Infinitely many bound states for Choquard equations with local nonlinearities

Nonlinear Analysis 189 (2019) 111583

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Infinitely many bound states for Choquard equations with local nonlinearities Xinfu Li a , Xiaonan Liu b , Shiwang Ma b ,∗ a b

School of Science, Tianjin University of Commerce, Tianjin 300134, China School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

article

info

Article history: Received 13 February 2019 Accepted 19 July 2019 Communicated by Vicentiu D Radulescu MSC: 35J20 35J60 Keywords: Multiplicity Choquard equations Bound states

abstract In this paper, we consider the following Choquard equation

{ (CH)

−∆u + u = (Iα ∗ |u|p )|u|p−2 u + V (x)|u|q−2 u u ∈ H 1 (RN ),

where N ≥ 3, α ∈ ((N −4)+ , N ), p ∈ [2, 2

N +α ), N −2

q ∈ (2,

in RN ,

(p−1)(N −2)α 2N , 1+ )∩(1+ N −2 N2

2N (N −α)+N (p−1)

) and Iα is the Riesz potential. Under some suitable decay (N −2)α assumptions but without any symmetry property on V (x), we prove that the problem has infinitely many solutions, whose energy can be arbitrarily large. © 2019 Published by Elsevier Ltd.

1. Introduction and main result In this paper, we consider the multiplicity of solutions for the following Choquard equations { p p−2 q−2 −∆u + u = (Iα ∗ |u| )|u| u + V (x)|u| u in RN , (CH) u ∈ H 1 (RN ), +α 2N where N ≥ 3, α ∈ (0, N ), p ∈ ( NN+α , N N −2 ), q ∈ (2, N −2 ) and Iα is the Riesz potential defined by

Iα (x) =

1 |x|

N −α

, x ∈ RN \ {0}.

The Choquard equation has many applications in Physics. For example, when N = 3, α = 2, p = 2 and V (x) ≡ 0, (CH) was investigated by Pekar in [25] to study the quantum theory of a polaron at rest. It was pointed in [15] that Choquard applied it as an approximation to Hartree–Fock theory of one component plasma. It also arises in multiple particles systems [12] and quantum mechanics [26]. ∗

Corresponding author. E-mail addresses: [email protected] (X. Li), [email protected] (X. Liu), [email protected] (S. Ma).

https://doi.org/10.1016/j.na.2019.111583 0362-546X/© 2019 Published by Elsevier Ltd.

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X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

When V (x) ≡ 0, the Choquard equation has been widely studied in many aspects, see the survey paper [24]. Precisely, the existence, radial symmetry and uniqueness up to translations of the ground state for (CH) were considered in Lieb [15] and the nondegenerate property was considered in Wei and Winter [34]. The regularity and radial symmetry of the ground states were established respectively in [8] and [21] as well. Later, Moroz and Van Schaftingen [22] obtained the regularity, positivity, radial symmetry and decay property at infinity of ground states of (CH) for the optimal range of parameters. Moroz and Van Schaftingen [23] considered the existence of ground states under the Berestycki–Lions type assumptions which could be proved to be almost necessary. The existence of sign-changing solutions was considered in [9] and [10]. For the equation with a potential function a(x), under some symmetry or periodicity assumptions on a(x), Ackermann [1] and Lions [17] considered the multiplicity of solutions to p

p−2

− ∆u + a(x)u = (Iα ∗ |u| )|u|

u in RN .

(1.1)

See also [9] and [37] for similar results. Recently, Liu et al. [19] obtained the existence of infinitely many solutions for (1.1), without symmetry or periodicity assumptions on the potential function a(x). We refer to Van Schaftingen and Xia [32] for the semiclassical problem. When V (x) ̸≡ 0, for the interaction of the nonlocal nonlinear term and the local nonlinear term, the study of (CH) is a bit difficult. When V (x) is a positive constant, the authors in [7,13,14,27,30,31] considered the existence of ground states under various conditions on N, α, p and q and the regularity is considered in [13] and [14]. Liu and Shi [20] studied the orbital stability of the standing waves. When V (x) depends on x, the authors in [14,30,33] studied the existence of a ground state and Vaira [31] considered the existence of a positive bound state when ground states may not exist. Zhong and Tang [38] investigated the existence of ground state sign-changing solutions for equation p

−∆u + (1 + λf (x))u = (Iα ∗ k|u| )k(x)|u|

p−2

2N −2

u + |u| N −2

u, x ∈ RN .

We should point out that recently the fractional problems have attracted many researchers’ interests, see [2–4,11,35] for example. As far as we know, there is no result about the existence of infinitely many solutions for (CH) without symmetry or periodicity assumptions on the potential function V (x). So in the present paper, we consider (CH) with potential function satisfying the following assumptions: (V1) V ∈ C 1 (RN , R) and V (x) ≥ 0 for any x ∈ RN . (V2) lim|x|→+∞ V (x) = V∞ > 0. N −α ∂V x (V3) lim|x|→+∞ |x| ∂x (x) = −∞, where x = |x| , x ̸= 0. (V4) There exists a constant c¯ > 1 such that, for |x| > c¯, |∇τx V (x)| ≤ −¯ c

∂V (x), ∂x

where ∇τx V (x) denotes the component of ∇V (x) which lies in the hyperplane orthogonal to x and containing x. Examples. 1. When α < N −1, we choose β > 0 such that N −α −β −1 > 0 and define V1 (x) ∈ C 1 (RN , R) −β satisfying V1 (x) ≥ 0 for x ∈ RN , V1 (x) = 0 for |x| < 1 and V1 (x) = 1+|x| for |x| > 2. By direct calculation, we have for |x| > 2, −β−2

∇V1 (x) = −β|x| Hence, V1 (x) satisfies (V1)–(V4).

x and

∂V1 x −β−1 (x) = ∇V1 (x) · = −β|x| . ∂x |x|

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

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2. When α < N − 1, we define V2 (x) ∈ C 1 (RN , R) satisfying V2 (x) ≥ 0 for x ∈ RN , V2 (x) = 1 for |x| < 1 and V2 (x) = 1 + (ln |x|)−1 for |x| > 2. By direct calculation, we have for |x| > 2, −2

∇V2 (x) = −(ln |x|)−2 |x|

x

and

∂V2 −1 (x) = −(ln |x|)−2 |x| . ∂x

Hence, V2 (x) satisfies (V1)–(V4). Our main result is as follows: ( −2)α +α 2N Assume that N ≥ 3, α ∈ ((N − 4)+ , N ), p ∈ [2, N ), q ∈ (2, ) ∩ 1 + (p−1)(N , N −2 N −2 N2 ) 2N (N −α)+N 2 (p−1) and (V1)–(V4) hold. Then problem (CH) has infinitely many solutions, whose energy 1+ (N −2)α can be arbitrarily large. Theorem 1.1.

The basic idea to prove Theorem 1.1 follows from that of Cerami et al. [5], which has already been applied to the Kirchhoff equation [36] and recently to the Choquard equation [19]. The outline of the method is as follows. Consider a sequence of balls Bρn (0) in RN with ρn → +∞ as n → +∞, and consider the related problems on these balls { p p−2 q−2 −∆u + u = (Iα ∗ |u| )|u| u + V (x)|u| u, in Bρn (0), (CHn ) u ≡ 0, on RN \ Bρn (0). Applying the classical mini–max arguments to the functionals corresponding to (CHn ), we may obtain the existence of infinitely many solutions to (CH). Precisely, let {un }n∈N be a sequence with un being a solution of (CHn ) for every n ∈ N and corresponding to mini–max classes of the same type, then we try to pass to the limit. But because of the lack of compactness, we do not know whether {un }n∈N converges strongly or not. So arguing indirectly we assume that {un }n∈N is not compact and obtain a decomposition of {un }n∈N . And then we use some uniform decay estimates on {un }n∈N and a local Pohoˇzaev-type equality to get a contradiction. Compared to [19], there is a difficulty in the present paper: For the interaction of the nonlocal p p−2 q−2 p term (Iα ∗|u| )|u| u and the local term V (x)|u| u, it is difficult to prove that {Iα ∗|un | }n∈N and {un }n∈N ∞ N are bounded in L (R ) which is a crucial step in the proof of the main result. To overcome the difficulty, we adopt a regularity result from [13], but with the restriction ) ( 2N (N − α) + N 2 (p − 1) (p − 1)(N − 2)α q ∈ 1+ ,1 + , N2 (N − 2)α see Lemma 3.3. This paper is organized as follows. In Section 2, we give some preliminaries and then the proof of Theorem 1.1 by using an argument of Morse index. In Section 3, by using the decomposition of the (PS) sequence, a local Pohoˇzaev-type equality and some decay estimates, we give the proof of a compactness result introduced in Section 2. 2. Proof of the main result In this section, we give some preliminaries and give the proof of Theorem 1.1. We use the following notations: − For 1 ≤ p ≤ +∞ and Ω ⊂ RN , the norm in Lp (Ω ) is denoted by | · |p,Ω , when Ω is a proper subset of N R , by | · |p when Ω = RN .

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

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− H01 (Ω ), Ω ⊂ RN , and H 1 (RN ) denote the Sobolev spaces obtained as closure of C0∞ (Ω ), C0∞ (RN ) respectively, with respect to the norms (∫ ∥u∥Ω =

)1/2 (∫ 2 2 (|∇u| + |u| )dx , ∥u∥ =

2

2

)1/2

(|∇u| + |u| )dx

.

RN

Ω

− H −1 (Ω ), Ω ⊂ RN , and H −1 (RN ) denote the dual spaces of H01 (Ω ) and H 1 (RN ) respectively. − If u ∈ H01 (Ω ), Ω ⊂ RN , we denote also by u its extension to RN made by setting u ≡ 0 outside Ω . − Br (x) denotes the ball in RN centered at x with radius r. − C, Ci , c, ci denote various positive constants. Next, we recall some basic lemmas. The following well known Hardy–Littlewood–Sobolev inequality can be found in [16]. Lemma 2.1. Let s, t > 1 and 0 < α < N with 1/s + (N − α)/N + 1/t = 2. Let u ∈ Ls (RN ) and v ∈ Lt (RN ). Then there exists a sharp constant C(N, α, s), independent of u and v, such that ⏐ ⏐∫ ∫ ⏐ ⏐ u(x)v(y) ⏐ ⏐ dxdy ⏐ ≤ C(N, α, s)∥u∥s ∥v∥t . ⏐ ⏐ ⏐ RN RN |x − y|N −α If s = t =

2N N +α ,

then C(N, α, s) = Cα (N ) = π

N −α 2

Γ ( α2 ) Γ ( N +α 2 )

{

Γ ( N2 ) Γ (N )

}− α

N

.

t

In particular, let u = v = |w| , then by Lemma 2.1, we know ∫ t t (Iα ∗ |w| )|w| dx RN

is well defined if w ∈ Lrt (RN ) for some r > 1 satisfying 2r + NN−α = 2. Thus, if w ∈ H 1 (RN ), by the Sobolev embedding theorem, ] [ N +α N +α , . t∈ N N −2 Remark 2.2. Iα ∗ v ∈ L

Nt N −αt

By the Hardy–Littlewood–Sobolev inequality above, for any v ∈ Lt (RN ) with t ∈ (1, N α ), (RN ) and ∥Iα ∗ v∥

Nt N −αt

≤ C(N, α, t)∥v∥t .

The following local Brezis–Lieb lemma can be found in [28]. Lemma 2.3. Let r ∈ (1, +∞). Assume {wn }n∈N is a bounded sequence in Lr (RN ) that converges to w almost everywhere. Then, for every t ∈ [1, r] we have ∫ r t t t lim ||wn | − |wn − w| − |w| | t dx = 0 n→+∞

and

∫ ||wn |

lim

n→+∞

RN

t−1

t−1

wn − |wn − w|

(wn − w) − |w|

t−1

r

w| t dx = 0.

RN

The following version of Brezis–Lieb lemma for the Riesz potential can be found in [22].

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

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2N t

Lemma 2.4. Let N ≥ 1, α ∈ (0, N ), t ∈ [1, ∞) and {un }n∈N be a bounded sequence in L N +α (RN ). If un → u a.e. on RN as n → +∞, then ∫ ∫ ∫ t t t t t t (Iα ∗ |un | )|un | dx − (Iα ∗ |un − u| )|un − u| dx → (Iα ∗ |u| )|u| dx RN

RN

RN

as n → +∞. The following Brezis–Lieb lemma can be found in [19]. 2N p

+α N +α (RN ). Lemma 2.5. Let N ≥ 3, α ∈ ((N −4)+ , N ), p ∈ [2, N N −2 ) and {un }n∈N be a bounded sequence in L N If un → u a.e. on R , then as n → +∞, p

(Iα ∗ |un | )|un |

p−2

p

un − (Iα ∗ |un − u| )|un − u|

p−2

p

(un − u) → (Iα ∗ |u| )|u|

p−2

u

in H −1 (RN ). +α 2N In the following, we give the variational setting for (CH). By (V1), (V2), p ∈ ( NN+α , N N −2 ), q ∈ (2, N −2 ), 1 N the Hardy–Littlewood–Sobolev inequality and the Sobolev imbedding theorem, the functional J : H (R ) → R of (CH) defined by ∫ ∫ ∫ 1 1 1 2 2 p p q J(u) = (|∇u| + |u| )dx − (Iα ∗ |u| )|u| dx − V (x)|u| dx 2 RN 2p RN q RN

is of class C 1 and ⟨J ′ (u), φ⟩ =

∫

∫

(∇u · ∇φ + uφ)dx − RN ∫ q−2 − V (x)|u| uφdx

p

(Iα ∗ |u| )|u|

p−2

uφdx

RN

RN

for any φ ∈ H 1 (RN ). Thus, the critical points of J are weak solutions of (CH). To prove Theorem 1.1, let {ρn }n∈N be an increasing sequence such that ρn → +∞ as n → +∞ and consider the following problem { p p−2 q−2 −∆u + u = (Iα ∗ |u| )|u| u + V (x)|u| u, in Bρn (0), (CHn ) u ≡ 0, in RN \ Bρn (0). It is clear that the corresponding functional Jn : H01 (Bρn (0)) → R of (CHn ) defined by ∫ ∫ ∫ p p 1 1 |u(y)| |u(x)| 2 2 Jn (u) = (|∇u| + |u| )dx − dydx 2 Bρn (0) 2p Bρn (0) Bρn (0) |x − y|N −α ∫ 1 q − V (x)|u| dx q Bρn (0) is even and of class C 1 . Moreover, ⟨Jn′ (u), ψ⟩ =

∫ (∇u · ∇ψ + uψ)dx Bρn (0)

∫

∫

− Bρn (0)

V (x)|u| Bρn (0)

p−2

|x − y|

Bρn (0)

∫ −

p

|u(y)| |u(x)| q−2

uψdx

u(x)ψ(x)

N −α

dydx

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

6

for any ψ ∈ H01 (Bρn (0)). We define the Nehari manifold Nn corresponding to (CHn ) by Nn := {u ∈ H01 (Bρn (0)) \ {0} : ⟨Jn′ (u), u⟩ = 0}. Then we have +α 2N Lemma 2.6. Let N ≥ 3, α ∈ (0, N ), p ∈ ( NN+α , N N −2 ), q ∈ (2, N −2 ), (V1) and (V2) hold, then we have (1) There exists a constant τ > 0 independent of n and u such that Jn (u) ≥ τ for any n ∈ N and u ∈ Nn . (2) Nn is a complete C 1,1 manifold. (3) Nn is a natural constraint, which means that any critical point of Jn |Nn ( Jn constrained on Nn ) is actually a free critical point of Jn .

Proof . (1) For any u ∈ Nn , we have En (u) := ⟨Jn′ (u), u⟩ = 0. By (V1), (V2), the Hardy–Littlewood–Sobolev inequality and the Sobolev imbedding theorem, we have ∫ ∫ ∫ ∫ p p |u(y)| |u(x)| q 2 2 dydx + V (x)|u| dx (|∇u| + |u| )dx = N −α Bρn (0) Bρn (0) Bρn (0) Bρn (0) |x − y| ∫ 2 p q ≤ ||u| χBρn (0) | 2N + V1 |u| dx N +α

2p

≤ C(∥|u|χBρn (0) ∥

q

+ ∥|u|χBρn (0) ∥ ) q

2p

= C(∥u∥Bρ

Bρn (0)

n (0)

+ ∥u∥Bρ

n (0)

)

with V1 = supx∈RN V (x) and C being a constant independent of n and u, which implies that there exists η > 0 such that ∥u∥Bρn (0) ≥ η for any n ∈ N and u ∈ Nn . For 2p ≤ q, we have 1 Jn (u) = Jn (u) − ⟨Jn′ (u), u⟩ 2p ( )∫ 1 1 2 2 − = (|∇u| + |u| )dx 2 2p Bρn (0) ( )∫ 1 1 q V (x)|u| dx, + − 2p q Bρn (0) and for 2p > q, 1 Jn (u) = Jn (u) − ⟨Jn′ (u), u⟩ q ( )∫ 1 1 2 2 = − (|∇u| + |u| )dx 2 q Bρn (0) )∫ ( ∫ p p 1 1 |u(y)| |u(x)| − dydx, + N −α q 2p Bρn (0) Bρn (0) |x − y| which imply that for any u ∈ Nn , {( Jn (u) ≥ min

1 1 − 2 2p

)

η2 ,

(

1 1 − 2 q

)

η2

} := τ > 0.

(2) It is obvious that En = ⟨Jn′ (u), u⟩ is of class C 1 . For any u ∈ Nn , ∫ ∫ ∫ p p |u(y)| |u(x)| 2 2 ′ ⟨En (u), u⟩ = 2 (|∇u| + |u| )dx − 2p dydx N −α Bρn (0) Bρn (0) Bρn (0) |x − y| ∫ q −q V (x)|u| dx Bρn (0) ∫ ∫ 2 2 q = (2 − 2p) (|∇u| + |u| )dx + (2p − q) V (x)|u| dx < 0 Bρn (0)

Bρn (0)

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

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for 2p < q, and ⟨En′ (u), u⟩

∫

2

= (2 − q)

2

(|∇u| + |u| )dx Bρn (0)

∫

∫

+ (q − 2p) Bρn (0)

p

|u(y)| |u(x)|

Bρn (0)

N −α

|x − y|

p

dydx < 0

for 2p ≥ q. Thus, Nn is a complete C 1,1 manifold. (3) Let u be a critical point of Jn constrained on Nn , then there exists a Lagrange multiplier λ ∈ R such that Jn′ (u) = λEn′ (u). Hence, 0 = ⟨Jn′ (u), u⟩ = λ⟨En′ (u), u⟩, which combined with ⟨En′ (u), u⟩ < 0 implies that λ = 0. Thus, u is a free critical point of Jn . □ By virtue of Lemma 2.6, we only need to find critical points of Jn |Nn . To this end, for any k ∈ N and n ∈ N, we consider Γkn = {A ⊂ Nn : A is compact, A = −A, γ(A) ≥ k} and define cnk = inf n sup Jn (u), A∈Γk u∈A

where γ(A) denotes the Krasnoselskii genus of A. According to Lemma 2.6 and the minimax principle (see [29]), {cnk }k∈N is a sequence of critical values of Jn |Nn and hence critical values of Jn . Since {ρn }n∈N is increasing with n, we know Γkn ⊂ Γkn+1 , which implies cnk ≥ cn+1 ≥ · · · ≥ τ > 0. k

(2.1)

Then we have (p−1)(N −2)α +α 2N Proposition 2.7. Assume that N ≥ 3, α ∈ ((N −4)+ , N ), p ∈ [2, N , 1+ N −2 ), q ∈ (2, N −2 )∩(1+ N2 2N (N −α)+N 2 (p−1) ) (N −2)α

and (V1)–(V4) hold. Let {ρn }n∈N be a sequence such that ρn → +∞ as n → +∞ and {un }n∈N be a sequence of nontrivial weak solutions of (CHn ) satisfying Jn (un ) ≤ c. Then {un }n∈N is relatively compact in H 1 (RN ). Since the proof of Proposition 2.7 is a bit lengthy, we will present the details in the next section. n From (2.1), ck := limn→+∞ cnk is well defined and ck+1 ≥ ck from the fact Γk+1 ⊂ Γkn . Moreover we have Proposition 2.8. Let N , α, p, q be as in Proposition 2.7 and (V1)–(V4) hold, then ck → +∞ as k → +∞. To prove Proposition 2.8, we need to use a Morse index argument. Definition 2.9. We call the augmented Morse index of a critical point u for a functional I, the number (possibly +∞) of eigenvalues of I ′′ (u) less than or equal to zero. +α 2N Under the conditions N ≥ 3, α ∈ (0, N ), p ∈ [2, N N −2 ), q ∈ (2, N −2 ), (V1)–(V2), the functionals J and Jn are of class C 2 . So the augmented Morse index for J and Jn are well defined, moreover we have +α 2N Lemma 2.10. Let N ≥ 3, α ∈ (0, N ), p ∈ [2, N N −2 ), q ∈ (2, N −2 ), (V1)–(V2) hold, then for any critical point u ∈ H 1 (RN ) of J, the augmented Morse index of u for J is finite.

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

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Proof . Suppose by contradiction that J ′′ (u) possesses infinitely many eigenfunctions vj , j ∈ N such that ∥vj ∥ = 1, (vi , vj ) = 0 for i ̸= j and ⟨J ′′ (u)vj , vj ⟩ ≤ 0, j ∈ N. (2.2) Direct calculation gives that ⟨J ′′ (u)vj , vj ⟩ =

∫

2

∫

2

(|∇vj | + |vj | )dx − (p − 1) R∫N

p

p−2

(Iα ∗ |u| )|u|

2

|vj | dx

RN p−2

−p

p−2

(Iα ∗ |u| uvj )|u| uvj dx ∫ q−2 2 − (q − 1) V (x)|u| |vj | dx.

(2.3)

RN

RN

Noting that vj ⇀ 0 as j → +∞ in H 1 (RN ), we obtain that ∫ p p−2 2 (Iα ∗ |u| )|u| |vj | dx → 0, RN

∫

p−2

(Iα ∗ |u|

uvj )|u|

p−2

uvj dx → 0

RN

and

∫ V (x)|u|

p−2

2

|vj | dx → 0

RN

as j → +∞. Inserting these into (2.3), we obtain that ∫ 2 2 ′′ ⟨J (u)vj , vj ⟩ ≥ (|∇vj | + |vj | )dx + oj (1) = 1 + oj (1), RN

which implies ⟨J ′′ (u)vj , vj ⟩ > 0 for j large enough. This contradicts to (2.2).

□

Now, we can give Proof of Proposition 2.8. Suppose by contradiction that ck → c0 < +∞ as k → +∞. Then there exists a k0 ∈ N such that for any k ≥ k0 we can find nk > 0 satisfying cnk < c0 + 1, for n ≥ nk . Then according to Theorem 2.3 of Chapter 2 in [6], for any k ≥ k0 , there exists a critical point wk ∈ H01 (Bρnk (0)) of Jnk such n that Jnk (wk ) = ck k and iJnk (wk ) ≥ k − 1, (2.4) where iJnk (wk ) denotes the augmented Morse index of wk for Jnk . It follows from Proposition 2.7 that wk → w strongly in H 1 (RN ) and w is a critical point of J. By Lemma 2.10, iJ (w) is well defined and finite. Thus, there exist a finite dimensional subspace M of H 1 (RN ) and a positive constant δ > 0 such that ⟨J ′′ (w)v, v⟩ ≥ δ∥v∥2 for any v ∈ M ⊥ .

(2.5)

On the other hand, when k is sufficiently large, we can always find vk ∈ M ⊥ (see (2.4)) such that ∥vk ∥ = 1 and (2.6) ⟨Jn′′k (wk )vk , vk ⟩ ≤ 0. Since wk → w and J is of class C 2 , we have for k sufficiently large, ∥J ′′ (w) − J ′′ (wk )∥ <

δ . 2

(2.7)

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

9

It follows from (2.5)–(2.7) that δ = δ∥vk ∥2 ≤ ⟨J ′′ (w)vk , vk ⟩ = ⟨(J ′′ (w) − Jn′′k (wk ))vk , vk ⟩ + ⟨Jn′′k (wk )vk , vk ⟩ ≤ ∥J ′′ (w) − Jn′′k (wk )∥∥vk ∥2 δ = ∥J ′′ (w) − J ′′ (wk )∥ < . 2 That is a contradiction and the proof is complete. Now, we are ready to give the proof of the main result. Proof of Theorem 1.1. By Lemma 2.6, for any k ∈ N, cnk are critical values of Jn for n ∈ N. So there exist critical points unk of Jn such that Jn (unk ) = cnk . According to (2.1) and Proposition 2.7, for any k ∈ N, {unk }n∈N is relatively compact and (up to a subsequence) converges strongly in H 1 (RN ) to some uk satisfying J ′ (uk ) = 0 and J(uk ) = lim cnk = ck . n→+∞

By Proposition 2.8, ck → +∞, which implies the conclusion. 3. Compactness results In this section, we prove Proposition 2.7. Firstly, we give a decomposition of a special (PS) sequence of J, whose proof is similar to that of Proposition 2.1 in [36] (see also Struwe [29]). 2N +α Proposition 3.1. Assume that N ≥ 3, α ∈ ((N − 4)+ , N ), p ∈ [2, N N −2 ), q ∈ (2, N −2 ) and (V1)–(V2) hold. Let {ρn }n∈N be a sequence such that ρn → +∞ as n → +∞ and {un }n∈N be a sequence of nontrivial solutions to (CHn ) satisfying Jn (un ) ≤ c. Then there exists u ∈ H 1 (RN ) satisfying J ′ (u) = 0, and either (Going if necessary to a subsequence, without loss of generality, we still denote by {un }n∈N .) (i) un → u in H 1 (RN ) as n → +∞, or (ii) there exist a positive number m ∈ N, sequences of points {ynk }n∈N ⊂ RN , k = 1, 2, . . . , m, nontrivial solutions w1 , w2 , . . . , wm of the following problem p

p−2

−∆w + w = (Iα ∗ |w| )|w| such that

w + V∞ |w|

q−2

w

in RN

m ∑

wk (· − ynk ) strongly in H 1 (RN ) as n → +∞, m k=1 ∑ ∥un ∥2 → ∥u∥2 + ∥wk ∥2 as n → +∞,

un → u +

k=1

|ynk | → +∞, |ynk − ynj | → +∞ as n → +∞ for 1 ≤ k ̸= j ≤ m. Proof . Since Jn′ (un ) = 0 and Jn (un ) ≤ c, we have 1 c + on (1)∥un ∥ ≥ Jn (un ) − ⟨Jn′ (un ), un ⟩ 2p ( )∫ 1 1 2 2 (|∇un | + |un | )dx = − 2 2p Bρn (0) ( )∫ 1 1 q + − V (x)|un | dx 2p q Bρn (0)

(3.1)

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

10

for 2p ≤ q, and 1 c + on (1)∥un ∥ ≥ Jn (un ) − ⟨Jn′ (un ), un ⟩ q ( )∫ 1 1 2 2 (|∇un | + |un | )dx = − 2 q Bρn (0) ( )∫ ∫ p p |un (y)| |un (x)| 1 1 + dydx − N −α q 2p |x − y| Bρn (0) Bρn (0)

(3.2)

for 2p > q, which imply that {un }n∈N is bounded in H 1 (RN ) since q > 2, p > 1 and V (x) ≥ 0. Then there exist u ∈ H 1 (RN ) such that un ⇀ u weakly in H 1 (RN ), un → u a.e. on RN and un → u strongly in ∞ N Ltloc (RN ) for t ∈ [1, N2N −2 ). For any φ ∈ C0 (R ), there exists n0 ∈ N such that suppφ ⊂ Bρn (0) for any n ≥ n0 . Then by Lemma 2.4 in [14], we can show that ∫ ∫ p p−2 0 = ⟨Jn′ (un ), φ⟩ = (∇un · ∇φ + un φ)dx − (Iα ∗ |un | )|un | un φdx RN ∫ RN q−2 − V (x)|un | un φdx N R ∫ ∫ p p−2 → (∇u · ∇φ + uφ)dx − (Iα ∗ |u| )|u| uφdx RN ∫ RN q−2 − V (x)|u| uφdx RN

as n → +∞, which implies that J ′ (u) = 0. In the following, we will prove that either (i) or (ii) holds. Step 1. Set u1n := un − u. Then u1n ⇀ 0 weakly in H 1 (RN ). By (V1) and (V2), Lemmas 2.3 and 2.5, we obtain, for any φ ∈ C0∞ (RN ), ⟨Jn′ (un ), φ⟩ = ⟨J ′ (un ), φ⟩ ∫ ∫ p p−2 = (∇un · ∇φ + un φ)dx − (Iα ∗ |un | )|un | un φdx RN ∫ RN q−2 − V (x)|un | un φdx N R ∫ = (∇u1n · ∇φ + ∇u · ∇φ + u1n φ + uφ)dx N R∫ ∫ p p−2 p p−2 − (Iα ∗ |u1n | )|u1n | u1n φdx − (Iα ∗ |u| )|u| uφdx N N R R ∫ ∫ q−2 q−2 − V (x)|u1n | u1n φdx − V (x)|u| uφdx + on (1)∥φ∥. RN

(3.3)

RN

By (V1) and (V2), {un }n∈N is bounded in H 1 (RN ) and u1n → 0 in Ltloc (RN ) for t ∈ [1, N2N −2 ), we have ∫ q−2 (V (x) − V∞ )|u1n | u1n φdx = on (1)∥φ∥. RN

Consequently, ∫ RN

q−2

V (x)|u1n | u1n φdx ∫ ∫ q−2 q−2 = V∞ |u1n | u1n φdx + (V (x) − V∞ )|u1n | u1n φdx RN ∫RN 1 q−2 1 = V∞ |un | un φdx + on (1)∥φ∥. RN

(3.4)

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

11

The combination of (3.3) and (3.4) implies that ′ ⟨Jn′ (un ), φ⟩ = ⟨J∞ (u1n ), φ⟩ + ⟨J ′ (u), φ⟩ + on (1)∥φ∥.

Hence, we have obtained that { ′ J∞ (u1n ) = on (1), ∥u1n ∥2 = ∥un ∥2 − ∥u∥2 + on (1). Set

∫

1

δ := lim sup sup n→+∞ y∈RN r If δ = 0, then RN |u1n | dx → 0 as n ′ this and ⟨J∞ (u1n ), u1n ⟩ = on (1), we get 1

∫

q

|u1n | dx.

B1 (y)

→ +∞ for r ∈ (2, N2N −2 ) follows from Lion’s lemma [18]. Combining 1 that ∥un ∥ → 0 as n → +∞, which implies that (i) holds and the

proof is complete. Otherwise, if δ 1 > 0, then there exists a sequence {yn1 }n∈N ⊂ RN such that ∫ δ1 q |u1n | dx > . 1) 2 B1 (yn ′ (u1n ) = on (1) Set wn1 := u1n (· + yn1 ). Then we can assume that wn1 ⇀ w1 ̸= 0 in H 1 (RN ). It follows from J∞ that ′ on (1) = ⟨J∞ (u1n (x)), φ(x − yn1 )⟩ ′ = ⟨J∞ (u1n (x + yn1 )), φ(x)⟩ ′ ′ = ⟨J∞ (wn1 (x)), φ(x)⟩ → ⟨J∞ (w1 (x)), φ(x)⟩ ′ (w1 ) = 0. It follows from u1n ⇀ 0 in H 1 (RN ) that {yn1 }n∈N is as n → +∞ for any φ ∈ C0∞ (RN ). Hence, J∞ N unbounded in R . Step 2. Set u2n = u1n − w1 (· − yn1 ) = un − u − w1 (· − yn1 ). Similarly, we get that

∥u2n ∥2 = ∥u1n (· + yn1 ) − w1 ∥2 = ∥u1n (· + yn1 )∥2 − ∥w1 ∥2 + on (1) = ∥u1n ∥2 − ∥w1 ∥2 + on (1) = ∥un ∥2 − ∥u∥2 − ∥w1 ∥2 + on (1) and ′ ′ ′ J∞ (u2n ) = J∞ (u1n ) − J∞ (w1 ) + on (1) → 0.

Similarly to Step 1, set

∫

2

δ := lim sup sup n→+∞ y∈RN 2

If δ = 0, then

∥u2n ∥

q

|u2n | dx.

B1 (y)

→ 0. Hence, { ∥un ∥2 → ∥u∥2 + ∥w1 ∥2 , un → u + w1 (· − yn1 ) strongly in H 1 (RN ).

Otherwise, if δ 2 > 0, then there exist {yn2 }n∈N ⊂ RN and w2 ∈ H 1 (RN ) such that wn2 := u2n (·+yn2 ) ⇀ w2 ̸= ′ 0 in H 1 (RN ) and J∞ (w2 ) = 0. It follows from u2n ⇀ 0 in H 1 (RN ) that |yn2 | → +∞ and |yn1 − yn2 | → +∞ as n → +∞. We repeat the argument by iteration. Similarly to the proof of (1) in Lemma 2.6, there exists a constant c0 > 0 such that for any nontrivial critical point w of J∞ , ∥w∥ > c0 . So the iteration must stop at some finite index m and the proposition is proved. □

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

12

In order to prove Proposition 2.7, we argue indirectly, then case (ii) in Proposition 3.1 occurs. In that case, we give some definitions which are useful for the estimates in the following. Definition 3.2. Let A be a subset of RN and v ∈ RN be a point such that v ̸∈ A. Then we define the set {w ∈ RN : w = v + λ(x − v), x ∈ A, λ ∈ R+ } by cone of vertex v generated by A. Under the assumptions of Proposition 3.1, when {un }n∈N is not relatively compact, we have {yni }n∈N with i = 1, . . . , m, m ≥ 1. Moreover for any n ∈ N, there exists 1 ≤ in ≤ m such that |ynin | = min1≤i≤m |yni |. Then we call a sequence {yn }n∈N a smallest sequence of {un }n∈N if it satisfies yn = ynin for all n ∈ N. We investigate a noncompact sequence {un }n∈N satisfying the assumptions of Proposition 3.1 and let {yn }n∈N be a smallest sequence of {un }n∈N . Firstly, for any n ∈ N, we construct a cone Cn of vertex y2n generated by a ball BRn (yn ) satisfying ∂Cn ∩ B rn (yni ) = ∅ for all 1 ≤ i ≤ m, where 2 { } 1 1 γ|yn | , 0 < γ < min , rn = 2m 5 4(¯ c + 1) and c¯ is the constant in (V4). To construct such a sequence of Cn , we consider C1,n which is the cone of vertex Obviously, ∂C1,n ∩ B rn (ynin ) = ∅. If

yn 2

generated by Brn (yn ).

2

∂C1,n ∩ B rn (yni ) = ∅ for any 1 ≤ i ≤ m, 2

then let C1,n be Cn and we have done. Otherwise, there exists some 1 ≤ j1 ≤ m such that ∂C1,n ∩ B rn (ynj1 ) ̸= 2 ∅. Then we define C2,n be the cone of vertex y2n generated by B2rn (yn ). Using the fact that |yn | ≤ |ynj1 | we obtain ∂C2,n ∩ B rn (ynj1 ) = ∅. If additionally there hold 2

∂C2,n ∩ B rn (yni ) = ∅ for any 1 ≤ i ≤ m, 2

then let C2,n be Cn . Otherwise, we repeat the procedure and after at most m steps, we can obtain Cn , the cone of vertex y2n generated by BRn (yn ). It is obvious that γ|yn | γ|yn | = rn ≤ Rn ≤ mrn = . 2m 2 Consequently, if we denote the width angle of Cn by θn , we have 2Rn [ γ ] sin θn = ∈ ,γ . |yn | m For any s ∈ R and all n ∈ N we define the cone Cs,n by Cs,n = Cn − s

yn , |yn |

and for any s ∈ R+ , n ∈ N, a region containing ∂Cn , S2s,n = Cs,n \C−s,n . Moreover, we define Sn = RN \

m ∑

B rn (yni ), 4

i=0

where yni is defined in Proposition 3.1 for 1 ≤ i ≤ m and yn0 = 0 for any n ∈ N. Next, we give some estimates for {un }n∈N .

(3.5)

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

13

Lemma 3.3. Let N , α, p, q be as in Proposition 2.7, (V1) and (V2) hold, {ρn }n∈N be a sequence such that ρn → +∞ as n → +∞ and {un }n∈N be a sequence of nontrivial weak solutions of (CHn ) satisfying p Jn (un ) ≤ c. Then {Iα ∗ |un | }n∈N and {un }n∈N are bounded in L∞ (RN ). To prove Lemma 3.3, we use the following result, whose proof is contained in [13]. 2N

N

2N

Lemma 3.4. Let N ≥ 3 and α ∈ (0, N ). If H, K ∈ L 2+α (RN ) + L α (RN ), M ∈ L∞ (RN ) + L 2 (RN ) and u ∈ H 1 (RN ) solves − ∆u + u = (Iα ∗ (Hu))K + M u in RN , (3.6) then u ∈ Lr (RN ) for r ∈ [2, N α that

2N N −2 ).

Moreover, there exists a positive constant C(r) independent of u such |u|r ≤ C(r)|u|2 .

Proof of Lemma 3.3. By (V1) and (V2), there exists V1 > 0 such that 0 ≤ V ≤ V1 in RN . Hence, |un | weakly satisfies p p−1 q−1 − ∆|un | + |un | ≤ (Iα ∗ |un | )|un | + V1 |un | , x ∈ RN . (3.7) Let vn be a weak solution of p

p−1

− ∆v + v = (Iα ∗ |un | )|un |

q−1

+ V1 |un |

,

(3.8)

then according to the maximum principle, we have vn > 0 and |un | ≤ vn in RN . In order to use Lemma 3.4, we rewrite (3.8) in the following form −∆vn + vn = (Iα ∗ (Hn vn ))Kn + Mn vn , in RN , p

q−1

+α and Mn = V1 |unv|n . Since |un | ≤ vn , vn > 0, p ∈ ( NN+α , N where Hn = |uvnn| , Kn = |un | N −2 ) and 2N q ∈ (2, N −2 ), there exists a constant C > 0 independent of n such that ( ( ) 2+α ) α 4 |Kn |, |Hn | ≤ C |un | N + |un | N −2 , |Mn | ≤ C 1 + |un | N −2 . p−1

It follows from the proof of Proposition 3.1 that {un }n∈N is bounded in H 1 (RN ). Thus, {Hn }n∈N and 2N N 2N {Kn }n∈N are bounded in L 2+α (RN )+L α (RN ), {Mn }n∈N is bounded in L∞ (RN )+L 2 (RN ). By Lemma 3.4, 2N r N we have {vn }n∈N is bounded in Lr (RN ) for r ∈ [2, N α N −2 ) and thus {un }n∈N is bounded in L (R ) for p p−1 2N 2 1 N 2N t N r ∈ [2, N }n∈N is α N −2 ). Consequently, {|un | }n∈N is bounded in L (R ) for t ∈ [ p , p α N −2 ), {|un | q−1 2 1 N 2N t1 N bounded in L (R ) for t1 ∈ [ p−1 , p−1 α N −2 ) and {|un | }n∈N is bounded in Lt2 (RN ) for ( ] 1 1 1 ∈ , 2 := B. 1 N 2N t2 q−1 α N −2 q−1 p

1 N 2N ∞ N Since p2 < N α < p α N −2 , we have {Iα ∗ |un | }n∈N is bounded in L (R ). Furthermore, it follows from p +α N t3 N p ∈ [2, N N −2 ) and Remark 2.2 that {Iα ∗ |un | }n∈N is bounded in L (R ) for t3 ∈ ( N −α , +∞]. By the p p−1 t4 N H¨ older inequality, {(Iα ∗ |un | )|un | }n∈N is bounded in L (R ) for ( ) 1 1 1 1 1 ∈ + 1 N 2N , N + 2 := A. t4 +∞ p−1 p−1 α N −2 N −α 2

−2)α (p−1) < q < 1 + 2N (N −α)+N Since 1 + (p−1)(N , we have A ∩ B ̸= ∅ and r0 ∈ A ∩ B for some r0 < 1. (N −2)α N2 By the standard iterating argument, we conclude that {vn }n∈N is bounded in L∞ (RN ) and then {un }n∈N is bounded in L∞ (RN ). The proof is complete.

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

14

Consequently, we have Corollary 3.5. Let N , α, p, q be as in Proposition 2.7, (V1)–(V2) hold and {un }n∈N be as that in Lemma 3.3, then we have −∆|un | ≤ c weakly holds for some positive constant c. Lemma 3.6. Let N , α, p, q be as in Proposition 2.7, (V1)–(V2) hold, {un }n∈N be as that in Lemma 3.3, {xn }n∈N be a sequence such that xn ∈ Sn . If {un }n∈N is not relatively compact, then for any h ∈ (0, 1) there holds lim sup |un (x)| = 0, n→+∞ B

hσn (xn ) (xn )

where σn (x) = min0≤i≤m |x − yni | and {yni }n∈N is defined in (3.5). Proof . Suppose by contradiction that there exist h ∈ (0, 1), δ > 0 and a sequence of points {zn }n∈N , such that zn ∈ Bhσn (xn ) (xn ) and |un (zn )| ≥ δ for n large enough. Thus by Corollary 3.5, we know ∫ 1 δ |un |dx > |Bρ (zn )| Bρ (zn ) 2 for small ρ. Here |Bρ (zn )| is the Lebesgue measure of Bρ (zn ) in RN . Therefore we have un (· + zn ) ⇀ v ̸= 0 in H 1 (RN ). On the other hand, according to the choices of σn , h and the fact un − u −

m ∑

wk (· − ynk ) → 0 in H 1 (RN ),

k=1

we have un (· + zn ) ⇀ 0 in H 1 (RN ). This is a contradiction and the proof is complete. □ Proposition 3.7. Let N , α, p, q be as in Proposition 2.7, (V1)–(V2) hold and {un }n∈N be as that in Lemma 3.6. Then for any β ∈ (0, 1) there exist a positive constant Cβ and nβ ∈ N such that |un (x)| ≤ Cβ e−βσn (x) for n > nβ and x ∈ Sn . Proof . Suppose by contradiction that there exists β ∈ (0, 1) such that for any k ∈ N, we can find nk ≥ k and xk ∈ Snk satisfying |unk (xk )| > ke−βσnk (xk ) . Choose h ∈ (β, 1) and β ′ ∈ (β, h). We claim that p

∆|unk (x)| ≥ |unk (x)| − (Iα ∗ |unk | )|unk (x)|

p−1

− V1 |unk (x)|

> (β ′ )2 h−2 |unk (x)| ≥ 0

q−1

(3.9)

+α weakly holds for x ∈ Bhσnk (xk ) (xk ). Indeed, when 2 < p < N N −2 , according to Lemmas 3.3 and 3.6 and q ∈ (2, N2N −2 ), we know (3.9) is true. When p = 2, it is shown in Proposition 3.7 in [19] that

lim

k→+∞ B

and then (3.9) holds.

sup hσn (xk ) (xk ) k

p

(Iα ∗ |unk | )(x) = 0

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

15

Noting that hσnk (xk ) > 1 for k large, therefore by (3.9) and the mean value inequality, we have 1 |unk (xk )| ≤ |B1 (xk )|

∫ B1 (xk )

|unk |dx.

Thus if there exists a constant C > 0 such that ∫ |unk |dx ≤ Ce−βσnk (xk ) ,

(3.10)

B1 (xk )

we will get a contradiction. Indeed, following the proof of Proposition 3.7 in [19] line to line, we can prove (3.10) holds and the proof is complete. □ Proposition 3.8. Let N , α, p, q be as in Proposition 2.7, (V1)–(V2) hold, {un }n∈N be as that in Lemma 3.6 and {yn }n∈N be a smallest sequence of {un }n∈N , then there exist positive constants β1 and C1 such that ∫ t |un | dx ≤ C1 e−β1 |yn | , t ∈ [2, +∞). Sn

Proof . It follows from Proposition 3.7 that for β ∈ (0, 1) and sufficiently large n, ∫

∫

t

|un | dx ≤ Cβ Sn

e

−βtσn (x)

dx ≤ Cβ

Sn

m ∫ ∑ Sn

i=0

∫ ≤ Cβ (m + 1)

i

e−βt|x−yn | dx

+∞ γ 8m |yn |

e−βtr rN −1 dr

≤ C1 e−β1 |yn | . The proof is complete.

□

Proposition 3.9. Let N , α, p, q be as in Proposition 2.7, (V1)–(V2) hold, {un }n∈N be as that in Lemma 3.6 and {yn }n∈N be a smallest sequence of {un }n∈N , then there exist positive constants β2 , C2 and a sequence {sn }n∈N ⊂ (− 12 , 12 ) such that for all n, ∫

2

|∇un | dσ ≤ C2 e−β2 |yn | .

∂Csn ,n

Proof . Firstly, for any n ∈ N, we define φn ∈ C ∞ (RN , [0, 1]) satisfying ⎧ ⎪ ⎨φn = 1 on S1,n ; supp(φn ) ⊂ S2,n ; ⎪ ⎩ ∆φn ≤ C, C ∈ R+ .

(3.11)

By Lemma 3.3, there exists a constant C1 > 0 such that −∆|un | + |un | ≤ C1 |un | where r = min{p, q} ≥ 2, which implies that ∫ ∫ −(∆|un |)|un |φn dx ≤ S2,n

S2,n

r−1

r

,

2

(C1 |un | − |un | )φn dx.

(3.12)

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

16

On the other hand, ∫

∫

∫ 2 |∇un | φn dx + (∇|un | · ∇φn )|un |dx S2,n S2,n ∫ ∫ 1 2 2 (∇(|un | ) · ∇φn )dx ≥ |∇un | dx + 2 S2,n S1,n ∫ ∫ 1 2 2 = (∆φn )|un | dx. |∇un | dx − 2 S2,n S1,n

−(∆|un |)|un |φn dx = S2,n

(3.13)

It follows from S2,n ⊂ Sn , (3.11)–(3.13) and Proposition 3.8 that for some positive constants C2 and β2 , ∫ ∫ ∫ 1 2 2 r 2 |∇un | dx ≤ (∆φn )|un | dx + (C1 |un | − |un | )φn dx 2 S1,n S2,n S2,n ∫ ∫ 1 2 r |un | dx + C1 ≤ C |un | dx 2 Sn Sn ≤ C2 e−β2 |yn | , which leads us to ∫

1 2

−1 2

∫

2

∫

|∇un | dσds =

sin θn

2

|∇un | dx ≤ C2 e−β2 |yn | .

S1,n

∂Cs,n

Here θn denotes the width angle of the cone Cn . Thus by using the integral mean value theorem and the fact γ sin θn ≥ m > 0, we get the conclusion. □ In the following, we denote the cone Csn ,n obtained in Proposition 3.9 by C˜n for n ∈ N. Define Dn = C˜n ∩ Bρn (0). Obviously, the boundary of Dn can be divided into two parts, ∂Dn = (∂Dn )i ∪ (∂Dn )e , where (∂Dn )i = ∂ C˜n ∩ Bρn (0) and (∂Dn )e = C˜n ∩ ∂Bρn (0). Moreover, we define yn yn = |yn | with {yn }n∈N being a smallest sequence of {un }n∈N . +α 2N Lemma 3.10. Let N ≥ 3, α ∈ ((N − 4)+ , N ), p ∈ [2, N N −2 ), q ∈ (2, N −2 ), V (x) satisfy (V1)–(V2) and {un }n∈N be nontrivial weak solutions of (CHn ) satisfying Jn (un ) ≤ c. If {un }n∈N is not relatively compact, then the following identity holds, ∫ 1 q − (∇V (x) · yn )|un | dx q Dn ∫ ∫ 1 1 2 2 |∇un | (yn · νn )dσ + |un | (yn · νn )dσ = 2 ∂Dn 2 ∂Dn ∫ ∫ 1 q − (∇un · νn )(∇un · yn )dσ − V (x)|un | (yn · νn )dσ q ∂Dn ∂Dn ∫ ∫ p |un (y)| p−2 − |u (x)| un (x)(∇un (x) · yn )dydx N −α n Dn Bρn (0)\Dn |x − y| ∫ ∫ p 1 |un (y)| p − dy|un (x)| (yn · νn )dσ, p ∂Dn Dn |x − y|N −α

where νn denotes the unit outward normal to ∂Dn .

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

Proof . Since un weakly solves (CHn ), we obtain ∫ (−∆un + un )(∇un · yn )dx Dn ∫ ( ) p p−2 q−2 = (Iα ∗ |un | )|un | un + V (x)|un | un (∇un · yn )dx.

17

(3.14)

Dn

According to Green’s identity, we have ∫ −∆un (∇un · yn )dx Dn ∫ ∫ = ∇un · ∇(∇un · yn )dx − Dn

(3.15) (∇un · νn )(∇un · yn )dσ.

∂Dn

Using the divergence theorem and the fact that yn does not depend on x, we have ∫ ∫ 1 2 ∇(|∇un | ) · yn dx ∇un · ∇(∇un · yn )dx = 2 Dn ∫Dn 1 2 = div(|∇un | yn )dx 2 Dn ∫ 1 2 = |∇un | (yn · νn )dσ. 2 ∂Dn Inserting (3.16) into (3.15), we get ∫ −∆un (∇un · yn )dx Dn ∫ ∫ 1 2 = |∇un | (yn · νn )dσ − (∇un · νn )(∇un · yn )dσ. 2 ∂Dn ∂Dn From the divergence theorem again, it follows that ∫ ∫ 1 2 un (∇un · yn )dx = ∇(|un | ) · yn dx 2 Dn Dn ∫ 1 2 |un | (yn · νn )dσ = 2 ∂Dn

(3.16)

(3.17)

(3.18)

and ∫

q−2

V (x)|un |

un (∇un · yn )dx

Dn

=

1 q

=−

∫

q

V (x)div(|un | yn )dx

(3.19)

Dn

1 q

∫

q

|un | (∇V (x) · yn )dx + Dn

1 q

∫

q

V (x)|un | (yn · νn )dσ. ∂Dn

Besides, ∫ Dn

p

p−2

(Iα ∗ |un | )|un | ∫ ∫ = Dn

un (∇un · yn )dx |un (y)|

N −α

|x − y| p |un (y)|

Bρn (0)\Dn

∫

∫

+ Dn

:= I1 + I2 ,

Dn

p

N −α

|x − y|

|un (x)|

|un (x)|

p−2

p−2

un (x)(∇un (x) · yn )dydx

un (x)(∇un (x) · yn )dydx

(3.20)

18

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

and from the divergence theorem, ∫ ∫ pI2 =

p

|un (y)|

p

div(|un (x)| yn )dydx Dn Dn |x − y| (∫ ) ∫ p |un (y)| p = div dy|un (x)| yn dx N −α Dn Dn |x − y| ) ) ∫ ( (∫ p |un (y)| p − ∇ dy · yn |un (x)| dx N −α Dn Dn |x − y| ∫ ∫ p |un (y)| p dy|un (x)| (yn · νn )dσ = N −α ∂Dn Dn |x − y| ∫ ∫ p p |un (y)| |un (x)| + (N − α) (x − y) · yn dydx. N −α+2 Dn Dn |x − y| N −α

Since yn does not depend on x and by the Fubini theorem, we obtain that ∫ ∫ p p |un (y)| |un (x)| (x − y) · yn dydx = 0. N −α+2 Dn Dn |x − y| Hence, I2 =

1 p

∫ ∂Dn

p

|un (y)|

∫ Dn

N −α

|x − y|

p

dy|un (x)| (yn · νn )dσ.

Inserting (3.21) into (3.20), we obtain ∫ p p−2 (Iα ∗ |un | )|un | un (∇un · yn )dx Dn ∫ ∫ p |un (y)| p−2 |u (x)| un (x)(∇un (x) · yn )dydx = N −α n |x − y| Dn Bρn (0)\Dn ∫ ∫ p 1 |un (y)| p dy|un (x)| (yn · νn )dσ. + N −α p ∂Dn Dn |x − y|

(3.21)

(3.22)

Inserting (3.17), (3.18), (3.19) and (3.22) into (3.14), we complete the proof. □ +α 2N Lemma 3.11. Let N ≥ 3, α ∈ ((N − 4)+ , N ), p ∈ [2, N N −2 ), q ∈ (2, N −2 ), V (x) satisfy (V1)–(V2), and {un }n∈N be as in Lemma 3.10, then the following inequality holds, ∫ 1 q − |un | (∇V (x) · yn ) dx q Dn ∫ ∫ 1 q ≤− (∇un · νn )(∇un · yn )dσ − V (x)|un | (yn · νn )dσ q (∂D ) (∂Dn )i ∫ ∫n i p |un (y)| p−2 − |u (x)| un (x)(∇un (x) · yn )dydx N −α n Dn Bρn (0)\Dn |x − y| ∫ ∫ p 1 |un (y)| p − dy|un (x)| (yn · νn )dσ, p (∂Dn )i Dn |x − y|N −α

where νn denotes the unit outward normal to ∂Dn . Proof . For all n, un weakly solves (CHn ), un = 0 on ∂Bρn (0) ⊃ (∂Dn )e , so ∇un and νn have the same direction, moreover on (∂Dn )e , yn · νn ≥ 0. Thus we obtain that ∫ 2 |un | (yn · νn )ds = 0, (∂Dn )e

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

∫

19

q

V (x)|un | (yn · νn )ds = 0, ∫ (∂Dn )e p |un (y)| p dy|un (x)| (yn · νn )dσ = 0, N −α (∂Dn )e Dn |x − y|

∫ and 1 2

∫

2

(∂Dn )e

∫

|∇un | (yn · νn )dσ − (∇un · νn )(∇un · yn )dσ (∂Dn )e ∫ 1 2 |∇un | (yn · νn )dσ ≤ 0, =− 2 (∂Dn )e

which combined with Lemma 3.10 give that ∫ 1 q (∇V (x) · yn )|un | dx − q Dn ∫ ∫ 1 1 2 2 |∇un | (yn · νn )dσ + |un | (yn · νn )dσ ≤ 2 (∂Dn )i 2 (∂Dn )i ∫ ∫ 1 q − (∇un · νn )(∇un · yn )dσ − V (x)|un | (yn · νn )dσ q (∂Dn )i (∂Dn )i ∫ ∫ p |un (y)| p−2 − |u (x)| un (x)(∇un (x) · yn )dydx N −α n Dn Bρn (0)\Dn |x − y| ∫ ∫ p 1 |un (y)| p − dy|un (x)| (yn · νn )dσ. p (∂Dn )i Dn |x − y|N −α From yn · νn ≤ 0 on (∂Dn )i , it follows that ∫ 2 |∇un | (yn · νn )dσ ≤ 0 (∂Dn )i

∫

2

|un | (yn · νn )dσ ≤ 0.

and

(3.23)

(3.24)

(∂Dn )i

The combination of (3.23) and (3.24) completes the proof. □ +α 2N Lemma 3.12. Let N ≥ 3, α ∈ ((N − 4)+ , N ), p ∈ [2, N N −2 ), q ∈ (2, N −2 ), V (x) satisfy (V1)–(V4), and {un }n∈N be as in Lemma 3.10, then for n large enough, we have ∫ ∫ 1 1 ∂V q q − (∇V (x) · yn )|un | dx ≥ − (x)|un | dx. q Dn 2q Dn ∂x

Proof . Let (yn )τx denote the component of yn which lies in the hyperplane orthogonal to x and containing x. By (V3) and (V4), for large n, we have −(∇V (x) · yn ) = −(∇V (x) · x)(yn · x) − (∇τx V (x) · (yn )τx ) ∂V ∂V ≥− (x)(yn · x) + c¯ (x)|(yn )τx | ∂x ∂x ∂V =− (x)(yn · x − c¯|(yn )τx |). ∂x We want to show that yn · x − c¯|(yn )τx | ≥ 12 in Dn . Firstly, we consider the case x ∈ B2Rn (yn ). By the definition of Rn , we have |x − yn | ≤ 2Rn ≤ γ|yn | for x ∈ B2Rn (yn ). Thus, yn · x = On the other hand,

yn yn + x − yn |yn | − |x − yn | |yn | − |x − yn | 1−γ · ≥ ≥ ≥ . |yn | |x| |x| |yn | + |x − yn | 1+γ ⏐ ⏐ ⏐ x ⏐⏐ |x − yn | |(yn )τx | = min |yn − λx| ≤ ⏐⏐yn − = ≤ γ. λ∈R |yn | ⏐ |yn |

20

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

Moreover, by homogeneity the above inequalities also hold for x in K (the cone of vertex 0 generated by B2Rn (yn )). By Dn ⊂ C˜n ⊂ K, we get that the two inequalities are also true for x ∈ Dn . Then from the choice of γ, it follows that 3(1 − γ) 1 1−γ − c¯γ ≥ ≥ . yn · x − c¯|(yn )τx | ≥ 1+γ 4(1 + γ) 2 The proof is complete. □ Now we are ready to prove the compactness result. Proof of Proposition 2.7. Suppose by contradiction that {un }n∈N is not relatively compact and {yn }n∈N is a smallest sequence of {un }n∈N . It follows from Lemmas 3.11 and 3.12 that for large n ∈ N, ∫ ∂V 1 q (x)|un | dx − 2q Dn ∂x ∫ ∫ 1 q V (x)|un | (yn · νn )dσ ≤− (∇un · νn )(∇un · yn )dσ − q (∂Dn )i (∂Dn )i ∫ ∫ (3.25) p |un (y)| p−2 |u (x)| u (x)(∇u (x) · y )dydx − n n n n N −α Dn Bρn (0)\Dn |x − y| ∫ ∫ p |un (y)| 1 p − dy|un (x)| (yn · νn )dσ. p (∂Dn )i Dn |x − y|N −α By the definition of C˜n , (∂Dn )i and Proposition 3.9, we know ⏐∫ ⏐ ∫ ⏐ ⏐ ⏐ ⏐ 2 (∇un · νn )(∇un · yn )dσ ⏐ ≤ |∇un | dσ ⏐ ⏐ (∂Dn )i ⏐ (∂Dn )i ∫ 2 |∇un | dσ ≤

(3.26)

C˜n

≤ C2 e−β2 |yn | . When n is sufficiently large, we have ∂ C˜n ⊂ Sn . And by Proposition 3.7, we obtain ⏐∫ ⏐ ∫ ⏐ ⏐ ⏐ ⏐ q q V (x)|un | (yn · νn )dσ ⏐ ≤ V1 |un | dσ ⏐ ⏐ (∂Dn )i ⏐ ∂ C˜n ∫ ≤ Cβ e−qβσn (x) dσ ∂ C˜n

≤

m ∫ ∑ i=0

i

e−qβ|x−yn | dσ.

∂ C˜n

For 1 ≤ k ∈ N and 0 ≤ i ≤ m, we define rn rn < |x − yni | ≤ 2k } 2 2 and denote by |Ak,i | the N − 1 dimensional measure of Ak,i . Then we have ( r )N −1 n |Ak,i | ≤ C 2k , 0 ≤ i ≤ m. 2 Consequently, ∫ ∞ ∫ ∑ i k−1 rn 2 dσ e−qβ|x−yn | dσ ≤ e−qβ2 Ak,i = {x ∈ ∂ C˜n : 2k−1

∂ C˜n

k=1 Ak,i ∞ ∑ −qβ2k−1 r2n

≤C ≤

e

k=1 C3 e−β3 |yn | .

(

2k

rn )N −1 2

(3.27)

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

21

Inserting this into (3.27), we obtain that ⏐ ⏐∫ ⏐ ⏐ ⏐ ⏐ q V (x)|un | (yn · νn )dσ ⏐ ≤ C3 e−β3 |yn | ⏐ ⏐ ⏐ (∂Dn )i

(3.28)

for some positive constants C3 and β3 . p Since {Iα ∗ |un | }n∈N is bounded in L∞ (RN ), similarly to the proof of (3.28), we obtain ∫ ∫ ∫ p |un (y)| p p dy|u (x)| (y · ν )dσ ≤ C |un | dσ ≤ C4 e−β4 |yn | . n n n N −α ∂ C˜n (∂Dn )i Dn |x − y| Let

{ G1 =

y : dist(y, ∂ C˜n ) <

1 rn − 4 2

}

{ and G2 =

y : dist(y, ∂ C˜n ) ≥

rn 1 − 4 2

(3.29)

} .

p Since {Iα ∗ |un | }n∈N is bounded in L∞ (RN ) and by the definition of Dn , C˜n and Sn , we know G1 ⊂ Sn , so it follows from the H¨ older inequality and Proposition 3.8 that ∫ ∫ p |un (y)| p−2 |u (x)| un (x)(∇un (x) · yn )dydx N −α n Dn ∩G1 Bρn (0)\Dn |x − y| ∫ p−1 ≤C |un (x)| |∇un (x)|dx Dn ∩G1 (3.30) (∫ )1/2 (∫ )1/2 2(p−1) 2 ≤C |un (x)| dx |∇un (x)| dx Dn ∩G1

Dn ∩G1

≤ C5 e−β5 |yn | . Besides, if x ∈ Dn ∩ G2 , y ∈ Bρn (0) \ Dn , then |x − y| ≥ r4n − 12 > r5n . Thus, ∫ ∫ p |un (y)| p−2 |u (x)| un (x)(∇un (x) · yn )dydx N −α n Dn ∩G2 Bρn (0)\Dn |x − y| ( )N −α ∫ ∫ 5 p p−1 ≤ |un (y)| |un (x)| |∇un (x)|dydx rn Dn ∩G2 Bρn (0)\Dn C ≤ N −α |yn |

(3.31)

+α ∞ N 1 N since p ∈ [2, N N −2 ) and {un }n∈N is bounded in L (R ) and H (R ). It follows from (3.30) and (3.31) that ∫ ∫ p |un (y)| p−2 |u (x)| un (x)(∇un (x) · yn )dydx N −α n Dn Bρn (0)\Dn |x − y| (3.32) C ≤ C5 e−β5 |yn | + . N −α |yn |

The combination of (3.25), (3.26), (3.28), (3.29) and (3.32) leads us to ∫ 1 ∂V C q − (x)|un | dx ≤ Ce−β|yn | + , N −α 2q Dn ∂x |yn | where C > 0 is a constant and β = min{β2 , β3 , β4 , β5 }. But on the other hand, let δn = 12 min0≤i̸=j≤m {|yni − ynj |, Rn }, then we have ∫ ∫ ∂V ∂V q q (x)|un | dx ≥ − (x)|un | dx − ∂x ∂x Dn ∩Bδn (yn ) Dn ( )∫ ∂V q ≥ inf − (x) |un | dx. ∂x Dn ∩Bδn (yn ) Dn ∩Bδ (yn ) n

(3.33)

22

X. Li, X. Liu and S. Ma / Nonlinear Analysis 189 (2019) 111583

By Bδn (yn ) ⊂ C˜n , we have Dn ∩ Bδn (yn ) = (C˜n ∩ Bρn (0)) ∩ Bδn (yn ) = Bρn (0) ∩ Bδn (yn ). Inserting this into the above inequality and using the fact un ∈ H01 (Bρn (0)), we deduce ( )∫ ∫ ∂V ∂V q q − (x)|un | dx ≥ inf − (x) |un | dx ∂x Dn ∩Bδn (yn ) Dn ∩Bδn (yn ) Dn ∂x ( )∫ ∂V q ≥ inf − (x) |un | dx ∂x Bδn (yn ) Dn ∩Bδn (yn ) ( )∫ ∂V q = inf − (x) |un | dx. ∂x Bδn (yn ) Bδ (yn )

(3.34)

n

By Proposition 3.1, the definition of yn and δn , we have ∫ q lim inf |un | dx ≥ c > 0. n→+∞

Bδn (yn )

Consequently, −

1 2q

∫ Dn

( ) ∂V ∂V q (x)|un | dx ≥ C inf − (x) . ∂x ∂x Bδn (yn )

(3.35)

From (3.33) and (3.35), we get for large n, ) ( ( ) ∂V 1 −β|yn | . inf − (x) ≤ C e + N −α ∂x Bδn (yn ) |yn |

(3.36)

For x ∈ Bδn (yn ), we have 21 |yn | ≤ |x| ≤ 2|yn |. Hence, for large n, ( ) N −α ∂V inf −|x| (x) ≤ C, ∂x Bδn (yn ) which contradicts to (V3). The proof is complete. Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 11571187) and Tianjin Municipal Education Commission with the Grant No. 2017KJ173 “Qualitative studies of solutions for two kinds of nonlocal elliptic equations”. References [1] N. Ackermann, On a periodic Schr¨ odinger equation with nonlocal superlinear part, Math. Z. 248 (2) (2004) 423–443. [2] V. Ambrosio, Zero mass case for a fractional Berestycki-lions-type problem, Adv. Nonlinear Anal. 7 (3) (2018) 365–374. [3] P. Belchior, H. Bueno, O.H. Miyagaki, G.A. Pereira, Remarks about a fractional Choquard equation: Ground state, regularity and polynomial decay, Nonlinear Anal. 164 (2017) 38–53. [4] G.M. Bisci, V.D. Radulescu, R. Servadei, Variational Methods for Nonlocal Fractional Problems (Vol. 162), Cambridge University Press, 2016. [5] G. Cerami, G. Devillanova, S. Solimini, Infinitely many bound states for some nonlinear scalar field equations, Calc. Var. 23 (2005) 139–168. [6] K.C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problem, Birkh¨ auser, Boston, 1993. [7] J.Q. Chen, B.L. Guo, Blow up solutions for one class of system of Pekar-Choquard type nonlinear Schr¨ odinger equation, Appl. Math. Comput. 186 (1) (2007) 83–92.

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