Standing waves with a critical frequency for nonlinear Choquard equations

Nonlinear Analysis 161 (2017) 87–107

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Nonlinear Analysis www.elsevier.com/locate/na

Standing waves with a critical frequency for nonlinear Choquard equations Jean Van Schaftingena, *, Jiankang Xiab a

Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium b Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, China

article

abstract

info

Article history: Received 28 November 2016 Accepted 26 May 2017 Communicated by Enzo Mitidieri

We study the nonlocal Choquard equation

(

)

− ε2 ∆uε + V uε = Iα ∗ |uε |p |uε |p−2 uε

MSC 2010: 35B05 35J60 Keywords: Nonlinear Choquard equations Nonlocal semilinear elliptic equation Semi-classical limit Variational methods

in RN

where N ≥ 1, Iα is the Riesz potential of order α ∈ (0, N ) and ε > 0 is a parameter. When the nonnegative potential V ∈ C(RN ) achieves 0 with a homogeneous behavior or on the closure of an open set but remains bounded away from 0 at infinity, we show the existence of groundstate solutions for small ε > 0 and exhibit the concentration behavior as ε → 0. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction and main results We are interested in the nonlinear Choquard equation ( p) p−2 − ε2 ∆uε + V uε = Iα ∗ |uε | |uε | uε

in RN

(Cε )

where the dimension N ∈ N∗ = {1, 2, . . .} of the Euclidean space RN is given and V ∈ C(RN , [0, +∞)) is an external potential. The function Iα : RN \ {0} → R is the Riesz potential of order α ∈ (0, N ), defined for each point x ∈ RN \ {0} by ( ) Γ N −α 2 Iα (x) = ( ) N N −α , 2α Γ α2 π 2 |x| where Γ is the classical Gamma function, and ε > 0 is a small parameter.

*

Corresponding author. E-mail addresses: [email protected] (J. Van Schaftingen), [email protected] (J. Xia).

http://dx.doi.org/10.1016/j.na.2017.05.014 0362-546X/© 2017 Elsevier Ltd. All rights reserved.

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The nonlocal semilinear equation (Cε ) with N = 3, α = 2 and p = 2 is known as the Choquard–Pekar equation. It appears in several physical contexts: standing waves for the Hartree equation, Pekar’s quantum physical model of a polaron at rest [32], Choquard’s model of an electron trapped in its own hole [21] and a model coupling the Schr¨ odinger equation of quantum mechanics and the classical Newtonian gravitational potential [13,18,19,26,33]. The above quantum models feature in their time-dependent form the quantum state of indistinguishable particles ψ : RN → C interacting with an external potential W and an attractive mean field potential 2 −Iα ∗ |ψ| : ( 2) iε∂t ψ = −ε2 ∆ψ + W ψ − Iα ∗ |ψ| ψ;

(1.1)

standing waves are solutions ψ of (1.1) of the form ψ(t, x) = e−iEt/ε u(x), where E ∈ R is a constant and the function u : RN → C satisfies the Choquard equation (Cε ) with p = 2 and V = W − E. The existence and qualitative properties of solutions of the Choquard equation (Cε ) have been studied mathematically for a few decades when ε is a fixed constant by variational methods [21,23,24,28] (see also the review [30] and the references therein). In quantum physical models, the parameter ε is an adimensionalized Planck constant which in the semiclassical limit r´egime is quite small. In general, one expects to recover some classical dynamics in this r´egime. In the formal limit α → 0, one has Iα → δ0 as distributions and the Choquard equation becomes the nonlinear Schr¨ odinger equation − ε2 ∆uε + V uε = |uε |

q−2

uε

in RN ,

whose semiclassical limit is well understood. Under the assumption that inf RN V > 0, that is, the frequency is subcritical E < inf RN W , solutions concentrating at critical points of the potential V have been constructed by topological and variational methods [4–6,12,17,31,34,38]. The remaining case inf RN V = 0 corresponds to the critical frequency E = inf RN W where the frequency E coincides with the bottom of the spectrum of the operator −∆ + W around infinity. This critical frequency case marks the boundary between the nice mathematically structure when inf RN V > 0 and the highly oscillating solutions that are expected when inf RN V < 0 that has been observed when N = 1 [16]. When inf RN V = 0 and V > 0 on RN , such constructions are still possible provided the function V does not decay too fast at infinity or the exponent q is large enough and the solutions have the same scaling [7,8,27]. When the potential V does vanish somewhere in RN , then the solutions exhibit a different concentration behavior with a scaling governed by the shape of the potential V and by the nonlinearity exponent q which was studied by J. Byeon and Z.-Q. Wang [9,10] (see also [15]). For the Choquard equation (Cε ), the semiclassical limit has been studied in the subcritical frequency case inf RN V > 0 [11,39] (with extensions to the quasilinear case [2,3] and to general nonlinearities [42]) and when inf RN V = 0 and V > 0 [29,35]. In this work we study a large class of potential V that vanishes somewhere on RN . Theorem 1.1. Let N ∈ N∗ , p ∈ (0, +∞) and V ∈ C(RN ). If N −2 1 N < < , N +α p N +α if V ≥ 0 on RN , if lim inf V (x) > 0, |x|→∞

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89

and if there exist γ > 0 and a nonempty set A ⊂ RN such that, for every x ∈ A there exists a positive γ-homogeneous function W ∈ C(RN ) such that lim

z→x

V (z) − W (z − x) = 0, γ |z − x|

and for every x ∈ RN \ A lim

z→x

V (z) γ = +∞, |z − x|

then, for sufficiently small ε > 0, equation (Cε ) has a positive groundstate solution uε . Moreover, there exist a point x∗ ∈ A, a positive γ-homogeneous function W∗ ∈ C(RN ) such that lim

z→x∗

V (z) − W∗ (z − x∗ ) = 0, γ |z − x∗ |

1 a groundstate v∗ ∈ HW (RN ) \ {0} of the problem ∗ ( p) p−2 − ∆v∗ + W∗ v∗ = Iα ∗ |v∗ | |v∗ | v∗

in RN ,

and a sequence (εn )n∈N in (0, +∞) converging to 0 such that, as n → ∞, ) ( α−γ 2 1 (RN ), εn(p−1)(γ+2) uεn x∗ + εnγ+2 · → v∗ in Hloc and ∫

2

RN

n→∞

= inf ∫

∫

1

2

|∇v∗ | + W∗ |v∗ | = lim

(

) pγ−α

2 γ+2 N + p−1

εn {∫

2

RN

2

ε2n |∇uεn | + V |uεn |

2

2

1 |∇v| + W |v| | v ∈ Hloc (RN ), ∫ ( p) p 2 2 Iα ∗ |v| |v| = |∇v| + W |v| , RN

RN

RN N

W ∈ C(R ) is positive and γ-homogeneous and there exists x ∈ RN such that } V (z) − W (z − x) lim < +∞. = 0 γ z→x |z − x| In the statement of Theorem 1.1 and in the sequel, a function W : RN → R is positive γ-homogeneous if for every y ∈ RN \ {0}, W (y) > 0 and if for every t ∈ [0, +∞) and every y ∈ RN , W (ty) = tγ W (y). For the nonlinear Schr¨ odinger equation, the semiclassical limit has been studied under similar asymptotic homogeneity conditions on the external potential V [9]. A simple example of potential V satisfying the assumptions of Theorem 1.1 is defined for each x ∈ RN by γ

V (x) =

|x| γ 1 + |x|

with A = {0}. Theorem 1.1 also covers several minimum points: if x1 , . . . , xℓ ∈ RN are distinct points and γ1 , . . . , γℓ ∈ (0, +∞), then the potential V defined for every x ∈ RN by V (x) =

ℓ ∏

γ

|x − xi | i γ 1 + |x − xi | i i=1

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J. Van Schaftingen, J. Xia / Nonlinear Analysis 161 (2017) 87–107

satisfies the assumptions of Theorem 1.1 with γ = max{γ1 , . . ., γℓ } and A = {xk | k ∈ {1, . . . , ℓ} and γk = γ}. Whereas previous studies of the semiclassical limit for the Choquard equation required the superquadraticity assumption p ≥ 2, with more delicate analysis and results when p = 2 [2,3,11,29,35,39,42], Theorem 1.1 holds in a range on p limited by the classical lower critical exponent NN+α , which is needed to deduce the well-definiteness of the associated functional by an application of the Hardy–Littlewood–Sobolev and Sobolev inequalities and which is required by suitable Pohoˇzaev identities [28]. The results of Theorem 1.1 can be restated in terms of convergence to minimizers of a concentration 1 1 function. Indeed, the limiting functional JW ∈ C 1 (HW (RN )) is defined for every v ∈ HW (RN ) by ∫ ∫ ( 1 1 2 2 p) p |∇v| + W |v| − Iα ∗ |v| |v| (1.2) JW (v) = 2 RN 2p RN 1 on the Hilbert space HW (RN ) obtained by completion of the set of smooth functions Cc∞ (RN ) endowed with the norm associated to the quadratic part of the functional JW : )1 (∫ 2 2 2 |∇u| + W |u| ∥u∥H 1 = ; W

RN

the limiting groundstate level E(W ) is defined by { } 1 ′ E(W ) = inf JW (v) | v ∈ HW (RN ) \ {0} and ⟨JW (v), v⟩ = 0

(1.3)

(this infimum is in fact always achieved and positive since the positive γ-homogeneous potential W is coercive [37]), then the function v∗ achieves the infimum in (1.3) with W = W∗ and { } C(x∗ ) = inf C(x) | x ∈ RN < +∞, where the concentration function C : RN → (0, +∞] is defined for each x ∈ RN by ⎧ E(W ) if W ∈ C(RN ) is positive and γ-homogeneous ⎪ ⎪ ⎪ ⎪ V (z) − W (z − x) ⎨ = 0, and lim γ z→x |z − x| C(x) = ⎪ ⎪ ⎪ V (z) ⎪ ⎩ +∞ if lim γ = +∞. z→x |z − x| Yang and Ding have proved the existence of a family of solutions under the assumptions of the theorem ∫ α 2 2 when N = 3 and that these solutions satisfy RN ε2 |∇uε | + V |uε | = o(ε1− p−1 ) [41]. The main difficulty in order to prove Theorem 1.1 is in the proof of the asymptotic lower bound on the energy ( pγ−α ) ∫ − 2 N + p−1 2 2 ε2 |∇uε | + V |uε | ≥ inf C, lim inf ε γ+2 ε→0

RN

RN

where we have two radically different behavior at points and these limits cannot be uniform. Our approach to this problem is to consider at every point all the homogeneous potentials that are asymptotically below the potential V ; at most points this class is unbounded, giving an infinity lower bound, and at the other points it is bounded and gives the lower bound. The case where the potential V vanishes on a large set has also been studied for the nonlinear Schr¨odinger equation [9], we consider such a case for the Choquard equation. Theorem 1.2. Let N ∈ N∗ , p ∈ (0, +∞) and V ∈ C(RN ). If N −2 1 N < < , N +α p N +α

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91

if V ≥ 0 on RN , if lim inf V (x) > 0, |x|→∞

and if there exists a bounded open set Ω with a smooth boundary such that { } ¯ ̸= ∅, x ∈ RN | V (x) = 0 = Ω then, for sufficiently small ε > 0, the Choquard equation (Cε ) has a positive groundstate solution uε ∈ H 1 (RN ). Moreover, there exist a groundstate v∗ ∈ H 1 (RN ) of the problem { ( p) p−2 −∆v∗ = Iα ∗ |v∗ | |v∗ | v∗ in Ω , v∗ = 0 on RN \ Ω , and a sequence (εn )n∈N in (0, +∞) converging to 0 such that, as n → ∞, − 1

εn p−1 uεn → v∗

in H 1 (RN ).

Theorem 1.2 is reminiscent of some results obtained for the problem ( p) p−2 − ∆uµ + (1 + µV )uµ = Iα ∗ |uµ | |uµ | uµ when µ → +∞ [1,25]. An example of potential satisfying the assumptions of Theorem 1.2 is the function V : RN → R defined for every x ∈ RN by { } 1 V (x) = max 0, 1 − |x| . The rest of the present paper is organized as follows. We study the existence of groundstate solutions for small parameters in Section 2, which completes the proof of the first part of Theorems 1.1 and 1.2. The asymptotics of Theorem 1.1 are obtained in Section 3, whereas those of Theorem 1.2 are the object of Section 4. 2. Existence of solutions Eq. (Cε ) is variational in nature, its weak solutions are, at least formally, critical points of the functional defined for every function u : RN → R by ∫ ∫ 1 1 2 2 p p 2 ε |∇u| + V |u| − (Iα ∗ |u| )|u| . (2.1) Iε (u) := 2 RN 2p RN The linear part of Eq. (Cε ) naturally induces a norm ∫ 2 2 2 ∥u ∥ε := ε2 |∇u| + V |u| . RN

The norms for various values of ε are all equivalent to each other. We define HV1 (RN ) to be the Hilbert space obtained by completion of the set of smooth test functions Cc∞ (RN ) with respect to any norm ∥·∥ε . Although it will not play any role in this work, using the continuity of V and the fact that lim inf |x|→∞ V (x) > 0, the space HV1 (RN ) could also be characterized as ∫ { } 2 1 N 1 N HV (R ) = u ∈ Hloc (R ) | V |u| < +∞ . RN

We recall how the space (R ) can be embedded continuously into the classical Sobolev space H 1 (RN ) equipped with the standard norm ∥ · ∥H 1 for fixed ε > 0, even though the potential V has a nontrivial set of zeroes. HV1

N

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Lemma 2.1. Let V : RN → [0, +∞). If lim inf |x|→∞ V (x) > 0, then for every ε > 0, there exists a constant C > 0 such that if u ∈ HV1 (RN ), then u ∈ H 1 (RN ) and ∫ ∫ 2 2 2 2 ε2 |∇u| + |u| ≤ C ε2 |∇u| + V |u| . RN

RN

Proof . Let µ ∈ (0, +∞) such that µ < lim inf |x|→∞ V (x). By definition of the limit, there exists R > 0 such that if x ∈ RN \ BR/2 , V (x) ≥ µ. (Here and in the sequel, we use the notation Br (a) to denote the ball in RN with radius r and centered at a and Br = Br (0).) By integration, we have immediately ∫ ∫ 1 2 2 V |u| . (2.2) |u| ≤ µ RN \BR/2 RN \BR/2 We take a function ψ ∈ C ∞ (RN ) such that 0 ≤ ψ ≤ 1 in RN , ψ(x) = 1 for each x ∈ BR/2 and ψ(x) = 0 for each x ∈ RN \ BR . Then, it follows from the classical Poincar´e inequality on the ball BR that ∫ ∫ ∫ 2 2 2 |u| ≤ |ψu| ≤ C1 |∇(ψu)| BR/2

BR

BR

∫ ≤ 2C1 RN

2

2C1 ∥∇ψ ∥L∞ |∇u| + µ 2

(2.3)

∫

2

V |u| . BR \BR/2

The conclusion then follows from the combination of the inequalities (2.3) and (2.2). □ By the classical Sobolev embedding of H 1 (RN ) to Lq (RN ), we deduce that the space HV1 (RN ) can be continuously embedded in Lq (RN ) when 12 ≥ 1q ≥ 12 − N1 . The well-definiteness, continuity and differentiability of the nonlocal term in the function Iε defined by (2.1) follows then from the classical Hardy–Littlewood–Sobolev inequality [22, Theorem 4.3] which states that Ns if α ∈ (0, N ), s ∈ (1, N/α) and if φ ∈ Ls (RN ), then Iα ∗ φ ∈ L N −αs (RN ) and ∥Iα ∗ φ∥

Ns

L N −αs

≤ CH ∥φ∥Ls ,

(2.4)

where the constant CH > 0 depends only on α, N , and s. A solution u is a groundstate of the Choquard equation (Cε ) if Iε (u) is the least among all nontrivial critical values of Iε , namely, u has the least energy among nontrivial solutions. A natural and well known method to search the groundstate is to minimize the functional Iε on the Nehari manifold (see [36]) of Eq. (Cε ) which is defined by { } Nε := u ∈ HV1 (RN ) | u ̸= 0 and ⟨Iε′ (u), u⟩ = 0 . The corresponding groundstate energy is described as cε := inf Iε (u). u∈Nε

N N −2 energy cε is positive and Lemma 2.2. Let p > 1 and 1/p ∈ ( N +α , N +α ). For given ε > 0, the groundstate ⏐ 1 Nε is a manifold of class C . Moreover, if u ∈ Nε is a critical point of Iε ⏐N , then Iε′ (u) = 0. ε

Proof . We fix ε > 0. If we define Gε (u) := ⟨Iε′ (u), u⟩, then for any u ∈ Nε , we have Gε (u) = 0, which, together with the Hardy–Littlewood–Sobolev inequality (2.4) and the Sobolev inequality implies that ∫ ( 1 p) p 2(p−1) 1= Iα ∗ |u| |u| ≤ C1 ∥u ∥ε , 2 ∥u ∥ε RN

J. Van Schaftingen, J. Xia / Nonlinear Analysis 161 (2017) 87–107

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where the constant C1 > 0 depends on ε. Hence, for any u ∈ Nε , we have Iε (u) =

(1 2

(1 1 1) 1 ) − p−1 2 ∥u ∥ε ≥ C1 − , 2p 2 2p

−

thus, cε > 0. Furthermore, since p > 1, for each u ∈ Nε , 2

1 − p−1

⟨Gε′ (u), u⟩ = −2(p − 1)∥u ∥ε ≤ −2(p − 1)C1

< 0,

(2.5)

Nε is thus an embedded submanifold of class C 1 by the implicit function theorem (see for example [20, Theorem 3.4.10], [43, Theorem 4.E]). ⏐ Let us now assume that the function u ∈ Nε is a critical point of the restricted functional Iε ⏐Nε , then there exists a Lagrange multiplier λε ∈ R, such that Iε′ (u) = λε G′ε (u). By testing this against u itself, we have λε ⟨G′ε (u), u⟩ = ⟨Iε′ (u), u⟩ = 0. We thus deduce by (2.5) that λε = 0 and the conclusion follows. □ We now prove the existence of groundstate solutions of (Cε ) for small parameters. N −2 N N Proposition 2.3. Let N ≥ 1, p > 1, 1/p ∈ ( N +α , N +α ) and let V ∈ C(R ). If

0 ≤ inf V < lim inf V (x), RN

|x|→∞

then, for sufficiently small ε > 0, the Choquard equation (Cε ) has a positive groundstate solution. Proposition 2.3 is a counterpart for the Choquard equation of Rabinotwitz’s existence result for the nonlinear Schr¨ odinger equation [34, Theorem 4.33]. Under the assumptions of Theorem 1.1, there exists x ∈ RN such that V (x) = 0, and the assumptions of Proposition 2.3 are satisfied. Proof of Proposition 2.3. By Ekeland’s variational principle [40], there exists a minimizing sequence (un )n∈N in Nε for cε , such that, as n → ∞, Iε (un ) → cε ,

and Iε′ (un ) − λn Gε′ (un ) → 0

( )′ in HV1 (RN ) .

We first observe that the sequence (un )n∈N is bounded, because p−1 1 2 ∥un ∥ε = Iε (un ) − ⟨Iε′ (un ), un ⟩ ≤ cε + on (1)∥un ∥ε . 2p 2p It follows then that λn ⟨Gε′ (un ), un ⟩ → 0 as n → ∞. By (2.5), we see that λn → 0 as n → ∞. Note that the ( )′ sequence (Gε′ (un ))n∈N is bounded in the dual space HV1 (RN ) , in fact, for every φ ∈ HV1 (RN ), we have |⟨Gε′ (un ), φ⟩|

⏐ ⏐ ∫ ⏐ ⏐ ( p) p−2 ⏐ = ⏐2(un |φ) + 2p Iα ∗ |un | |un | un φ⏐⏐ RN ( ) ≤ C2 ∥un ∥ε + C3 ∥un ∥ε2p−1 ∥φ∥ε ≤ C4 ∥φ∥ε .

( )′ Hence, Iε′ (un ) → 0 as n → ∞ in HV1 (RN ) .

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Up to a subsequence we can assume that un ⇀ u weakly in HV1 (RN ) and un → u almost everywhere in R as n → ∞. If u ̸= 0, we reach the conclusion. Indeed, Iε′ (u) = 0, which, together with the weakly lower semi-continuity of the norm, implies that (1 1) 1 2 ∥un ∥ε − cε + on (1)∥un ∥ε = Iε (un ) − ⟨Iε′ (un ), un ⟩ = 2p 2 2p (1 1 1) 2 ≥ ∥u ∥ε = Iε (u) − ⟨Iε′ (u), u⟩ = Iε (u) ≥ cε , − 2 2p 2p N

that is, the function u is a minimizer for cε and is thus a groundstate of the Choquard equation (Cε ) by Lemma 2.2. 2N p/(N +α) In order to conclude, we assume by contradiction that u = 0. We have then un → 0 in Lloc (RN ) N as n → ∞. We choose R > 0, δ > 0, µ > ν > 0 and x∗ ∈ R , such that V (x) ≥ µ if |x| ≥ R,

and V (x) ≤ ν

for every x ∈ Bδ (x∗ ),

and a cut-off function η ∈ C ∞ (RN ) such that 0 ≤ η ≤ 1 in RN , η = 0 in BR and η = 1 on RN \ B2R . We define the function vn = ηun . We have by our contradiction assumption, as n → ∞, ∫ ∫ 2 2 2 2 2 ε |∇vn | + µ|vn | ≤ ε2 |∇vn | + V |vn | N N R ∫R 2 2 ≤ ε2 |∇un | + V |un | + on (1), RN

and ∫

(

p)

∫

p

Iα ∗ |vn | |vn | =

RN

(

p) p Iα ∗ |un | |un | + on (1).

RN

If we define the functional Iεµ : H 1 (RN ) → R for v ∈ H 1 (RN ) ∫ ∫ ( 1 1 2 2 p) p ε2 |∇v| + µ|v| − Iα ∗ |v| |v| , Iεµ (v) = 2 RN 2p RN and if we take tn ∈ (0, +∞) such that ∫ ∫ 2 2 2 2 2p tn ε |∇vn | + µ|vn | = tn RN

(

p) p Iα ∗ |vn | |vn | ,

RN

then lim supn→∞ tn ≤ 1 and thus cε = lim Iε (un ) ≥ lim inf Iε (tn un ) ≥ lim inf Iεµ (tn vn ) ≥ cµε , n→∞

n→∞

n→∞

(2.6)

where { } cµε = inf Iεµ (v) | v ∈ H 1 (RN ) \ {0} and ⟨Iεµ ′ (v), v⟩ = 0 . Since ν < µ, we have cν1 < cµ1 and there exists a function φ ∈ Cc∞ (RN ) \ {0}, such that ⟨I1ν′ (φ), φ⟩ = 0 and I1ν (φ) < cµ1 . −α

We next let φε (x) = ε 2(p−1) φ(ε−1 (x − x∗ )), and we observe that ⟨Iεν′ (φε ), φε ⟩ = 0 and Iεν (φε ) < cµε .

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On the other hand, by the definition of cε , we have (∫ ) p 2 2 p−1 2 ) ε |∇φ | + V |φ | ε ε N R 1 − cε ≤ max Iε (tφε ) = ) 1 . t≥0 2 2p (∫ p p p−1 (I ∗ |φ | ).|φ | α ε ε RN (1

When ε is small enough so that the inclusion ε supp φ ⊂ Bδ holds, we have V ≤ ν on the set x∗ + ε supp φ and we conclude that (∫ ) p 2 2 p−1 2 ) (1 ε |∇φ | + ν|φ | ε ε N R 1 − = Iεν (φε ) < cµε , cε ≤ ) 1 2 2p (∫ p p p−1 (I ∗ |φε | )|φε | RN α which contradicts the lower bound (2.6). □

3. Asymptotics for potential with homogeneous zeroes 3.1. Asymptotic upper bound We define the upper concentration function C¯ : RN → R for x ∈ RN by } { ¯ ¯ ¯)|W ¯ ∈ C(RN ) is positive and γ-homogeneous and lim sup V (z) − W (zγ − x) ≤ 0 . C(x) = inf E(W |z − x| z→x ¯ The quantity E(W ) was defined above in (1.3) as the groundstate energy of the limiting functional J ¯

W

1 N defined in (1.2) on the weighted Sobolev space HW ¯ (R ). The assumptions of Theorem 1.1 ensure that C¯ = C everywhere in RN . Indeed, given x ∈ RN , if γ ¯ satisfying the condition and thus C(x) ¯ limz→x V (x)/|z − x| = +∞, then there is no function W = +∞ = N C(x). Otherwise, there exists a positive γ-homogeneous function W ∈ C(R ) such that

V (z) − W (z − x) = 0, γ |z − x| ¯ ¯ ∈ C(RN ) is positive and γ-homogeneous and thus we have C(x) = E(W ) ≥ C(x). Moreover, if the function W and if ¯ (z − x) V (z) − W ≤ 0, lim sup γ |z − x| z→x ¯ ), so that by taking the infimum, C(x) ≤ C(x). ¯ ¯ in RN , and thus E(W ) ≤ E(W then W ≤ W lim

z→x

To alleviate the notation, we fix for the rest of this section 2 κ= . γ+2 Proposition 3.1. One has lim sup ε→0

cε (

pγ−α

κ N + p−1

ε

¯ ) ≤ inf C. RN

¯ ∈ C(RN ) be a positive γ-homogeneous function such that Proof . Let x∗ ∈ RN , let W ¯ (x − x∗ ) V (x) − W lim sup ≤0 γ |x − x∗ | x→x∗ and let φ ∈ Cc∞ (RN ) \ {0}. For every ε > 0, we define the function φε : RN → R for each x ∈ RN by (x − x ) γ−α κ ∗ φε (x) = ε 2(p−1) φ . εκ

(3.1)

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We observe that by homogeneity and scaling, we have for each ε > 0, ∫ ( pγ−α ) ∫ κ N + p−1 2 2 |∇φ| , ε2 |∇φε | = ε N N R R ∫ ( pγ−α ) ∫ V (x∗ + εκ y) κ N + p−1 2 2 V |φε | = ε |φ(y)| dy, εκγ N N R R ∫ ( pγ−α ) ∫ ( ( κ N + p−1 p) p p) p Iα ∗ |φε | |φε | = ε Iα ∗ |φ| |φ| . RN

RN

¯ is γ-homogeneous and satisfies the condition (3.1), we have for each y ∈ RN , Since the function W lim sup ε→0

κ ¯ (εκ y) W V (x∗ + εκ y) ¯ (y), ¯ (y) + |y|γ lim sup V (x∗ + ε y) − ≤W =W γ κγ κ ε |ε y| ε→0

uniformly when y stays in the support of φ, which is compact by assumption. Thus by Lebesgue’s dominated convergence theorem, we have ∫ ∫ 1 2 ¯ |φ|2 . lim sup ( pγ−α ) W V |φε | ≤ κ N + p−1 N N ε→0 R R ε For every ε > 0, we fix tε ∈ (0, +∞) in such a way that ∫ ∫ 2 2 t2ε ε2 |∇φε | + V |φε | = t2p ε RN

(

p) p Iα ∗ |φε | |φε | ,

RN

and we observe that lim supε→0 tε ≤ t∗ , where t∗ ∈ (0, +∞) is characterized by ∫ ∫ ( p) p 2 ¯ |φ|2 = t2p Iα ∗ |φ| |φ| . t2∗ |∇φ| + W ∗ RN

RN

We have then lim sup ε→0

cε (

pγ−α

κ N + p−1 ε

) ≤ lim sup ε→0

I (tφε ) ( ε pγ−α ) = lim sup

sup

t∈(0,+∞) εκ N + p−1

ε→0

Iε (tε φε ) ( pγ−α )

κ N + p−1 ε

≤ JW ¯ (t∗ φ) =

sup

t∈(0,+∞)

JW ¯ (tφ).

Since the left-hand side is independent of the function φ, taking the infimum with respect to φ and by 1 N density of the set of smooth test functions Cc∞ (RN ) in the weighted space HW ¯ (R ), we have lim sup ε→0

cε (

pγ−α

κ N + p−1 ε

¯ ). ) ≤ E(W

¯ , by taking now the infimum with respect to Since the left-hand side does not depend on the potential W N ¯ suitable positive γ-homogeneous functions W ∈ C(R ), we deduce that cε ¯ ∗ ). lim sup ( pγ−α ) ≤ C(x κ N + p−1 ε→0 ε Since the point x∗ ∈ RN is arbitrary, the conclusion follows.

□

3.2. Asymptotic lower bound and behavior of solutions We define the lower concentration function C : RN → R by { } V (z) − W (z − x) C(x) = sup E(W ) | W ∈ C(RN ) is positive and γ-homogeneous and lim inf ≥ 0 , γ z→x |z − x| where the quantity E(W ) was defined in (1.3).

J. Van Schaftingen, J. Xia / Nonlinear Analysis 161 (2017) 87–107

97

γ

Under the assumptions of Theorem 1.1, we have C = C. Indeed, if limz→x V (x)/|z − x| = +∞, then we can take any positive and γ-homogeneous function in the definition of the lower concentration function C and thus C(x) = +∞. Otherwise, there exists a positive and γ-homogeneous function W ∈ C(RN ) such that lim

z→x

V (z) − W (z − x) =0 γ |z − x|

and thus C(x) ≤ C(x). Moreover, if lim inf z→x

V (z) − W (z − x) ≥ 0, γ |z − x|

then W ≤ W on RN and by monotonicity of E, we have E(W ) ≤ E(W ) = C(x); it follows then that C(x) ≤ C(x). Proposition 3.2. Let (εn )n∈N be a sequence of positive numbers converging to 0 and (un )n∈N be solutions in HV1 (RN ) of problem ( Cεn ). If lim inf n→∞

1 (

pγ−α

κ N + p−1 εn

) Iεn (un ) < +∞,

then up to a subsequence, there exist R∗ > 0 and x∗ ∈ RN such that ∫ 1 2 ( γ−α ) |un | > 0 lim n→∞ κ N + p−1 Bεκ R∗ (x∗ ) n εn and 1

C(x∗ ) ≤ lim inf

(

n→∞

pγ−α

κ N + p−1

) Iεn (un ).

εn

If moreover W is a positive γ-homogeneous function such that lim

x→x∗

V (x) − W (x − x∗ ) = 0, γ |x − x∗ |

1 then there exists v∗ ∈ HW (RN ) \ {0} such that κ

γ−α

εn 2(p−1) un (x∗ + εκn ·) → v∗

1 in Hloc (RN ),

v∗ is a weak solution to p

− ∆v∗ + W v∗ = (Iα ∗ |v∗ | )|v∗ |

p−2

v∗

and C(x∗ ) ≤ JW (v∗ ) ≤ lim inf n→∞

1 (

pγ−α

κ N + p−1 εn

) Iεn (un ).

In order to prepare the proof of Proposition 3.2, we first give a lower bound on the potential V . Lemma 3.3. Let V ∈ C(RN ) and γ > 0. If V ≥ 0 on RN , lim inf V (x) > 0 |x|→∞

98

J. Van Schaftingen, J. Xia / Nonlinear Analysis 161 (2017) 87–107

and if for each x ∈ RN lim inf z→x

V (z) γ > 0, |z − x|

then there exist k ∈ N, a1 , . . . , ak ∈ RN , µ > 0 and ν > 0 such that for each x ∈ RN , { γ γ} V (x) ≥ min µ, ν|x − a1 | , . . ., ν|x − ak | . γ

In the statement of Lemma 3.3, the condition lim inf z→x V (z)/|z − x| > 0 allows this inferior limit to be +∞; in fact this will be the case at all but a finite number of points of RN . Proof of Lemma 3.3. We define the set K = V −1 ({0}). Since the function V is continuous and since γ lim inf |x|→∞ V (x) > 0, the set K is compact. If x ∈ K, we have lim inf z→x V (z)/|z − x| > 0 and there exists thus δ > 0 such that Bδ (x) ∩ K = {x}. Hence, the set K is finite and can be written as K = {a1 , . . . , ak } with k ∈ N and a1 , . . . , ak ∈ RN . Moreover, there exist ρ > 0 and ν > 0 such that if j ∈ {1, . . . , k} and γ x ∈ Bρ (aj ), then V (x) ≥ ν|x − aj | . Since lim inf |x|→∞ V (x) > 0, there exists µ > 0 such that V (x) ≥ µ for ⋃k N every x ∈ R \ j=1 Bρ (aj ). The conclusion follows. □ Thanks to Lemma 3.3, we establish a uniform estimate on rescaled balls of RN , which is very useful in our subsequent arguments. Lemma 3.4. There exists a positive number C, such that if ε is sufficiently small, then for every x ∈ RN and every u ∈ H 1 (Bεκ (x)), we have ∫ ∫ 2 2 2 2 ε2κ |∇u| + |u| ≤ Cε−κγ ε2 |∇u| + V |u| . Bεκ (x)

Bεκ (x)

Proof . Let x ∈ RN . By the Minkowski, Poincar´e and Cauchy–Schwarz inequalities (see for example [14]), we first see that (∫ ) 21 (∫ ) 12 (∫ ) 12 2 2 2 |u| ≤ |u − u ¯| + |¯ u| Bεκ (x) Bεκ (x) Bεκ (x) (3.2) 1 (∫ )2 ∫ 1 2 κ |u|, + ≤ C1 ε |∇u| 1 Bεκ (x) |Bεκ | 2 Bεκ (x) where the constant C1 only depends on the dimension N , and u ¯ denotes the average of the function u on the ball Bεκ (x): ∫ 1 u ¯ := u. |Bεκ (x)| Bεκ (x) Here and in the sequel, we use the notation |A| to denote the Lebesgue’s measure of the subset A of RN . By Lemma 3.3, we observe that, if λ ≤ µ, |{z ∈ RN | V (z) < λ}| ≤ k|B1 |

( λ ) Nγ µ

.

If we take λ = µ((1/4k)1/N εκ )γ , we have, if ε is small enough, |{z ∈ RN | V (z) < C2 εκγ }| ≤

|Bεκ | . 4

(3.3)

J. Van Schaftingen, J. Xia / Nonlinear Analysis 161 (2017) 87–107

99

We have thus, by (3.3) and by the Cauchy–Schwarz inequality ∫ ∫ ∫ 1 C3 2 V |u| + |u| ≤ |u| (C2 εκγ )1/2 Bεκ (x) Bεκ (x) Bεκ (x)∩V −1 ([0,C2 εκγ )) 1 (∫ ) 12 ) 12 1/2 (∫ C3 |Bεκ | 2 |Bεκ | 2 2 V |u| ≤ |u| + . 2 (C2 εκγ )1/2 Bεκ (x) Bεκ (x) In view of (3.2) we obtain finally (∫ |u| Bεκ (x)

2

) 12

) 12 ) 12 (∫ (∫ C3 1 2 2 ≤ C1 ε |∇u| + V |u| + |u| 2 Bεκ (x) (C2 εκγ )1/2 Bεκ (x) Bεκ (x) ) 12 ) 12 (∫ (∫ 1 C4 2 2 2 ε2 |∇u| + V |u| + |u| . ≤ γ/(γ+2) 2 Bεκ (x) ε Bεκ (x)

The conclusion follows.

κ

(∫

2

) 12

□

1 (RN ) on a ball gives a control in H 1 on Finally, we recall how similarly to Lemma 2.1, a control in HW the same ball.

Lemma 3.5. If W ∈ C(RN ) is positive and γ-homogeneous, then there exists a constant C > 0 such that if R > 0 and v ∈ H 1 (BR ), then ∫ ∫ 2 W |v| 2 2 |v| ≤ C R2 |∇v| + . Rγ BR BR Proof . By scaling of the inequality and by γ-homogeneity of the potential W , we can assume without loss of generality that R = 1. We choose ψ ∈ Cc∞ (B1 ) such that ψ = 1 on B1/2 . By the Poincar´e inequality with Dirichlet boundary conditions on the ball B1 and since W is bounded from below on B1 \ B1/2 , we have by Weierstrass’ theorem, that ∫ ∫ ∫ ∫ 2 2 2 2 2 2 2 |v| = ψ |v| + (1 − ψ )|v| ≤ C1 |∇(ψv)| + (1 − ψ 2 )|v| B1 B1 B1 B1 \B1/2 ∫ ∫ ∫ 2 2 2 2 2 ≤ 2C1 |∇v| + (2C1 ∥∇ψ ∥∞ +1) |v| ≤ C2 |∇v| + W |v| . □ B1 \B1/2

B1

B1

We are now in a position to prove Proposition 3.2. Proof of Proposition 3.2. By taking if necessary a subsequence, we can assume that 1

lim inf

(

pγ−α κ N + p−1

n→∞

RN

1 q

(

2

pγ−α

κ N + p−1 εn

n→∞

εn

We also observe that for each n ∈ N, ∫

1

) Iεn (un ) = lim sup

2

ε2n |∇un | + V |un | =

) Iεn (un ) < +∞.

2p Iε (un ). p−1 n

(3.4)

By the scaled version of the classical Sobolev embedding theorem, we have, for each q > 1 with ∈ ( 21 − N1 , 12 ), that for every x ∈ RN (∫ |un | Bεκ (x) n

q

) 2q

κN ( 2 q −1)

∫

2

2

ε2κ n |∇un | + |un | ,

≤ C1 εn

Bεκ (x) n

J. Van Schaftingen, J. Xia / Nonlinear Analysis 161 (2017) 87–107

100

here the Sobolev embedding constant C1 is independent of the point x ∈ RN , which, together with Lemma 3.4, implies that (∫ |un |

q

) 2q ≤

κ( 2N −γ−N ) C2 εn q

∫

2

2

ε2n |∇un | + V |un |

Bεκ (x)

Bεκ (x)

n

n

and then ∫

q

|un | ≤

κ( 2N −γ−N ) C2 εn q

(∫ |un |

Bεκ (x)

q

)1− 2q ∫

Bεκ (x)

n

2

2

ε2n |∇un | + V |un | .

(3.5)

Bεκ (x)

n

n

By integration of both sides on (3.5) with respect to x over RN , we get ∫

∫

q

|un | dx ≤ RN

κ( 2N −γ−N ) C 2 εn q

(

∫

q

x∈RN

n

∫

2

RN

Bεκ (x) n

2

ε2n |∇un | + V |un | dx.

|un |

sup

Bεκ (x)

)1− 2q ∫

Bεκ (x) n

By interchanging the integrals, we conclude that ∫

q

|un | ≤

κ( 2N −γ−N ) C 2 εn q

(

RN

∫

q

2

|un |

sup x∈RN

)1− 2q ∫

Bεκ (x) n

RN

2

ε2n |∇un | + V |un | ,

the constant C2 depends neither on the point x ∈ RN nor on the parameter εn > 0 provided that εn is small enough. Since by assumption for every n ∈ N the function un is a solution of the Choquard equation (Cεn ), we deduce from the Hardy–Littlewood–Sobolev inequality (2.4) that ∫ 2 2 ε2n |∇un | + V |un | N R (∫ ) N +α ∫ N 2N p ( p) p N +α Iα ∗ |un | |un | ≤ C3 = |un | RN RN ( ) N +α ( )1− N +α ∫ ∫ N Np 2N p κ( N +α 2 2 p −γ−N ) 2 ≤ C4 εn sup |un | N +α εn |∇un | + V |un | ; x∈RN

Bεκ (x) n

RN

by the boundedness assumption on the sequence and by (3.4), we then arrive at ∫ 2N p 1 ( ) lim inf sup |un | N +α > 0. N p γ−α n→∞ Bεκ (x) x∈RN κ N + N +α p−1 n εn Hence, there exists a sequence of points (xn )n∈N in the space RN such that ∫ 2N p 1 ( ) lim inf |un | N +α > 0. N p γ−α n→∞ κ N + N +α p−1 Bεκ (xn ) n εn Since

Np N +α

− 1 > 0, 1 −

(N −2)p N +α

(3.6)

> 0, ( Np ) ( (N − 2)p ) 2p −1 + 1− = N +α N +α N +α

(3.7)

( Np )( 2) ( (N − 2)p ) 2 −1 1− + 1− = , N +α N N +α N

(3.8)

and

J. Van Schaftingen, J. Xia / Nonlinear Analysis 161 (2017) 87–107

101

by a scaling of the endpoint Gagliardo–Nirenberg interpolation inequality on the ball and by Lemma 3.4, we have ∫ 2N p |un | N +α Bεκ (xn ) n ( ) ( ) (∫ ) N2 N p −1 (∫ ) N2 1− (N −2)p N +α N +α 2 2 2 ≤ C5 |∇un | + ε−2κ |un | (3.9) n |un | Bεκ (xn ) Bεκ (xn ) n n ( ( ) ) ( ∫ ) N2 N p −1 (∫ ) N2 1− (N −2)p N +α N +α 1 2 2 2 ≤ C6 2 ε2n |∇un | + V |un | |un | . εn Bεκ (xn ) Bεκ (xn ) n

n

Thus, by (3.4), by the boundedness assumption on the energy and by (3.6), we deduce from (3.9) that ( ) ) ( ) ( N p ) (∫ ( ) N2 1− (N −2)p γ−α N p γ−α N +α κ N −2+ p−1 N −1 κ N + N +α p−1 2 2 N +α ≤ C 7 εn |un | εn Bεκ (xn ) n

and we then have in view of the identities (3.7) and (3.8) that ∫ 1 2 |un | > 0. lim inf ( γ−α ) n→∞ κ N + p−1 Bεκ (xn ) n εn On the other hand we have ∫ 1 lim sup ( pγ−α ) n→∞

κ N + p−1 εn

1

2

V |un | ≤ lim sup n→∞

Bεκ (xn ) n

(

) pγ−α

κ N + p−1 εn

∫ RN

(3.10)

2

2

ε2n |∇un | + V |un | < +∞.

(3.11)

By combining (3.10) and (3.11), we deduce that lim sup n→∞

1 εκγ n

inf

V < +∞.

(3.12)

Bεκ (xn ) n

We claim that there exists a point x∗ ∈ RN such that V (x∗ ) = 0 and up to a subsequence, the sequence (xn )n∈N satisfies the condition that lim sup n→∞

|xn − x∗ | < +∞. εκn

(3.13)

In fact, by (3.12), there is a sequence yn ∈ Bεκn (xn ) such that lim sup n→∞

1 V (yn ) < +∞. εκγ n

By Lemma 3.3, this implies that γ

lim sup n→∞

min1≤i≤k (|yn − ai | ) < +∞, εκγ n

where {a1 , . . . , ak } = V −1 ({0}). Thus, up to a subsequence, there exists a point x∗ ∈ {a1 , . . . , ak } such that lim sup n→∞

|yn − x∗ | < +∞, εκn

and thus lim sup n→∞

|xn − x∗ | |xn − x∗ | ≤ 1 + lim sup < +∞. εκn εκn n→∞

J. Van Schaftingen, J. Xia / Nonlinear Analysis 161 (2017) 87–107

102

In particular by (3.10), there exists R∗ > 0 such that ∫ 1 lim inf ( γ−α ) n→∞

κ N + p−1 εn

2

|un | > 0.

(3.14)

BR∗ εκ (x∗ ) n

We define now for each n ∈ N the rescaled function vn : RN → R for each y ∈ RN by α−γ

κ

vn (y) = εn 2(p−1) un (x∗ + εκn y). Let W ∈ C(RN ) be a positive γ-homogeneous function such that lim inf x→x∗

V (x) − W (x − x∗ ) ≥ 0. γ |x − x∗ |

We observe that since W is positive, this is equivalent to having lim sup x→x∗

W (x − x∗ ) ≤ 1. V (x)

(3.15)

We now compute for each R > 0 and n ∈ N, )∫ ( ∫ 1 W (x − x∗ ) 2 2 2 2 |∇vn | + W |vn | ≤ ( pγ−α ) ε2n |∇un | + V |un | , sup κ N + γ−α V (x) N |x−x∗ |≤Rεκ BR R n εn and thus in view of (3.15), for every R > 0, ∫ 2 2 |∇vn | + W |vn | ≤ lim lim sup n→∞

1

n→∞

BR

(

) pγ−α

κ N + γ−α εn

∫ RN

2

2

ε2n |∇un | + V |un | < +∞.

(3.16)

By Lemma 3.5, the sequence (vn )n∈N is bounded in H 1 (BR ). By weak compactness and by a diagonal argument, there exists a function v∗ : RN → R such that for each R > 0, one has v∗ ∈ H 1 (BR ) and the sequence (vn )n∈N converges weakly to v∗ in the space H 1 (BR ). By the lower semicontinuity of the norm, by (3.16) and by (3.4) and by the boundedness assumption, we have ∫ ∫ 2 2 2 2 |∇v∗ | + W |v∗ | = lim |∇v∗ | + W |v∗ | R→∞ N R BR ∫ 2 2 ≤ lim lim inf |∇vn | + W |vn | (3.17) R→∞ n→∞ BR ∫ 1 2 2 ≤ lim sup ( pγ−α ) ε2n |∇un | + V |un | < +∞. κ N + N n→∞ R γ−α εn Moreover, in view of Rellich’s compact embedding theorem, the sequence (vn )n∈N converges strongly to v∗ in L2 (BR ) and thus in view of (3.14), ∫ ∫ 2 2 |v∗ | = lim |vn | > 0. B R∗

n→∞

BR∗

We observe that for each n ∈ N, the function vn satisfies the equation ( p) p−2 − ∆vn + Vn vn = Iα ∗ |vn | |vn | vn , where the rescaled potential Vn is defined for each y ∈ RN by Vn (y) =

) 1 ( V x∗ + εκn y . εκγ n

(3.18)

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103

In order to pass to the limit in (3.18), we consider a test function φ ∈ Cc∞ (RN ). We first have by the weak convergence on balls ∫ ∫ lim ∇vn · ∇φ = ∇v∗ · ∇φ. (3.19) n→∞

RN

RN

If we assume that φ ≥ 0, since for each n ∈ N the function vn is nonnegative, we deduce by Fatou’s lemma that ∫ ∫ lim inf V n vn φ ≥ W v∗ φ. (3.20) n→∞

RN

RN

We finally study the Riesz potential term. We take R > 0 large enough such that supp φ ⊂ BR . Since vn ⇀ v∗ in H 1 (BR ), thus we have, as n → ∞, )( 1 1) 1 ( p−2 p−2 − , (3.21) χB2R |vn | vn → χB2R |v∗ | v∗ in Lq (RN ) with > p − 1 q 2 N where χBR denotes the characteristic function of the ball BR . By the Hardy–Littlewood–Sobolev inequality (2.4), we know that, as n → ∞, (1 1 1) α p p − (3.22) Iα ∗ (χB2R |vn | ) → Iα ∗ (χB2R |v∗ | ) in Lq (RN ) with > p − . q 2 N N p

p

2N

Moreover, since the sequence (|vn | )n∈N converges weakly to |v∗ | in L N +α (RN ) we have [29, Proposition 3.4], as n → ∞, ( ( p) p) Iα ∗ (1 − χB2R )|vn | → Iα ∗ (1 − χB2R )|v∗ |

in L∞ (BR ).

(3.23)

Summarizing (3.22) and (3.23), we obtain that, as n → ∞, p

p

Iα ∗ |vn | → Iα ∗ |v∗ |

in Lq (BR ) with

(1 1 1) α >p − − . q 2 N N

In view of (3.21), we get that (

( p) p−2 p) p−2 Iα ∗ |vn | |vn | vn → Iα ∗ |v∗ | |v∗ | v∗

in Lq (BR ),

as n → ∞, with q ≥ 1 and (1 (1 1) α 1) 1 >p − − + (p − 1) − . q 2 N N 2 N Since supp φ ⊂ BR , we have ∫ lim n→∞

(

p)

Iα ∗ |vn | |vn |

p−2

∫ vn φ =

RN

(

p) p−2 Iα ∗ |v∗ | |v∗ | v∗ φ.

(3.24)

RN

By Eq. (3.18), and by the limits (3.19), (3.20) and (3.24), we have for every φ ∈ Cc∞ (RN ) with φ ≥ 0 that, ∫ ∫ ( p) p−2 ∇v∗ · ∇φ + W v∗ φ ≤ Iα ∗ |v∗ | |v∗ | v∗ φ. (3.25) RN

RN

By (3.17), the function v∗ is an admissible test function and thus ∫ ∫ ( 2 2 p) p |∇v∗ | + W |v∗ | ≤ Iα ∗ |v∗ | |v∗ | . RN

RN

J. Van Schaftingen, J. Xia / Nonlinear Analysis 161 (2017) 87–107

104

There exists thus t∗ ∈ (0, 1] such that ∫ ∫ 2 2 2 2p t∗ |∇v∗ | + W |v∗ | = t∗ RN

(

p) p Iα ∗ |v∗ | |v∗ | .

RN

We have then ∫ ( 1 ) 2p p) p t∗ Iα ∗ |v∗ | |v∗ | − E(W ) ≤ JW (t∗ v∗ ) = 2 2p N ∫ R (1 ( 1) p) p Iα ∗ |vn | |vn | ≤ lim inf lim inf ≤ − n→∞ 2 2p n→∞ RN (1

1 (

pγ−α

κ N + p−1 εn

) Iεn (un ).

In the case where there exists a positive γ-homogeneous function W ∈ C(RN ) such that lim

x→x∗

V (x) − W (x − x∗ ) = 0, γ |x − x∗ |

we observe that equality holds in (3.20) with a limit and thus equality holds also in (3.25), so that the additional conclusion follows. □ Proof of Theorem 1.1. This follows from Propositions 2.3, 3.1 and 3.2. In fact, we have C(x∗ ) = C(x∗ ) ≤ lim inf n→∞

cεn (

pγ−α

κ N + p−1 εn

) ≤ lim sup n→∞

cεn (

pγ−α

κ N + p−1 εn

¯ ) ≤ inf C(x) ≤ C(x∗ ), x∈RN

where cεn = Iεn (uεn ) =

(1 2

−

1) 2p

∫

2

2

ε2n |∇uεn | + V |uεn | .

RN

We thus deduce that C(x∗ ) = JW∗ (v∗ ) = which yields the conclusion.

(1 2

−

1) 2p

∫

2

2

|∇v∗ | + W∗ |v∗ | , RN

□

4. Asymptotics for a potential vanishing on an open set This last section is devoted to the proof of Theorem 1.2 which covers the case where the potential vanishes on the closure of smooth bounded open set. Proof of Theorem 1.2. The existence of solutions for every ε ∈ (0, ε0 ) follows immediately from Proposition 2.3 with ε0 > 0. We define the auxiliary functional Kε ∈ C 1 (HV1 (RN )) for each v ∈ HV1 (RN ) by ∫ ∫ ( V 2 1 1 2 p) p |∇v| + 2 |v| − Iα ∗ |v| |v| , Kε (v) = 2 RN ε 2p RN and we observe that for every u ∈ HV1 (RN ), ( ) 2p 1 Kε ε− p−1 u = ε− p−1 Iε (u). Hence, we define for every ε > 0, the function 1

vε = ε− p−1 uε .

J. Van Schaftingen, J. Xia / Nonlinear Analysis 161 (2017) 87–107

105

We also consider the functional K∗ ∈ C 1 (E) defined for each v ∈ E by ∫ ∫ ( 1 1 2 p) p |∇v| − Iα ∗ |v| |v| , K∗ (v) = 2 Ω 2p Ω where { } E = u ∈ HV1 (RN ) | u = 0 in RN \ Ω . We observe that for every v ∈ E, we have K∗ (v) = Kε (v), and thus, for every ε > 0, since uε is a groundstate, { } 1 N ′ Kε (vε ) = inf Kε (v) 0 ε (v), v⟩ = } { | v ∈ HV (R ) \ {0} and ⟨K ≤ c∗ = inf K∗ (v) | v ∈ E \ {0} and ⟨K∗′ (v), v⟩ = 0 . We deduce therefrom that for every ε > 0, we have ∫ V 2p 2 2 |∇vε | + 2 |vε | ≤ c∗ . ε p−1 RN On the other hand, by Lemma 2.1, we have if ε ≤ ε0 , ∫ ∫ ∫ V V 2 2 2 2 2 2 |∇vε | + |vε | ≤ C1 |∇vε | + 2 |vε | ≤ C1 |∇vε | + 2 |vε | ε ε N N N R R R 0

(4.1)

and thus ∫

2

ε→0

2

|∇vε | + |vε | < +∞.

lim sup RN

It follows that there exists a sequence (εn )n∈N in (0, +∞) converging to 0 such that the sequence (vεn )n∈N converges weakly in H 1 (RN ) to some function v∗ ∈ H 1 (RN ). By Rellich’s compactness theorem, this sequence also converges strongly in L2loc (RN ). ¯ ⊂ U , then, since lim inf |x|→∞ V (x) > 0, we have inf N V > 0, and If the set U ⊂ RN is open and if Ω R \U thus for every εn > 0, ∫ ∫ V ε2n 2 2 2 |∇vεn | + 2 |vεn | , |vεn | ≤ inf V ε N N N R R \U n R \U so that ∫ lim

n→∞

RN \U

2

|vεn | = 0.

It follows thus that the sequence (vεn )n∈N converges strongly to v∗ in L2 (RN ). By the Gagliardo–Nirenberg– Sobolev interpolation inequality we have ( ) ( ) ∫ (∫ ) N2 N p −1 (∫ ) N2 1− (N −2)p 2N p N +α N +α 2 2 2 |∇(vεn − v∗ )| + |vεn − v∗ | |vεn − v∗ | , |vεn − v∗ | N +α ≤ C2 RN

RN

RN

2N p

so that in view of (4.1), the sequence (vεn )n∈N converges also strongly to v∗ in L N +α (RN ). Moreover, we ¯. also have v∗ = 0 on RN \ Ω In view of the Hardy–Littlewood–Sobolev inequality (2.4), the classical Sobolev inequality and of (4.1), we have, for each n ∈ N, ∫ (∫ ) Np (∫ ) Np 2N p V 2 2 N +α 2 2 N +α N +α |vεn | ≤ C3 |∇vεn | + |vεn | ≤ C4 |∇vεn | + 2 |vεn | εn RN RN RN (∫ ( ) Np (∫ 2N p )p p) p N +α = C4 Iα ∗ |vεn | |vεn | ≤ C5 |vεn | N +α . RN

RN

106

J. Van Schaftingen, J. Xia / Nonlinear Analysis 161 (2017) 87–107

Since p > 1, we deduce that ∫

∫

2N p

2N p

|v∗ | N +α = lim

n→∞

RN

RN

|vεn | N +α > 0,

and thus v∗ ̸= 0. If we consider now a test function w ∈ E, we have for every n ∈ N, ∫ ∫ ( V p) p−2 0= ∇vεn · ∇w + 2 vεn w − Iα ∗ |vεn | |vεn | vεn w RN ∫RN ∫ εn ( p) p−2 = ∇vεn · ∇w − Iα ∗ |vεn | |vεn | vεn w. Ω

Ω 2N p

By the weak convergence of the sequence (vεn )n∈N in H 1 (RN ), by its strong convergence in L N +α (RN ) and by the Hardy–Littlewood–Sobolev inequality (2.4), we deduce that ∫ ∫ ( p) p−2 ∇v∗ · ∇w − Iα ∗ |v∗ | |v∗ | v∗ w Ω Ω (∫ ) ∫ ( p) p−2 = lim ∇vεn · ∇w − Iα ∗ |vεn | |vεn | vεn w = 0. n→∞

Ω

Ω

In view of the regularity assumptions on the set Ω and by classical regularity theory, the function v∗ satisfies the announced equation. We also have ∫ ∫ ∫ ( V 2 2 2 p) p |∇vεn | + 2 |vεn | = lim Iα ∗ |vεn | |vεn | lim sup |∇vεn | ≤ lim n→∞ n→∞ ε n→∞ RN RN RN n∫ ∫ ( p) p 2 = Iα ∗ |v∗ | |v∗ | = |∇v∗ | . RN

RN

This implies that the sequence (vεn )n∈N converges strongly in H 1 (RN ) to v∗ and that the function v∗ is a groundstate of the limiting equation. □ Acknowledgments J. Van Schaftingen was supported by the Projet de Recherche (Fonds de la Recherche Scientifique—FNRS) T.1110.14 “Existence and asymptotic behavior of solutions to systems of semilinear elliptic partial differential equations”. Jiankang Xia is partially supported by NSF of China (NSFC-11271201) and he acknowledges the support of the China Scholarship Council and the hospitality of the Universit´e catholique de Louvain (Institut de Recherche en Math´ematique et en Physique). References

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