a.e. in Ω.

(2.25)

Set vn = un − u0. Since X0 is a Hilbert space, we have

‖un‖2X0
= ‖vn‖2X0

+ ‖u0‖2X0
+ o(1). (2.26)

By Brezis-Lieb’s Lemma, we have∫
Ω

Q(x)(u+
n )pdx =

∫
Ω

Q(x)(v+
n )pdx+

∫
Ω

Q(x)(u+
0 )pdx+ o(1). (2.27)

By (2.26) and (2.27), we have

β − Iλ(u0) =
1

2
‖vn‖2X0

− 1

p

∫
Ω

Q(x)(v+
n )pdx+ o(1), (2.28)

and

‖vn‖2X0
−
∫

Ω

Q(x)(v+
n )pdx+ ‖u0‖2X0

− λ
∫

Ω

(u+
0 )qdx−

∫
Ω

Q(x)(u+
0 )pdx = o(1).

By (2.24) and (2.25), we have

0 = lim
n→∞

〈I ′λ(un), u0〉 = ‖u0‖2X0
− λ

∫
Ω

(u+
0 )qdx−

∫
Ω

Q(x)(u+
0 )pdx, (2.29)

consequently,

‖vn‖2X0
−
∫

Ω

Q(x)(v+
n )pdx = o(1). (2.30)

Now, we assume that

‖vn‖2X0
→ b,

∫
Ω

Q(x)(v+
n )pdx→ b, as n→∞. (2.31)

By (2.5) and (2.31), we obtain

‖vn‖2X0
≥ Ss

(∫
RN
|vn|pdx

)2/p

≥ Ss
(∫

RN
Q(x)(v+

n )pdx
)2/p

.

Passing to the limit, we have b ≥ Ssb2/p. This implies that b = 0 or b ≥ Sp/(p−2)
s =

S
N/(2s)
s . If b = 0, the proof is complete. Assume that b ≥ S

N/(2s)
s . By (2.29) and

(2.31), we have

β =
(1

2
− 1

p

)
b+ I(u0) ≥ s

N
SN/(2s)s + αλ.

which implies a contradiction. Hence, b = 0, that is un → u0 in X0 as n→∞. �

Recall that

Ss := inf
v∈Hs(RN )\{0}

∫
R2N |v(x)− v(y)|2K(x− y) dx dy( ∫

RN |v|pdx
)2/p .



EJDE-2021/23 FRACTIONAL LAPLACE PROBLEMS WITH CRITICAL GROWTH 11

It is well known from [26] that the infimum in formula above is attained at ũ, where

ũ(x) =
κ

(µ2 + |x− x0|2)
N−2s

2

, x ∈ RN , (2.32)

with κ ∈ R \ {0}, µ > 0 and x0 ∈ RN fixed constants. We suppose κ > 0 for our
convenience. Equivalently, the function ū is defined as

ū =
ũ

‖ũ‖Lp(RN )

is such that

Ss =

∫
R2N

|ū(x)− ū(y)|2K(x− y) dx dy.

The function

u∗(x) = ū
( x

S
1/(2s)
s

)
, x ∈ RN ,

is a solution of

(−∆)su = |u|p−2u in RN . (2.33)

Now, we consider the family of function Uε defined as

Uε(x) = ε−(N−2s)/2u∗(x/ε), x ∈ RN ,

for any ε > 0. The function Uε is a solution of problem (2.33) and satisfies∫
R2N

|Uε(x)− Uε(y)|2K(x− y) dx dy =

∫
RN
|Uε(x)|pdx = SN/(2s)s . (2.34)

Fix a maximum point ai of Q, where 1 ≤ i ≤ m. Let ηi ∈ C∞ be such that
0 ≤ ηi ≤ 1 in RN , ηi(x) = 1 if |x− ai| < ρ0/2; ηi(x) = 0 if |x− ai| ≥ ρ0. For every
ε > 0 we define the function

uε,i(x) = ηi(x)Uε(x− ai), x ∈ RN . (2.35)

In what follows we suppose that up to a translation x0 = 0 in (2.32). From [26] we
have the following estimates∫

R2N

|uε,i(x)− uε,i(y)|2K(x− y) dx dy = SN/(2s)s +O(εN−2s), (2.36)∫
RN
|uε,i|pdx = SN/(2s)s +O(εN ), (2.37)

where Cs is a positive constant depending on s.

Lemma 2.9. We have(∫
Ω

Q(x)(u+
ε,i)

pdx
)2/p

=
(∫

Ω

upε,idx
)2/p

+ o(εσ). (2.38)

Proof. It is easy to see that∣∣ ∫
Ω

[Q(x)− 1](u+
ε,i)

pdx
∣∣ ≤ ∫

Ω

|Q(x)−Q(ai)|upε,idx

=

∫
Bρ0 (ai)

|Q(x)−Q(ai)|upε,idx.

By (H2), for any γ > 0 there exists δ ∈ (0, ρ) such that

|Q(x)−Q(ai)| < γ|x− ai|σ for all |x− ai| < δ.



12 Y. ZHANG, Q. LI, L. PANG EJDE-2021/23

Recall that

Uε(x) = κ̃ε−(N−2s)/2
(
µ2 +

∣∣ x

εS
1/(2s)
s

∣∣2)−(N−2s)/2

.

By using the shorthand notation ρε := ρ0/(εS
1/(2s)
s µ), δε := δ/(εS

1/(2s)
s µ) and the

change of variable, we have∫
Bρ0 (ai)

|Q(x)−Q(ai)|upε,idx

= κ̃S
N
2s
s µ−N

∫
Bρε (0)

|Q(µS
1
2s
s εx+ ai)−Q(ai)|ηpi (µS

1
2s
s εx+ ai)(1 + |x|2)−Ndx

= κ̃S
N
2s
s µ−N

(∫
Bδε (0)

+

∫
Bρε (0)\Bδε (0)

)∣∣Q(µS
1
2s
s εx+ ai)−Q(ai)

∣∣
× ηpi (µS

1
2s
s εx+ ai)(1 + |x|2)−Ndx

≤ Cγεσ
∫
Bδε (0)

|x|σ

(1 + |x|2)N
dx+ C

∫
Bρε (0)\Bδε (0)

1

(1 + |x|2)N
dx

= CNωNγε
σ

∫ δε

0

rσ+N−1

(1 + r2)N
dr + CNωN

∫ ρε

δε

rN−1

(1 + r2)N
dr

≤ C ′γεσ + C ′εN ,

where C,C ′ > 0 are constants independent of ε, and ωN denotes the volume of the
unit ball in RN . Consequently,

lim sup
ε→0

ε−σ
∣∣ ∫

Ω

[Q(x)− 1](u+
ε,i)

pdx
∣∣ ≤ C ′γ.

The arbitrariness of γ implies (2.38). �

The following lemma is a key for proving our main result.

Lemma 2.10. There exist ε0 > 0 such that for ε < ε0 and λ ∈ (0,Λ),

sup
t≥0

Iλ(uλ + tuε,i) < β∗ uniformly in i, (2.39)

where uλ is the positive solution obtained in Proposition 2.7.

Proof. Since Iλ is continuous in X0 and uε,i is uniformly bounded in X0, there
exists t1 > 0 such that for t ∈ [0, t1],

Iλ(uλ + tuε,i) < αλ +
s

N
S
N
2s
s .

Direct computations show that

Iλ(uλ + tuε,i) =
1

2
‖uλ‖2X0

+ t(uλ, uε,i)X0 +
t2

2
‖uε,i‖2X0

− λ

q

∫
Ω

(uλ + tuε,i)
qdx− 1

p

∫
Ω

Q(x)(uλ + tuε,i)
pdx.

(2.40)

From (2.37), we have ∫
Ω

upε,idx ≥
1

2
S
N
2s
s



EJDE-2021/23 FRACTIONAL LAPLACE PROBLEMS WITH CRITICAL GROWTH 13

for ε small enough. Note that the last term in (2.40) satisfies

1

p

∫
Ω

Q(x)(uλ + tuε,i)
pdx ≥ tp

p

∫
Ω

Q(x)
(uλ
t

+ uε,i

)p
dx

≥ min
x∈Ω̄

Q(x)
tp

p

∫
Ω

upε,idx

≥ min
x∈Ω̄

Q(x)
S
N
2s
s

2p
tp.

Thus, I(uλ + tuε,i) → −∞ as t → ∞ uniformly in ε and i. Consequently, there

exists t2 > t1 such that Iλ(uλ + tuε,i) < αλ + s
N S

N
2s
s for t ≥ t2. Then, we only need

to verify that inequality

sup
t1≤t≤t2

Iλ(uλ + tuε,i) < β∗ uniformly in i,

for ε small enough.
From now on, we assume that t ∈ [t1, t2]. Then there exists a constant C > 0

such that ∫
Ω

Q(x)(uλ + tuε,i)
pdx

≥
∫

Ω

Q(x)upλdx+ tp
∫

Ω

Q(x)upε,idx+ pt

∫
Ω

Q(x)up−1
λ uε,idx

+ p tp−1

∫
Ω

Q(x)up−1
ε,i uλdx− Ct

p/2

∫
Ω

Q(x)u
p/2
λ u

p/2
ε,i dx.

(2.41)

We have used the following inequality (see [4]) for r > 2, there exists a constant Cr
(depending on r) such that

(α+ β)r ≥ αr + βr + r
(
αr−1β + αβr−1

)
− Crαr/2βr/2 ∀α, β > 0.

Using that uλ is a positive solution of (1.1), and (2.41), (2.36), and by Lemma 2.9,
we have

Iλ(uλ + tuε,i)

≤ 1

2
‖uλ‖2X0

+ t(uλ, uε,i)X0
+
t2

2
‖uε,i‖2X0

− λ

q

∫
Ω

(uλ + tuε,i)
qdx

− 1

p

∫
Ω

Q(x)upλdx−
1

p
tp
∫

Ω

Q(x)upε,idx− t
∫

Ω

Q(x)up−1
λ uε,idx

− tp−1

∫
Ω

Q(x)up−1
ε,i uλdx+ Cpt

p/2

∫
Ω

Q(x)u
p/2
λ u

p/2
ε,i dx

=
1

2
‖uλ‖2X0

+ λt

∫
Ω

uq−1
λ uε,idx+

t2

2
‖uε,i‖2X0

− λ

q

∫
Ω

(uλ + tuε,i)
qdx

− 1

p

∫
Ω

Q(x)upλdx−
1

p
tp
∫

Ω

Q(x)upε,idx− t
p−1

∫
Ω

Q(x)up−1
ε,i uλdx

+ Cpt
p/2

∫
Ω

Q(x)u
p/2
λ u

p/2
ε,i dx

= Iλ(uλ) + λt

∫
Ω

uq−1
λ uε,idx+

t2

2
‖uε,i‖2X0

− λ

q

∫
Ω

(uλ + tuε,i)
qdx



14 Y. ZHANG, Q. LI, L. PANG EJDE-2021/23

+
λ

q

∫
Ω

uqλdx−
1

p
tp
∫

Ω

Q(x)upε,idx− t
p−1

∫
Ω

Q(x)up−1
ε,i uλdx

+ Cpt
p/2

∫
Ω

Q(x)u
p/2
λ u

p/2
ε,i dx

= Iλ(uλ) +
t2

2
‖uε,i‖2X0

− λ

q

∫
Ω

[
(uλ + tuε,i)

q − uqλ − qtu
q−1
λ uε,i

]
dx

− 1

p
tp
∫

Ω

Q(x)upε,idx− t
p−1

∫
Ω

Q(x)up−1
ε,i uλdx+ Cpt

p/2

∫
Ω

Q(x)u
p/2
λ u

p/2
ε,i dx

≤ Iλ(uλ) +
t2

2
‖uε,i‖2X0

− tp

p

∫
Ω

Q(x)upε,idx− t
p/2
(
t
p−2
2

∫
Ω

Q(x)up−1
ε,i uλdx

− Cp
∫

Ω

Q(x)u
p/2
λ u

p/2
ε,i dx

)
≤ Iλ(uλ) + S

N
2s
s

( t2
2
− tp

p

)
− tp/2

(
t
p−2
2

∫
Ω

up−1
ε,i Q(x)uλdx

− Cp
∫

Ω

Q(x)u
p/2
λ u

p/2
ε,i dx

)
+O(εN−2s) + o(εσ)

≤ Iλ(uλ) +
s

N
S
N
2s
s − tp/2

(
t
p−2
2

∫
Ω

Q(x)up−1
ε,i uλdx− Cp

∫
Ω

Q(x)u
p/2
λ u

p/2
ε,i dx

)
+O(εN−2s) + o(εσ). (2.42)

Here we have used the elementary inequality: (α+β)q ≥ αq + qαq−1β for α, β > 0.
Now, we estimate the third term in (2.42). There exists a constant C1 > 0

independent of i such that Q(x)uλ(x) ≥ C1 for all x ∈ Bρ0/2(ai). Then∫
Ω

Q(x)up−1
ε,i uλdx ≥ C1

∫
Bρ0/2(ai)

Up−1
ε (x− ai)dx ≥ C1ε

N−2s
2 . (2.43)

Direct computations show that there exists a constant C2 > 0 independent of i
such that∫

Ω

Q(x)u
p/2
λ u

p/2
ε,i dx ≤ C2

∫
Bρ0 (ai)

Up/2ε (x− ai)dx ≤ C2ε
N
2 | ln ε|. (2.44)

By (2.42), (2.43) and (2.44), we have

sup
t1≤t≤t2

I(uλ + tuε,i) < I(uλ) +
s

N
S
N
2s
s

for ε small enough. �

We define X+ := {u ∈ X0 : u+ 6≡ 0}, and

A1 :=
{
u ∈ X+

0 :
1

‖u‖X0

t−
( u

‖u‖X0

)
> 1
}
,

A2 :=
{
u ∈ X+

0 :
1

‖u‖X0

t−
( u

‖u‖X0

)
< 1
}
.

Following the idea in [29], we have the following results.

Lemma 2.11. Assume that λ ∈ (0,Λ). We have

(i) X+
0 = A1 ∪ A2 ∪N−λ ,

(ii) N+ ⊂ A1,
(iii) there exists tε,i > 1 such that uλ + tε,iuε,i ∈ A2 for each 1 ≤ i ≤ m,



EJDE-2021/23 FRACTIONAL LAPLACE PROBLEMS WITH CRITICAL GROWTH 15

(iv) there exists sε,i ∈ (0, 1) such that uλ+sε,itε,iuε,i ∈ N−λ for each 1 ≤ i ≤ m,

(v) α−λ < α+
λ + s

N S
N
2s
s .

Proof. (i) Let

S :=
{
u ∈ X+

0 :
1

‖u‖X0

t−
( u

‖u‖X0

)
= 1
}
.

It suffices to prove that N−λ = S. Let v = u/‖u‖X0 for u ∈ N−λ . By Lemma

2.3, there exists t−(v) > 0 such that t−(v)v ∈ N−λ , that is t−(v)
‖u‖X0

u ∈ N−λ . Since

u ∈ N−λ , we have t−(v) = ‖u‖X0 . Hence, we obtain N−λ ⊂ S. On the other hand,
let u ∈ S. Then,

u = t−
( u

‖u‖X0

) u

‖u‖X0

∈ N−.

Thus, S ⊂ N−λ . Consequently, N−λ = S.

(ii) For any u ∈ N+
λ , let v = u

‖u‖X0
. By Lemma 2.3, there exists t−(v) > 0 such

that t−(v)v ∈ N−, that is

1

‖u‖X0

t−
( u

‖u‖X0

)
u ∈ N−.

Hence,

t−(u) =
1

‖u‖X0

t−
( u

‖u‖X0

)
.

By Lemma 2.3, we have 1 = t+(u) < tmax(u) < t−(u). Therefore, N+ ⊂ A1.
(iii) Firstly, we claim that there exists a positive constant C independent of i

such that

sup
t≥0

t−
( uλ + tuε,i
‖uλ + tuε,i‖X0

)
< C.

Assume by contradiction that there exists s sequence {tn,i} such that tn,i → +∞
and t−(vn,i)→ +∞ as n→∞, where

vn,i :=
uλ + tnuε,i

‖uλ + tnuε,i‖X0

.

Since t−(vn,i)vn,i ∈ N−λ , by Lebesgue Dominated Convergence Theorem, we have∫
Ω

(v+
n,i)

pdx =
1

‖t−1
n,iuλ + uε,i‖pX0

∫
Ω

(t−1
n,iuλ + uε,i)

pdx

→
∫

Ω
upε,idx

‖uε,i‖pX0

=
S
N
2s
s +O(εN )

[S
N
2s
s +O(εN−2s)]p/2

as n→∞. Thus

Iλ(t−(vn,i)vn,i)

=
1

2
(t−(vn,i))

2 − λ

q
(t−(vn,i))

q

∫
Ω

(v+
n,i)

qdx− (t−(vn,i))
p

p

∫
Ω

(v+
n,i)

pdx

→ −∞



16 Y. ZHANG, Q. LI, L. PANG EJDE-2021/23

as n → ∞, which is impossible since I is bounded from below on Nλ by Lemma
2.1. Set

tε,i =
‖uλ‖X0 +

(
‖uλ‖2X0

+ |C2 − ‖uλ‖2X0
|
)1/2

‖uε,i‖X0

+ 1.

Then

‖uλ + tε,iuε,i‖2X0
= ‖uλ‖2X0

+ t2ε,i‖uε,i‖2X0
+ 2tε,i (uλ, uε,i)X0

> ‖uλ‖2X0
+
∣∣C2 − ‖uλ‖2X0

∣∣
≥ C2 >

[
t−
( uλ + tuε,i
‖uλ + tuε,i‖X0

)]2
.

Hence, we obtain uλ + tε,iuε,i ∈ A2.
(iv) Define γi : [0, 1]→ R as

γi(s) :=
1

‖uλ + stε,iuε,i‖X0

t−
( uλ + stε,iuε,i
‖uλ + stε,iuε,i‖X0

)
for all s ∈ [0, 1].

Note that γ(s) is a continuous function of s. Since γ(0) > 1 and γ(1) < 1 there
exists sε,i ∈ (0, 1) such that γ(sε,i) = 1, that is uλ + sε,itε,iuε,i ∈ N−λ .

(v) By Lemma 2.10 and (iv), we have α−λ < α+
λ + s

N S
N
2s
s . �

Let

P = {ai : 1 ≤ i ≤ m} and Pρ0 = ∪mi=1Bρ0(ai).

Let r0 = max1≤i≤m |ai| + ρ0. We minimize the energy functional Iλ on some
submanifolds ofNλ. To this end, we define a barycenter map (cf. [8]) K : X0\{0} →
RN as

K(u) =

∫
Ω
χ(x)|u|pdx∫
Ω
|u|pdx

,

where

χ(x) =

{
x, |x| ≤ r0,

r0x/|x|, |x| > r0.

Lemma 2.12. K(t−(uε,i)uε,i) → ai as ε → 0. In particular, there exists ε1 ∈
(0, ε0) such that if ε ∈ (0, ε1), then K(t−(uε,i)uε,i) ∈ Pρ0 for each 1 ≤ i ≤ m.

Proof. Direct computations imply that

K(t−(uε,i)uε,i) =

∫
Ω
χ(x)ηpi (x)Upε (x− ai)dx∫
Ω
ηpi (x)Upε (x− ai)dx

=

∫
Bρ0 (ai)

χ(x)ηpi (x)Upε (x− ai)dx∫
Bρ0 (ai)

ηpi (x)Upε (x− ai)dx

= ai +
ε
∫
Bρ0/ε(0)

xηpi (εx+ ai)
(
µ2 + | x

S
1/(2s)
s

|2
)−N

dx∫
Bρ0/ε(0)

ηpi (εx+ ai)
(
µ2 + | x

S
1/(2s)
s

|2
)−N

dx
.

Since
∫
Bρ0/ε(0)

xηpi (εx+ ai)(µ2 + | x

S
1/(2s)
s

|2)−Ndx is bounded and∫
Bρ0/ε(0)

ηpi (εx+ ai)(µ2 + | x

S
1/(2s)
s

|2)−Ndx



EJDE-2021/23 FRACTIONAL LAPLACE PROBLEMS WITH CRITICAL GROWTH 17

is bounded away from zero, we have

K(t−(uε,i)uε,i)→ ai

as ε → 0. Consequently, there exists ε1 ∈ (0, ε0) such that K(t−(uε,i)uε,i) ∈ Pρ0
for any ε ∈ (0, ε1) and each 1 ≤ i ≤ m. �

For each 1 ≤ i ≤ m, we define

Oλ,i = {u ∈ N−λ : |K(u)− ai| < ρ0},
∂Oλ,i = {u ∈ N−λ : |K(u)− ai| = ρ0},

βλ,i = inf
u∈Oλ,i

Iλ(u), β̃λ,i = inf
u∈∂Oλ,i

Iλ(u).

Consider the critical problem

(−∆)su = |u|p−2u in Ω,

u = 0 in RN \ Ω.
(2.45)

We define the energy functional J : X0 → R associated with the critical problem
(2.45) as

J(u) =
1

2

∫
R2N

(u(x)− u(y))2K(x− y) dx dy − 1

p

∫
Ω

|u|pdx.

Set

M(Ω) = {u ∈ X0 \ {0} : 〈J ′(u), u〉 = 0}, γ(Ω) = inf
u∈M(Ω)

J(u).

Similarly, we define J∞ : Ḣs(RN )→ R as

J∞(u) =
1

2

∫
R2N

(u(x)− u(y))2K(x− y) dx dy − 1

p

∫
RN
|u|pdx,

where Ḣs(RN ) denotes the space of functions u ∈ Lp(RN ) such that
∫
R2N (u(x) −

u(y))2K(x− y) dx dy <∞. Set

M(RN ) = {u ∈ Ḣs(RN ) : 〈J∞(u), u〉 = 0}, γ(RN ) = inf
u∈M(RN )

J∞(u).

It is easy to see that γ(RN ) = s
N S

N
2s
s . The following results corresponds to the

classical results of [9, 28].

Lemma 2.13. (i) γ(Ω) = γ(RN ) and γ(Ω) is never achieved except when Ω =
RN ;

(ii) γ(Ω) = α0.

Proof. (i) Since M(Ω) ⊂M(RN ), we have γ(RN ) ≤ γ(Ω). Conversely, let {un} ⊂
Ḣs(RN ) be a minimizing sequence for γ(RN ). By density of C∞0 (RN ) in Ḣs(RN )
we may assume that un ∈ C∞0 (RN ). We can choose yn ∈ RN and λn > 0 such that

uyn,λnn (·) := λ
N−2s

2
n un(λn ·+yn) ∈ C∞0 (Ω).

Since

‖uyn,λnn ‖X0
= ‖un‖Ḣ(RN ),

∫
Ω

|uyn,λnn |pdx =

∫
RN
|un|pdx,

we obtain γ(Ω) ≤ γ(RN ). Thus, γ(Ω) = γ(RN ).



18 Y. ZHANG, Q. LI, L. PANG EJDE-2021/23

Assume by contradiction that Ω 6= RN and u ∈ X0 is a minimizer for γ(Ω). Let
t > 0 such that t|u| ∈ M(Ω). Then

t =
( ∥∥|u|∥∥2

X0∫
Ω
|u|pdx

) 1
p−2 ≤

( ‖u‖2X0∫
Ω
|u|pdx

) 1
p−2

= 1.

Consequently,

γ(Ω) ≤ J(t|u|) = tp
(1

2
− 1

p

) ∫
Ω

|u|pdx ≤ γ(Ω).

Thus, t = 1 and |u| ∈ M(Ω) is another minimizer for γ(Ω). For this reason we
assume straight away that u ≥ 0. Clearly, u ∈ RN is a minimizer for J∞. Therefore,
we obtain that J ′∞(u) = 0. So that u is a solution of

(−∆)su = up in RN .
By maximum principle [27, Proposition 2.2.8], u > 0 in RN . This is a contradiction.

(ii) For every u ∈ N0, one sees immediately that tu ∈ M(Ω) for some t > 0.
Indeed, tu ∈M(Ω) is equivalent to

‖tu‖2X0
=

∫
Ω

|tu|pdx,

which has solution

t =
( ‖u‖2X0∫

Ω
|u|pdx

) 1
p−2

> 0.

Since u ∈ N0 and maxx∈Ω̄Q(x) = 1, we have

‖u‖2X0
=

∫
Ω

Q(x)(u+)pdx ≤
∫

Ω

|u|pdx,

which implies t ≤ 1. Therefore,

γ(Ω) ≤ J(tu) =
(1

2
− 1

p

)
‖tu‖2X0

≤
(1

2
− 1

p

)
‖u‖2X0

.

By the arbitrariness of u ∈ N0, we have γ(Ω) ≤ α0.
By (2.36) and (2.38), we have

‖uε,i‖2X0( ∫
Ω
Q(x)(u+

ε,i)
pdx
)2/p = Ss +O(εN−2s) + o(εσ). (2.46)

Direct computations show that

sup
t≥0

(a
2
t2 − b

p
tp
)

=
s

N

( a

b2/p

)N/(2s)
for any a, b > 0. By (2.46), we obtain that

sup
t≥0

I0(tuε,i) =
s

N

( ‖uε,i‖2X0( ∫
Ω
Q(x)(u+

ε,i)
pdx
)2/p)N/(2s)

≤ s

N
SN/(2s)s +O(εN−2s) + o(εσ).

(2.47)

Let tε,i > 0 be such that tε,iuε,i ∈ N0. Then, by (2.47), we have

α0 ≤ I0(tε,iuε,i) ≤ sup
t≥0

I0(tuε,i) ≤
s

N
SN/(2s)s +O(εN−2s) + o(εσ).

Passing to the limit, we obtain that α0 ≤ γ(Ω). Thus γ(Ω) = α0. �



EJDE-2021/23 FRACTIONAL LAPLACE PROBLEMS WITH CRITICAL GROWTH 19

To show that βλ,i are (PS) values, we need the following Palais-Smale decom-
position theorem, see [20, Theorem 1.1] or [19, Theorem 4].

Theorem 2.14. Assume that {un} is a (PS)c sequence in X0 for J . Then there
exists a (possibly trivial) solution u0 ∈ X0 to problem (2.45) such that, up to a
subsequence, un ⇀ u0 in X0. Moreover, either the convergence is strong, or there
exist ` ∈ N, nontrivial solutions u1, . . . , u` ∈ Ḣs(RN ) to the equations

(−∆)su = |u|p−2u in RN , (2.48)

or

(−∆)su = |u|p−2u in RN+ , u = 0 in RN \ RN+ , (2.49)

sequences of points x1
n, . . . , x

`
n ⊂ Ω, and finitely many sequences of numbers r1

n, . . . ,
r`n ⊂ (0,+∞) converging to zero such that, up to a subsequence,

ujn := (rjn)
N−2s

2 un(xjn + rjnx) ⇀ uj in Ḣs(RN ),

for j = 1, . . . , `, and

lim
n→∞

∥∥un − u0 −
∑̀
j=1

(rjn)
2s−N

2 uj
(x− xjn

rjn

)∥∥
Ḣ(RN )

= 0,

lim
n→∞

‖un‖2X0
=
∑̀
j=0

‖uj‖2
Ḣs(RN )

,

lim
n→∞

J(un) = J(u0) +
∑̀
j=1

J∞(uj),

∣∣ ln rin
rjn

∣∣+
∣∣xin − xjn

rin

∣∣→∞ as n→∞, for i 6= j, i, j = 1, . . . , `.

Lemma 2.15. There exists δ0 > 0 such that if u ∈ N0 and I0(u) ≤ α0 + δ0, then
K(u) ∈ Pρ0/2.

Proof. Assume by contradiction that there exists a sequence {un} ⊂ N0 such that

I0(un) = α0 + o(1) and K(un) 6∈ Pρ0/2,

for all n ∈ N. Let sn > 0 be such that snun ∈M(Ω). By Lemma 2.13, We obtain

γ(Ω) ≤ J(snun) ≤ I0(snun) ≤ sup
s≥0

I0(sun) = I0(un) = γ(Ω) + o(1).

Then, sn = 1 + o(1) and J(snun) = γ(Ω) + o(1). By Ekeland’s variational
principle[14], there exists a sequence {vn} ⊂ M(Ω) such that

J ′(vn)→ 0, J(vn)→ γ(Ω), ‖vn − snun‖X0 → 0.

By Lemma 2.13 and Theorem 2.14, there exists a (possibly trivial) solution v0 ∈ X0

to problem (2.45) such that vn ⇀ v0 in X0, and there exist ` ∈ N, nontrivial

solutions v1, . . . , v` ∈ Ḣs(RN ) to (2.48) or (2.49), sequences of points x1
n, . . . , x

`
n ⊂

Ω and finitely many sequences of numbers r1
n, . . . , r

`
n ⊂ (0,+∞) converging to zero

such that, up to a subsequence,

vn = v0 +
∑̀
j=1

(rjn)
2s−N

2 vj
(x− xjn

rjn

)
+ o(1) in Ḣs(RN ), (2.50)



20 Y. ZHANG, Q. LI, L. PANG EJDE-2021/23

and

J(vn) = J(v0) +
∑̀
j=1

J∞(vj) + o(1). (2.51)

If v0 6= 0 or ` > 1, then by (2.51), we have J(vn)→ J(v0) +
∑`
j=1 J∞(vj) > γ(Ω),

which is a contradiction. Thus, by (2.50),

vn = (r1
n)

2s−N
2 v1

(x− x1
n

r1
n

)
+ o(1) in Ḣs(RN ). (2.52)

By following the argument in [20, Theorem 1.1], we have dist(x1
n, ∂Ω)/r1

n → ∞ as
n→∞. We may assume x1

n → x1
0 ∈ Ω̄ since Ω is bounded. By Lebesgue dominated

convergence theorem, we obtain that

SN/(2s)s ≤
∫

Ω

Q(x)|vn|pdx+ o(1)

= (r1
n)−N

∫
Ω

Q(x)
∣∣v1
(x− x1

n

r1
n

)∣∣pdx+ o(1)

=

∫
Ωn

Q(xr1
n + x1

n)|v1(x)|pdx+ o(1)

= Q(x1
0)

∫
RN
|v1(x)|p 1Ωndx+

∫
RN

[
Q(xr1

n + x1
n)

−Q(x1
0)
]
|v1(x)|p 1Ωndx+ o(1)

= Q(x1
0)

∫
RN
|v1(x)|pdx+ o(1),

where 1Ωn is the indicator function,

1Ωn :=

{
1, if x ∈ Ωn,

0, if x 6∈ Ωn,

Ωn := {x ∈ RN : xr1
n + x1

n ∈ Ω} → RN as n → ∞. Thus, we obtain that x1
0 ∈ P.

Consequently,

K(un) =

∫
Ω
χ(x)

∣∣v1
(x−x1

n

r1n

)∣∣pdx∫
Ω

∣∣v1
(x−x1

n

r1n

)∣∣pdx + o(1)

=

∫
Ωn
χ(xr1

n + x1
n)|v1(x)|pdx∫

Ωn
|v1(x)|pdx

+ o(1)

→ x1
0 ∈ Pρ0/2

as n→∞. We get a contradiction. �

Lemma 2.16. There exists Λ∗ ∈
(
0, q2Λ

)
such that if λ ∈ (0,Λ∗) and u ∈ N−λ with

Iλ(u) ≤ s
N S

N/(2s)
s + δ0

2 (δ0 is the constant from Lemma 2.15), then K(u) ∈ Pρ0/2.

Proof. Fix any u ∈ N−λ with Iλ(u) ≤ s
N S

N/(2s)
s + δ0

2 , and let

t(u) =
( ‖u‖2X0∫

Ω
Q(x)(u+)pdx

)1/(p−2)

.



EJDE-2021/23 FRACTIONAL LAPLACE PROBLEMS WITH CRITICAL GROWTH 21

Clearly,t(u)u ∈ N0. For λ ∈ (0,Λ0), by (2.1), we have(1

2
− 1

p

)
‖u‖2X0

≤ Iλ(u) + λ
(1

q
− 1

p

)
|Ω|

p−q
p S−q/2s ‖u‖qX0

≤ s

N
SN/(2s)s +

δ0
2

+ Λ0

(1

q
− 1

p

)
|Ω|

p−q
p S−q/2s ‖u‖qX0

.

Thus, there exists a constant C1 independent of λ and u such that ‖u‖X0 ≤ C1.
By (2.3), we obtain

2− q
p− q

≤
∫

Ω
Q(x)(u+)pdx

‖u‖2X0

.

Consequently,

t(u) ≤
(p− q

2− q

)1/(p−2)

.

Since t−(u) = 1 and tmax < t(u) (tmax is defined in Lemma 2.3), by Lemma 2.3, we
have

s

N
SN/(2s)s +

δ0
2
≥ Iλ(u) = sup

t≥tmax

Iλ(tu)

≥ Iλ(t(u)u)

≥ I0(t(u)u)− λ

q

∫
Ω

(t(u)u+)qdx.

Thus, we obtain

I0(t(u)u) ≤ s

N
SN/(2s)s +

δ0
2

+
λ

q

∫
Ω

(t(u)u+)qdx

≤ s

N
SN/(2s)s +

δ0
2

+
λ

q
|Ω|

p−q
p S−q/2s tq(u)‖u‖qX0

≤ s

N
SN/(2s)s +

δ0
2

+
λ

q
|Ω|

p−q
p S−q/2s Cq1

(p− q
2− q

)q/(p−2)
.

Consequently, there exists Λ∗ ∈ (0, qΛ/2) such that

I0(t(u)u) ≤ s

N
SN/(2s)s + δ0

for λ ∈ (0,Λ∗). By Lemma 2.15, we have

K(t(u)u) =

∫
RN χ(x)|t(u)u|pdx∫

RN |t(u)u|pdx
∈ Pρ0/2

for λ ∈ (0,Λ∗). Meanwhile, K(u) ∈ Pρ0/2 for λ ∈ (0,Λ∗). �

Lemma 2.17. For each u ∈ Nλ, there exist ε > 0 and a differential function
η : Bε(0) ⊂ X0 → (0,+∞) such that

η(0) = 1, η(w)(u− w) ∈ Nλ for w ∈ Bε(0),

〈η′(0), w〉 =
2(u,w)X0 − λq

∫
Ω

(u+)q−1wdx− p
∫

Ω
Q(u+)pdx

(2− q)‖u‖2X0
− (p− q)

∫
Ω
Q(u+)pdx

(2.53)

for all w ∈ X0.

Since the proof of the above Lemma is similar to that of Lemma 2.5, we omit it.

Lemma 2.18. For each 1 ≤ i ≤ m, there exists a (PS)βλ,i sequence {uin} ⊂ Oλ,i
for Iλ.



22 Y. ZHANG, Q. LI, L. PANG EJDE-2021/23

Proof. By Lemma 2.16, we have

β̃λ,i ≥
s

N
SN/(2s)s +

δ0
2

(2.54)

for all λ ∈ (0,Λ∗). By Lemma 2.10, we have

βλ,i ≤ α−λ < β∗ (2.55)

for all λ ∈ (0,Λ∗). For each 1 ≤ i ≤ m, by (2.54) and (2.55), we have

βλ,i < β̃λ,i (2.56)

for all λ ∈ (0,Λ∗). Then
βλ,i = inf

u∈Oλ,i∪∂Oλ,i
Iλ(u)

for all λ ∈ (0,Λ∗). By Lemma 2.17 and Ekeland’s variational principle, we can
prove Lemma 2.18. The rest of proof is similar to that of Lemma 2.6, we omit
it. �

Proof of Theorem 1.1. For each 1 ≤ i ≤ m, by Lemma 2.18, there exists a (PS)βλ,i
sequence {uin} ⊂ Oλ,i for Iλ. Since Iλ satisfies the (PS)β condition for β < β∗,
by (2.55), Iλ has at least m critical points in N−λ for λ ∈ (0,Λ∗). Consequently,

problem (1.1) has m positive solutions. Furthermore, since u ∈ N+
λ is a solution of

(1.1), as shown in Theorem 1.1, problem (1.1) has m+ 1 positive solutions. �

3. Critical and convex case q > 2

This section is devoted to the study of problem (1.1) when the exponent satisfies
2 < q < p. As the energy functional Iλ is not bounded below on X0, it is useful to
consider the functional on the Nehari manifold

Nλ = {u ∈ X0 \ {0} : 〈I ′λ(u), u〉 = 0}

=
{
u ∈ X0 \ {0} : ‖u‖2X0

= λ

∫
Ω

(u+)qdx+

∫
Ω

Q(x)(u+)pdx
}
.

Now, we give some properties of Nλ.

Lemma 3.1. The functional Iλ is coercive and bounded from below on Nλ.

Proof. For every u ∈ Nλ, we have

Iλ(u) =
(1

2
− 1

q

)
‖u‖2X0

+
(1

q
− 1

p

) ∫
Ω

Q(x)(u+)pdx > 0, (3.1)

since 2 < q < p. Thus, Iλ is coercive and bounded from below on Nλ. �

Lemma 3.2. For each u ∈ X+
0 , there exists t(u) > 0 such that t(u)u ∈ Nλ and

Iλ(t(u)u) = supt≥0 Iλ(tu).

Proof. We define γ(t) = Iλ(tu) for t ≥ 0. It is easy to see that there exists t(u) > 0
such that γ′(t) > 0 for t ∈ (0, t(u)) and γ′(t) < 0 for t ∈ (t(u),+∞). Then
supt≥0 γ(t) is attained at some t(u) > 0. This implies that γ′(t(u)) = 0. Conse-
quently, t(u)u ∈ Nλ. �

To prove the existence of positive solutions, we claim that Iλ satisfies the (PS)β

condition in X0 for β < s
N S

N
2s
s .

Lemma 3.3. Iλ satisfies the (PS)β condition in X0 for β < s
N S

N
2s
s .



EJDE-2021/23 FRACTIONAL LAPLACE PROBLEMS WITH CRITICAL GROWTH 23

The proof of the above lemma is similar to that of Lemma 2.8, we omit it. Direct
computation yields the following estimates.

Lemma 3.4. There exists a positive constant Cr such that

∫
Ω

|uε,i|rdx ≥


Crε

N− (N−2s)r
2 , if r > N

N−2s ,

Crε
N
2 | ln ε|, if r = N

N−2s ,

Crε
(N−2s)r

2 , if r < N
N−2s .

(3.2)

Next, we want to obtain an estimate of supt≥0 Iλ(tuε,i).

Lemma 3.5. There exists ε0 ∈ (0, ρ0/2) such that if ε ∈ (0, ε0), then

sup
t≥0

Iλ(tuε,i) <
s

N
S
N
2s
s uniformly in i. (3.3)

Proof. Since Iλ is continuous in X0 and uε,i is uniformly bounded in X0, there
exists t1 > 0 such that for t ∈ [0, t1],

Iλ(tuε,i) <
s

N
S
N
2s
s .

By (2.37), we have ∫
Ω

upε,idx ≥
1

2
S
N
2s
s

for ε small enough. Thus, I(tuε,i) → −∞ as t → ∞ uniformly in ε and i. Conse-

quently, there exists t2 > t1 such that Iλ(tuε,i) <
s
N S

N
2s
s for t ≥ t2. Then, we only

need to verify that inequality

sup
t1≤t≤t2

Iλ(tuε,i) <
s

N
S
N
2x
s uniformly in i,

for ε small enough.
From now on, we assume that t ∈ [t1, t2]. Since N ≥ 4, we obtain that q > 2 >
N

N−2s . Consequently, by Lemma 3.4,∫
Ω

|uε,i|qdx ≥ CqεN−
(N−2s)q

2 . (3.4)

By (2.36), (2.38) and (3.4), we have

Iλ(tuε,i) =
1

2
t2‖uε,i‖2X0

− λ

q
tq
∫

Ω

|uε,i|qdx−
1

p
tp
∫

Ω

Q(x)|uε,i|pdx

= S
N
2s
s

( t2
2
− tp

p

)
− λ

q
tq
∫

Ω

|uε,i|qdx+O(εN−2s) + o(εσ)

≤ s

N
S
N
2s
s − Cqtq1

λ

q
εN−

(N−2s)q
2 +O(εN−2s) + o(εσ).

Since N ≥ 4 and σ = N − (N−2s)q
2 , we can choose ε0 > 0 small enough such that

−Cqtq1
λ

q
εN−

(N−2s)q
2 +O(εN−2s) + o(εσ) < 0

for all ε ∈ (0, ε0). Thus, we obtain (3.3). �

By Lemma 3.2, there exists t(uε,i) > 0 such that t(uε,i)uε,i ∈ Nλ. Then we have
the following lemma.



24 Y. ZHANG, Q. LI, L. PANG EJDE-2021/23

Lemma 3.6. K(t(uε,i)uε,i)→ ai as ε→ 0. In particular, there exists ε1 ∈ (0, ε0)
such that if ε ∈ (0, ε1), then K(t(uε,i)uε,i) ∈ Pρ0 for each 1 ≤ i ≤ m.

The proof of the above lemma is similar to that of Lemma 2.12, we omit it. For
each 1 ≤ i ≤ m, we define

Gλ,i = {u ∈ Nλ : |K(u)− ai| < ρ0},
∂Gλ,i = {u ∈ Nλ : |K(u)− ai| = ρ0},

δλ,i = inf
u∈Gλ,i

Iλ(u), δ̃λ,i = inf
u∈∂Gλ,i

Iλ(u).

Lemma 3.7. There exists δ0 > 0 such that if u ∈ N0 and I0(u) ≤ α0 + δ0, then
K(u) ∈ Pρ0/2.

The proofof the above lemma is similar to that of Lemma 2.15, we omit it.

Lemma 3.8. There exists Λ∗ > 0 such that if λ ∈ (0,Λ∗) and u ∈ Nλ with

Iλ(u) ≤ s
N S

N/(2s)
s + δ0 (δ0 is the constant from Lemma 3.7), then K(u) ∈ Pρ0/2.

Proof. Fix any u ∈ Nλ with Iλ(u) ≤ s
N S

N/(2s)
s + δ0

2 , and let

t(u) =
( ‖u‖2X0∫

Ω
Q(x)(u+)pdx

)1/(p−2)

.

Clearly, t(u)u ∈ N0. Since

Iλ(v) ≥ 1

2
‖v‖2X0

− λ

q
|Ω|

p−q
p S−q/2s ‖v‖qX0

− 1

p
S−p/2s ‖v‖pX0

, ∀v ∈ X0,

there exist positive numbers d1 and d2 such that Iλ(v) ≥ d2 if ‖v‖X0
= d1.

Obviously, there exists t0 > 0 such that ‖t0u‖ = d1. By Lemma 3.2, we have

0 < d2 ≤ Iλ(t0u)

≤ sup
t≥0

Iλ(tu)

= Iλ(u)

=
(1

2
− 1

p

)
‖u‖2X0

− λ
(1

q
− 1

p

) ∫
Ω

(u+)qdx

≤
(1

2
− 1

p

)
‖u‖2X0

.

Consequently, there exists a constant C1 independent of λ and u such that ‖u‖X0
≥

C1. On the other side, we have

s

N
SN/(2s)s +

δ0
2
≥ Iλ(u)

=
(1

2
− 1

q

)
‖u‖2X0

+
(1

q
− 1

p

) ∫
Ω

Q(x)(u+)pdx

≥
(1

2
− 1

q

)
‖u‖2X0

.

Thus, there exists a constant C2 independent of λ and u such that ‖u‖X0
≤ C2.

Moreover, ∫
Ω

Q(x)(u+)pdx = ‖u‖2X0
− λ

∫
Ω

(u+)qdx



EJDE-2021/23 FRACTIONAL LAPLACE PROBLEMS WITH CRITICAL GROWTH 25

≥ C1 − λ|Ω|
p−q
p S−q/2s ‖u‖qX0

≥ C1 − λ|Ω|
p−q
p S−q/2s C2.

It follows that there exists Λ > 0 such that for λ ∈ (0,Λ),∫
Ω

Q(x)(u+)pdx ≥ C1 − Λ|Ω|
p−q
p S−q/2s C2 > 0.

Hence, there exists a constant C3 > 0 independent of u such that t(u) ≤ C3 for
λ ∈ (0,Λ).

By Lemma 3.2, we have

s

N
SN/(2s)s +

δ0
2
≥ Iλ(u) = sup

t≥0
Iλ(tu)

≥ Iλ(t(u)u)

≥ I0(t(u)u)− λ

q

∫
Ω

(t(u)u+)qdx.

Thus, we obtain

I0(t(u)u) ≤ s

N
SN/(2s)s +

δ0
2

+
λ

q

∫
Ω

(t(u)u+)qdx

≤ s

N
SN/(2s)s +

δ0
2

+
λ

q
|Ω|

p−q
p S−q/2s tq(u)‖u‖qX0

≤ s

N
SN/(2s)s +

δ0
2

+
λ

q
|Ω|

p−q
p S−q/2s Cq1

(p− q
2− q

)q/(p−2)
.

Consequently, there exists Λ∗ ∈ (0,Λ) such that

I0(t(u)u) ≤ s

N
SN/(2s)s + δ0

for λ ∈ (0,Λ∗). By Lemma 3.7, we have

K(t(u)u) =

∫
RN χ(x)|t(u)u|pdx∫

RN |t(u)u|pdx
∈ Pρ0/2

for λ ∈ (0,Λ∗). Meanwhile, K(u) ∈ Pρ0/2 for λ ∈ (0,Λ∗). �

According to Lemma 3.5 and 3.6, there exists ε1 > 0 such that

δλ,i ≤ Iλ(t(uε,i)uε,i) <
s

N
S
N
2s
s (3.5)

for all ε ∈ (0, ε1). By Lemma 3.8, we obtain

δ̃λ,i ≥
s

N
S
N
2s
s +

δ0
2

(3.6)

for λ ∈ (0,Λ∗). By (3.5) and (3.6), we obtain δ̃λ,i > δλ,i for λ ∈ (0,Λ∗). Thus,

δλ,i = inf
u∈Gλ,i

Iλ(u).

Consequently, similar to that of Lemma 2.18, we obtain the following lemma.

Lemma 3.9. For each 1 ≤ i ≤ m, there exists a (PS)δλ,i sequence {uin} ⊂ Gλ,i for
Iλ.



26 Y. ZHANG, Q. LI, L. PANG EJDE-2021/23

Proof of Theorem 1.2. For each 1 ≤ i ≤ m, by Lemma 3.9, there exists a (PS)δλ,i

sequence {uin} ⊂ Gλ,i for Iλ. Since Iλ satisfies the (PS)β condition for β < s
N S

N
2s
s ,

by (3.5), Iλ has at least m critical points in Nλ for λ ∈ (0,Λ∗). Consequently,
problem (1.1) has m positive solutions. �

Acknowledgments. The authors are grateful to Professor Zhaosheng Feng and
the referees for their careful reading and valuable comments and suggestions, which
improved this article. This work is partially supported by the National Natural
Science Foundation of China (No. 11871315) and the Natural Science Foundation
of Shanxi Province of China (No. 201901D111021).

References

[1] A. Ambrosetti, H. Brezis, G. Cerami; Combined effects of concave and convex nonlinearities

in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.

[2] A. Ambrosetti, J. Garcia, I. Peral; Multiplicity reults for some nonlinear elliptic equations,
J. Func. Anal., 137 (1996), 219-242.

[3] J. H. Bae, Y. H. Kim; Multiple solutions for discontinuous elliptic problems involving the

fractional Laplacian, Electron. J. Differential Equations, 2019 (18) (2019), 1-16.
[4] A. Bahri, Y. Li; On a min-max procedure for the existence of a positive solution for certain

scalar field equations in RN , Rev. Mat. Iberoam., 6 (1990), 1-15.

[5] B. Barrios, E. Colorado, A. de Pablo, U. Sanchez; On some critical problems for the fractional
Laplacian operator, J. Differ. Equ., 252 (2012), 6133-6162.

[6] B. Barriosa, E. Colorado, R. Servadeid, F. Soria; A critical fractional equation with concave-

convex power nonlinearities, Ann. I. H. Poincaré-AN, 32 (2015), 875-900.
[7] T. Bartsch, M. Willem; On an elliptic equation with concave and convex nonlinearities, Proc.

Amer. Math. Soc., 123 (1995), 3555-3561.
[8] V. Benci, G. Cerami; Positive solutions of semilinear elliptic equation in exterior domins,

Arch. Ration. Mech. Anal., 99 (1987), 283-300.

[9] H. Brézis, L. Nirenberg; Positive solutions of nonlinear elliptic equations involving critical
Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.

[10] K. J. Brown, T. Wu; A fibering map approach to a semilinear elliptic boundary value problem,

Elec. J. Differ. Equ., 2007 (2007), 1-9.
[11] X. Cabré, J. Tan; Positive solutions of nonlinear problems involving the square root of the

Laplacian, Adv. Math., 224 (2010), 2052-2093.

[12] M. Chhetri, P. Girg, E. Hollifield; Existence of positive solutions for fractional Laplacian
equations: theory and numerical experiments, Electron. J. Differential Equations, 2020 (81)

(2020), 1-31.

[13] R. Cont, P. Tankov; Financial Modeling with Jump Processes, Chapman & Hall/CRC Financ.
Math. Ser., Chapman and Hall/CRC, BocaRaton, Fl, 2004.

[14] I. Ekeland; On the variational principle, J. Math. Anal. Appl., 17 (1974), 324-353.

[15] A. Fiscella, R. Servadei, E. Valdinoci; Density properties for fractional Sobolev spaces, Ann.
Acad. Sci. Fenn. Math., 40 (2015), 235-253.

[16] J. Garcia, I. peral; Multiplicity of solutions for elliptic problems with critical exponent or
with a nonsymmetric term, Trans. Amer. math. Soc., 323 (1991), 877-895.

[17] A. Majda, E. Tabak; A two-dimensional model for quasigeostrophic flow: comparison with
the two-dimensional Euler flow, in: Nonlinear Phenomena in Ocean Dynamics, Los Alamos,
NM, 1995, Physica D, 98 (1996), 515-522.

[18] E. D. Nezza, G. Palatucci, E. Valdinoci; Hitchhikers guide to the fractional Sobolev spaces,

Bull. Sci. Math., 136 (2012), 521-573.
[19] G. Palatucci, A. Pisante; Improved Sobolev embeddings, profile decomposition, and concen-

tration compactness for fractional Sobolev spaces, Calc. Var. Partial Differ. Equ., 50 (2014),
799-829.

[20] G. Palatucci, A. Pisante; A global compactness type result for Palais-Smale seuqences in
fractional spaces, Nonlinear Anal., 117 (2015), 1-7.

[21] X. Ros-Oton, J. Serra; The Pohozaev identity for the fractional Laplacian, Arch. Ration.
Mech. Anal., 213 (2014), 587-628.



EJDE-2021/23 FRACTIONAL LAPLACE PROBLEMS WITH CRITICAL GROWTH 27

[22] S. Secchi, N. Shioji, M. Squassina; Coron problem for fractional equations, Differ. Integral

Equ., 28 (2015), 103-118.

[23] R. Servadei, E. Valdinoci; Lewy-Stampacchia type estimates for variational inequalities driven
by (non)local operators, Rev. Mat. Iberoam., 29 (2013), 1091-1126.

[24] R. Servadei, E. Valdinoci; Variational methods for the non-local operators of elliptic type,

Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
[25] R. Servadei, E. Valdinoci; A Brezis-Nirenberg result for non-local critical equations in low

dimension, Comm. Pure Appl. Anal., 12 (2013), 2445-2464.

[26] R. Servadei, E. Valdinoci; The Brezis-Nirenberg result for the fractional Laplacian, Tran.
Amer. Math. Soc., 367 (2015), 67-102.

[27] L. Silvestre; Regularity of the obstacle problem for a fractional power of the Laplace operator,

Comm. Pure Appl. Math., 60 (2007), 67-112.
[28] M. Struwe; Variational methods, applications to nonlinear partial differential equations and

Hamilton systems, Fourth Edition, Springer-Verlag, Berlin, 2008.
[29] G. Tarantello; On nonhomogeneous elliptic equations invoving critical Sobolev exponent, Ann.

Inst. H. Poincaré Anal. Non Lineairé, 9 (1992), 281-304.

[30] E. Valdinoci; From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp.
Mat. Apl. Se MA, 49 (2009), 33-44.

[31] L. Vlahos, H. Isliker, Y. Kominis, K. Hizonidis; Normal and a nomalous diffusion: atutorial,

in: T. Bountis(Ed.), Order and Chaos, 10th Volume, Patras University Press, 2008.

Yajing Zhang (corresponding author)
School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

Email address: zhangyj@sxu.edu.cn

Qiaoqin Li

School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

Email address: 1525760982@qq.com

Lu Pang

School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China
Email address: 757005378@qq.com


	1. Introduction
	2. Critical and concave case 1<q<2
	2.1. Preliminaries and Nehari manifold
	2.2. Existence of a ground state solution
	2.3. Proof of Theorem ??

	3. Critical and convex case q>2
	Acknowledgments

	References