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A note on Kirchhoff-type equations with Hartree-type nonlinearities
Nonlinear Analysis 99 (2014) 35–48
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A note on Kirchhoff-type equations with Hartree-type nonlinearities Dengfeng Lü ∗ School of Mathematics and Statistics, Hubei Engineering University, Hubei 432000, PR China
article
info
abstract
Article history: Received 15 October 2013 Accepted 24 December 2013 Communicated by S. Carl
We consider the following Kirchhoff-type equation in R3
MSC: 35J60 35Q55 35J10
where a > 0, b ≥ 0 are constants, α ∈ (0, 3), p ∈ (2, 6 − α), µ > 0 is a parameter and g (x) is a nonnegative continuous potential satisfying some conditions. By using the Nehari manifold and the concentration compactness principle, we establish the existence of ground state solutions for the equation if the parameter µ is large enough. Moreover, some concentration behaviors of these solutions as µ → +∞ are discussed. © 2014 Elsevier Ltd. All rights reserved.
− a+b
Keywords: Kirchhoff-type equation Hartree-type nonlinearity Variational methods Ground state solution
1 p ∗ | u | |u|p−2 u, |∇ u|2 dx ∆u + (1 + µg (x))u = |x|α R3
1. Introduction and main results In this paper, we are concerned with the existence and concentration of ground state solutions for the following Kirchhoff-type equation in R3 :
− a + b
R3
|∇ u| dx 1u + Vµ (x)u = 2
1
|x|α
∗ | u|
p
|u|p−2 u,
(Pµ )
u ∈ H (R ), 1
3
where a > 0, b ≥ 0 are constants, Vµ (x) = 1 + µg (x), µ > 0 is a parameter and g (x) is a continuous potential function on R3 , α ∈ (0, 3), p ∈ (2, 6 − α) and ∗ is a notation for the convolution of two functions in R3 . It is pointed out in [1] that such equations arise in various models of physical and biological systems, for example, problem (Pµ ) is related to the stationary analogue of the equation
∂ 2u ρ 2 − ∂t
L 2 2 ∂ u ∂u + = 0, dx h 2L 0 ∂ x ∂ x2
P0
E
(1.1)
which was proposed by Kirchhoff in [2] as an extension of the classical D’Alembert wave equation for free vibrations of elastic strings. Kirchhoff’s model considers the changes in the length of the string produced by transverse vibrations. The
∗
Fax: +86 712 2345434. E-mail address: [email protected].
0362-546X/$ – see front matter © 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.12.022
36
D. Lü / Nonlinear Analysis 99 (2014) 35–48
parameters in Eq. (1.1) have the following meanings: ρ is the mass density, P0 is the initial tension, h is the area of the crosssection, E is the Young modulus of the material, and L is the length of the string. After the pioneer work of Lions [3], where a functional analysis approach was proposed, problem (1.1) began to call attention of several researchers. In recent years, many authors studied the following Kirchhoff-type equation
2 − a+b |∇ u| dx 1u + V (x)u = f (x, u) R3 u ∈ H 1 (R3 ), u > 0
in R3 ,
(1.2)
in R , 3
1
where f is a C function and satisfies certain conditions. Some interesting results were obtained. For example, in [4], Wu obtained existence results for nontrivial solutions and a sequence of high energy solutions for problem (1.2) by applying the symmetric mountain pass theorem. Subsequently, Liu and He [5] proved the existence of infinitely many high energy solutions for (1.2) when f is a subcritical nonlinearity which needs not satisfy the usual Ambrosetti–Rabinowitz-type growth conditions. We also note that several existence results have been obtained for (1.2) on a bounded domain Ω ⊂ R3 , see [1,6–8] and we also refer the reader to [9–13] for more results. On the other hand, in many physical applications, the Hartree-type nonlinearities appear naturally, that is f (x, u) = (K (x) ∗ G(u))g (u), where G ∈ C 1 (R, R) and g = G′ . The function K (x) here is usually called the response function. In (Pµ ), if we set a = 1, b = 0, µ = 0, α = 1 and p = 2, it reduces to the following equation
− 1u + u =
1
|x|
u,
2
∗ | u|
u ∈ H 1 (R3 ).
(1.3)
Eq. (1.3) is usually called the Choquard equation which arises in various branches of mathematical physics, such as the quantum theory of large systems of nonrelativistic bosonic atoms and molecules, physics of multiple-particle systems, etc., see for example [14]. Eq. (1.3) was proposed by Choquard in 1976 as an approximation to Hartree–Fock theory for one component plasma [15]. It was also proposed by Penrose [16] as a model for the self-gravitational collapse of a quantum mechanical wave function. Lieb [15] and Lions [17] obtained the existence of solutions for (1.3) by using variational methods. Further results for related problems may be found in [18–22] and the references therein. Recently, Ma and Zhao [23] considered the generalized Choquard equation
− 1u + u =
1
|x|α
p
∗ | u|
|u|p−2 u,
u ∈ H 1 (RN ),
(1.4)
where p ≥ 2. Under some assumptions on N , α and p, they proved that every positive solution of (1.4) is radially symmetric and monotone decreasing about some point. More recently, Clapp and Salazar [24] gave the existence of positive and sign changing solutions of (1.4) when RN and u are replaced by bounded domains Ω and W (x)u respectively. Moroz and Schaftingen [25] showed the regularity, positivity and radial symmetry of the ground state solutions for the optimal range of parameters, and also obtained decay asymptotics at infinity for them. Motivated by the works we mentioned above, in this paper, we study the existence and concentration of ground state solutions for a class of Kirchhoff-type equations involving Such problems are often referred to as Hartree-type nonlinearities. being nonlocal because of the appearance of the terms ( R3 |∇ u|2 dx)1u and R3 |x1|α ∗ |u|p |u|p dx which imply that problem (Pµ ) is no longer a pointwise identity. This phenomenon provokes some mathematical difficulties, which make the study of such problems particularly interesting. To the best of our knowledge, it seems that there is almost no work on the existence of ground state solutions to Eq. (Pµ ), which is just our aim. The main difficulties when dealing with this problem lie in the presence of the nonlocal terms and the lack of compactness due to the unboundedness of the domain R3 . Different from the results mentioned above, by exploiting the Nehari manifold method and the concentration compactness principle, we get the ground state solutions to problem (Pµ ) and the asymptotic behavior of these solutions as µ → +∞. Before stating our main results, we need to introduce some hypotheses on the potential function g (x): (g1 ) g (x) ∈ C (R3 , R) and g (x) ≥ 0 for all x ∈ R3 ; (g2 ) Ω = int g −1 (0) is nonempty with smooth boundary and Ω = g −1 (0); (g3 ) there exists M > 0 such that L({x ∈ R3 |g (x) ≤ M }) < ∞, where L denotes the Lebesgue measure in R3 . This kind of hypotheses was first introduced by Bartsch and Wang [26] in the study of a nonlinear Schrödinger equation. The hypotheses (g1 )–(g3 ) imply that Vµ (x) represents a potential well whose depth is controlled by µ and Vµ (x) is called a steep potential well if µ is large. It is worth mentioning that we do not impose any other hypotheses on the behavior of g (x) for |x| → ∞. The main results in this paper show that the following Kirchhoff-type equation
1 2 p − a+b |∇ u| dx 1u + u = ∗ |u| |u|p−2 u, α | x | Ω u ∈ H01 (Ω ) plays a special role: it can be seen as the limit problem for (Pµ ) as µ → +∞, where Ω = int g −1 (0).
(P∞ )
D. Lü / Nonlinear Analysis 99 (2014) 35–48
37
Our main results for (Pµ ) are the following: Theorem 1.1. Assume that α ∈ (0, 3), p ∈ (2, 6 − α) and (g1 )–(g3 ) are satisfied. Then there exists µ∗ > 0 such that, for all µ ≥ µ∗ , the problem (Pµ ) has at least one ground state solution uµ in H 1 (R3 ). Furthermore, for any sequence µn → +∞, uµn converges in H 1 (R3 ) along a subsequence to a ground state solution of (P∞ ). Theorem 1.2. Let α ∈ (0, 3), p ∈ (2, 6 − α) and (g1 )–(g3 ) hold. Assume that uµn be a sequence of solutions of (Pµn ) with µn → +∞ and such that lim infn→∞ Φµn (uµn ) < ∞, here Φµ is defined in (2.3). Then uµn → u¯ in H 1 (R3 ) such that u¯ is a nontrivial solution of (P∞ ). Remark 1.1. If we let a = 1 and b = 0, (Pµ ) reduces to the Choquard equation, thus our results cover the case for the Choquard equation. The rest of this paper is organized as follows. In Section 2, we introduce some notations, set the variational framework for problem (Pµ ) and present some preliminary results about the Nehari manifold. In Section 3, we prove some important lemmas that will be used for the proofs of the main results. In Section 4, we prove the main results. 2. Variational setting and preliminaries In this paper we will use the following notations:
• • • • • •
C , C1 , C2 , . . . denote various positive constants whose exact values are not important. → (respectively ⇀) denotes strong (respectively weak) convergence. on (1) denotes a quantity such that on (1) → 0 as n → ∞. Br denotes a ball centered at the origin with radius r > 0. The dual space of a Banach space H will be denoted by H −1 . ⟨·, ·⟩ denotes the duality pairing between H −1 and H .
Throughout this paper we suppose that (g1 )–(g3 ) are satisfied. H 1 (R3 ) is the usual Sobolev space endowed with the standard inner product
⟨u, v⟩ =
∇ u∇v + uv dx, R3
and the associated norm ∥u∥2 = ⟨u, u⟩. Let H := {u ∈ H 1 (R3 ) : the inner product
(u, v)H =
R3
g (x)|u|2 dx < +∞} be the Hilbert space equipped with
∇ u∇v + g (x)uv dx R3
and endowed with the equivalent norm ∥u∥2µ = R3 (a|∇ u|2 + Vµ (x)|u|2 )dx. It is clear that, for each q ∈ [2, 2∗ ], there exists aq > 0 (independent of µ) such that if µ ≥ 1 then
|u|q ≤ aq ∥u∥µ ,
∀ u ∈ H,
(2.1)
1 = 6 and |u|q = ( R3 |u|q dx) q is the usual norm in Lq (R3 ). Set where 2∗ = N2N −2 1 |u(x)|p |u(y)|p p p D(u) = ∗ | u | | u | dx = dxdy. α |x − y|α R3 |x| R3 R3 Using the Hardy–Littlewood–Sobolev inequality (see [27, Theorem 4.3])
R3
R3
φ(x)ψ(y) dxdy ≤ C (κ)|φ|r |ψ|s , |x − y|κ
where 0 < κ < 3, 1 < r , s < ∞ and
|D(u)| ≤ C (α)
|u|
6p 6−α
1 r
+
1 s
+
κ 3
∀ φ ∈ Lr (R3 ), ∀ ψ ∈ Ls (R3 ), = 2, we have the estimate of the D(u) as
6−α 3 = C (α)|u|2p pr
dx
R3
(2.2)
6 for every u ∈ H 1 (R3 ), where C (α) is a positive constant and r = 6−α . For using the Sobolev embedding, we let
6p 6−α
∈ (2, 2 ), ∗
that is, p ∈
6−α 3
,6 − α .
38
D. Lü / Nonlinear Analysis 99 (2014) 35–48
By (2.2) we know that D(u) is well-defined in H . Moreover, similar to the proof of Lemma 2.5 in [22], we can get that D(u) ∈ C 1 (H , R). Let
Φµ (u) =
1 2
b
2
∥ u∥ µ +
2
2
|∇ u| dx
4
−
R3
1
1
2p
R3
|x|α
∗ |u|
|u|p dx.
p
(2.3)
In view of the hypotheses (g1 )–(g3 ), (2.2) and D(u) ∈ C 1 (H , R), the functional Φµ (u) is well-defined and Φµ (u) ∈ C 1 (H , R). Hence, the solutions of problem (Pµ ) are the critical points of the energy functional Φµ (u). Moreover, the functional Φµ (u) satisfies the mountain-pass geometry. Lemma 2.1. Let α ∈ (0, 3), p ∈ (2, 6 − α) and (g1 )–(g3 ) hold. Then the functional Φµ (u) satisfies the following conditions. (i) There exist θ , ρ > 0 such that Φµ (u) ≥ θ > 0 for all ∥u∥µ = ρ . (ii) There exists e ∈ H with ∥e∥µ > ρ such that Φµ (e) ≤ 0. Proof. (i) By (2.2) and (2.1), we have that
Φµ (u) =
≥ ≥
1 2 1 2 1 2
2
b
2
2
|∇ u| dx
∥ u∥ µ +
4
∥u∥2µ −
2p
−
R3
C0
1
1
2p
R3
|x|α
p
|u|p dx
∗ | u|
|u|2p pr
∥u∥2µ − C1 ∥u∥2p µ,
(2.4)
where C1 > 0 is independent of µ. Since p > 2, thus we can choose some θ , ρ > 0 such that Φµ (u) ≥ θ > 0 for all
∥u∥µ = ρ .
(ii) First we note that, for each µ > 0, Φµ (0) = 0. Furthermore, since p > 2, we have that
lim Φµ (tu) = lim
t →+∞
t →+∞
t2 2
∥u∥2µ +
bt 4
2
4
|∇ u|2 dx
−
R3
t 2p
1
2p
R3
|x|α
∗ |u|p |u|p dx = −∞.
Hence, we can choose t0 > 0 large enough such that ∥t0 u∥µ > ρ and Φµ (t0 u) < 0. Let e = t0 u, then (ii) holds.
In order to find the weak solutions of problem (Pµ ), we consider the Nehari manifold
Mµ = u ∈ H \ {0} : γ (u) = 0 ,
where
γ ( u) =
⟨Φµ′ (u), u⟩
2
2
= ∥u∥µ + b
2
|∇ u| dx
1
R3
|x|α
−
R3
p
∗ | u|
|u|p dx.
We define the minimax cµ as cµ = inf Φµ (u).
(2.5)
u∈Mµ
Now we are ready to prove some results concerning the Nehari manifold Mµ . Lemma 2.2. For any u ∈ Mµ there exist σ , δ > 0 such that ∥u∥µ ≥ σ and ⟨γ ′ (u), u⟩ ≤ −δ . Proof. For any u ∈ Mµ , by (2.2) and (2.1), we have 0 = ⟨Φµ′ (u), u⟩ 2
= ∥u∥µ + b
2
2
|∇ u| dx R3
1
R3
|x|α
−
2 2p ≥ ∥u∥2µ − C0 |u|2p pr ≥ ∥u∥µ − C2 ∥u∥µ .
p
∗ | u|
|u|p dx
D. Lü / Nonlinear Analysis 99 (2014) 35–48
39
Note that p > 2, thus there exists σ > 0 such that ∥u∥µ ≥ σ . Furthermore,
⟨γ (u), u⟩ = 2∥u∥µ + 4b ′
2
2
2
|∇ u| dx
− 2p
R3
= (2 − 2p)∥u∥µ + (4b − 2bp)
p
|x|α 2
R3
2
1
∗ | u|
|u|p dx
|∇ u|2 dx R3
≤ −(2p − 2)σ 2 < 0. By Lemma 2.2, Mµ is a smooth manifold in H . It is easy to see that Φµ is well-defined and smooth on Mµ . Moreover, we can proceed analogously to the proof of Theorem 4.3 in [28] to show that if u is a critical point of Φµ constrained to Mµ , then u is a nontrivial solution for (Pµ ). Lemma 2.3. Φµ is bounded from below by a positive constant for all u ∈ Mµ . Proof. Let u ∈ Mµ , by the definition of Mµ and Lemma 2.2 we obtain
Φµ (u) =
1 2
=
∥ u∥ µ + 1 2
>
b
2
1 2
−
−
1
4
2
|∇ u| dx
−
R3
2
∥ u∥ µ +
2p 1
2
b
1
2p
R3
|x|α
∗ | u|
p
|u|p dx
2
b
−
4
1
2
|∇ u| dx
2p
R3
σ 2 > 0.
2p
Lemma 2.4. For any 0 ̸= u ∈ H , there is a unique tu > 0 such that tu u ∈ Mµ . Moreover, Φµ (tu u) = maxt >0 Φµ (tu). Proof. Let u ∈ H \ {0} be fixed and for t > 0, we consider the fibering maps f : t → Φµ (tu) defined by f (t ) := Φµ (tu) =
t2 2
2
∥ u∥ µ +
bt 4
2
2
|∇ u| dx
4
−
t 2p
1
2p
R3
|x|α
R3
p
∗ | u|
|u|p dx.
First we claim that f (t ) > 0 for t > 0 small. Indeed, by (2.4), we have that f (t ) ≥
t2 2
∥u∥2µ − C1 t 2p ∥u∥2p µ,
so f (t ) > 0 if t > 0 is small enough. Using the similar argument as in the proof of Lemma 2.1(ii), we see that f (t ) → −∞ as t → +∞. Hence, there exists tu > 0 such that f (t ) has a positive maximum and f ′ (tu ) = 0. On the other hand, we notice that if tu ∈ Mµ , then t must be such that 0 = ⟨Φµ′ (tu), tu⟩ = t 2 ∥u∥2µ + bt 4
2
|∇ u|2 dx
− t 2p
1
R3
|x|α
R3
∗ |u|p |u|p dx
and setting
β = ∥u∥µ , 2
η=
2
2
|∇ u| dx R3
,
ϑ=
1
R3
|x|α
p
∗ | u|
|u|p dx,
we are led to find a positive solution of t 2 (β + bηt 2 − ϑ t 2p−2 ) = 0 with β, η, ϑ > 0 and b ≥ 0. Obviously, since p > 2, the equation β + bηt 2 − ϑ t 2p−2 = 0 has a unique solution t = tu > 0. We observe that f ′ (t ) = 1t ⟨Φµ′ (tu), tu⟩. So tu u ∈ Mµ and Φµ (tu u) = maxt >0 Φµ (tu). The proof is completed. 3. The (PS )c condition In this section, the main goal is to show that functional Φµ (u) satisfies the (PS )c condition. Recall that, for a given functional I ∈ C 1 (H , R), we say that a sequence {zn } ⊂ H is a (PS )c sequence if it satisfies I (zn ) → c and I ′ (zn ) → 0
40
D. Lü / Nonlinear Analysis 99 (2014) 35–48
as n → ∞. Moreover, if any (PS )c sequence has a convergent subsequence, then we say that I satisfies the (PS )c condition. In this paper, we will take I = Φµ (u). Lemma 3.1. Under the assumptions of Theorem 1.1 and letting {un } be a (PS )c sequence for Φµ (u), we have (i) {un } is bounded in H ; (ii) either c ≥ c0 for some c0 > 0 independent of µ or c = 0. Proof. (i) Let {un } be a (PS )c sequence for Φµ (u), i.e.,
Φµ (un ) = c + on (1) and Φµ′ (un ) = on (1). Note that p > 2 and b ≥ 0, then we have, c + on (1) −
1 2p
on (∥un ∥µ ) = Φµ (un ) −
=
2
≥
1
1 2
−
−
1 2p
⟨Φµ′ (un ), un ⟩
1
∥un ∥2µ +
2p
b 4
−
b
2
2p
|∇ un |2 dx R3
1 2p
∥un ∥2µ .
Thus
2
∥un ∥µ ≤ c
1 2
−
1
−1 ,
2p
(3.1)
for n large enough and therefore (i) holds. (ii) Since Φµ′ (un ) = on (1), we have that on (∥un ∥µ ) = ⟨Φµ′ (un ), un ⟩ 2
2
= ∥un ∥µ + b
2
|∇ un | dx
1
R3
|x|α
−
R3
p
∗ | un |
|un |p dx
≥ ∥un ∥2µ − C2 ∥un ∥2p µ, since p > 2, there exists σ1 > 0(σ1 < 1) such that
⟨Φµ′ (un ), un ⟩ ≥ Now, if c <
(p−1)σ12 2p
∥un ∥2µ ,
for ∥un ∥µ < σ1 .
(3.2)
and {un } is a (PS )c -sequence of Φµ , then by (3.1), we get that
lim ∥un ∥2µ ≤
n→∞
1 4
2pc p−1
< σ12 .
Hence, ∥un ∥µ < σ1 for n large, then by (3.2), we have 1 4
∥un ∥2µ ≤ ⟨Φµ′ (un ), un ⟩ = on (1)∥un ∥µ ,
which implies ∥un ∥µ → 0 as n → ∞ and c = 0, it follows that (ii) holds for c0 =
(p−1)σ12 2p
.
Next, we recall that pointwise convergence of a bounded sequence implies weak convergence (see [29, Proposition 5.4.7]). Lemma 3.2. Let N ≥ 3, q ∈ (1, +∞) and {un } be a bounded sequence in Lq (RN ). If un → u almost everywhere in RN as n → ∞, then un ⇀ u weakly in Lq (RN ). Then we have the following Brézis–Lieb type lemma for the nonlocal term D(u). Lemma 3.3. Let α ∈ (0, 3), p ∈ ( 6−α , 6 − α). If {un } ⊂ H is a bounded sequence such that un → u almost everywhere in R3 3 as n → ∞, then the following hold. (i) D(un ) − D(un − u) → D(u) as n → ∞. (ii) D′ (un ) − D′ (un − u) → D′ (u) in H −1 as n → ∞.
D. Lü / Nonlinear Analysis 99 (2014) 35–48
41
Proof. The proof is analogous to that of Lemma 3.5 in [18], but we exhibit it here for completeness. First, similarly to the proof of the Brézis–Lieb Lemma [30], we know that 6
|un − u|p − |un |p → |u|p in L 6−α (R3 ) as n → ∞.
(3.3)
By the Hardy–Littlewood–Sobolev inequality, we can deduce that 1
∗ (|un − u|p − |un |p ) →
|x|α
1
6
∗ |u|p in L α (R3 ) as n → ∞.
|x|α
(3.4)
We notice that
1
|x|α
R3
=
∗ | un |
p
|un | dx −
1
1
p
|x|α
R3
|x|α
R3
p
∗ | un − u|
|un − u|p dx
∗ (|un | − |un − u| ) (|un | − |un − u| )dx + 2 p
p
p
p
1
R3
|x|α
∗ (|un | − |un − u| ) |un − u|p dx. p
p
(3.5)
On the other hand, by Lemma 3.2, we have that 6
|un − u|p ⇀ 0 in L 6−α (R3 ) as n → ∞.
(3.6)
Combining (3.3)–(3.6), conclusion (i) holds. The proof of conclusion (ii) is similar to (i), and is omitted here.
Lemma 3.4. Under the assumptions of Theorem 1.1, let µ > 0 be fixed and {un } be a (PS )c -sequence of Φµ . Then up to a subsequence un ⇀ u in H with u being a weak solution of (Pµ ). Moreover, Φµ (un − u) → c − Φµ (u) and Φµ′ (un − u) → 0 as n → ∞. Proof. By Lemma 3.1(i), we know that {un } is bounded in H . Therefore, there is a subsequence of {un } such that un ⇀ u in H as n → ∞. In order to see that u is a critical point of Φµ we recall that un ⇀ u in H , un → u, a.e. in R3 , un → u in Lsloc (R3 ), 2 ≤ s < 6. Moreover, there exists A ∈ R, such that
|∇ un |2 dx → A as n → ∞. R3
Then by Fatou’s lemma we get that
R3
|∇ u|2 dx ≤ A.
We claim that R3 |∇ u|2 dx = A. Arguing by contradiction, we assume that H , for any ϕ ∈ H , we have
a∇ u∇ϕ dx + R3
R3
(1 + µg (x))uϕ dx + Ab
1
R3
|x|α
∇ u∇ϕ dx − R3
R3
|∇ u|2 dx < A. By Φµ′ (un ) → 0 and un ⇀ u in p
∗ | u|
|u|p−2 uϕ dx = 0.
Then ⟨Φµ′ (u), u⟩ < 0. On the other hand, by Lemma 2.4 it is easy to get that ⟨Φµ′ (tu), tu⟩ > 0 for t > 0 is small enough. Hence, there exists t0 ∈ (0, 1) satisfying ⟨Φµ′ (t0 u), t0 u⟩ = 0. Moreover, Φµ (t0 u) = max0≤t ≤1 Φµ (tu). So 1
cµ ≤ Φµ (t0 u) = Φµ (t0 u) −
= <
t02 4 1 4
2
∥ u∥ µ + 2
∥ u∥ µ +
= lim inf
1 4
n→∞
1 4
1 4
−
−
1
2p
R3
2
∥un ∥µ +
1 4
1 4
−
|x|α
R3
1
= lim inf Φµ (un ) −
1
2p
t0
2p
n→∞
⟨Φµ′ (t0 u), t0 u⟩ 4
∗ |u|p |u|p dx
1
p
∗ | u| |x|α
1
2p
R3
|u|p dx 1
|x|α
∗ | un |
⟨Φµ′ (un ), un ⟩
p
= cµ ,
p
|un | dx
42
D. Lü / Nonlinear Analysis 99 (2014) 35–48
which is a contradiction. Then
|∇ u|2 dx = A = lim
n→∞
R3
|∇ un |2 dx.
R3
(3.7)
Thus for any ϕ ∈ H , we have
⟨Φµ′ (u), ϕ⟩ = lim ⟨Φµ′ (un ), ϕ⟩ = 0, n→∞
which shows that u is a weak solution of (Pµ ). Now we consider a new sequence un = un − u. By the Brézis–Lieb lemma [30], we have that
∥ un ∥2µ = ∥un ∥2µ − ∥u∥2µ + on (1), 2 2 |∇ un |2 dx
(3.8)
|∇ un |2 dx
=
R3
2
+ on (1).
|∇ u|2 dx
−
R3
R3
(3.9)
Next we prove that
Φµ ( un ) = c − Φµ (u) as n → ∞
(3.10)
Φµ′ ( un ) → 0 as n → ∞.
(3.11)
and
Using (3.8) and (3.9) we get that
Φµ ( un ) =
=
1 2 1 2
b
2
∥ un ∥ µ +
2
2
|∇ un | dx
4 1
b
2
4
∥un ∥2µ − ∥u∥2µ +
−
−
R3
1
1
2p
R3
|x|α
1
2p
R3
|∇ un |2 dx
p
∗ | un | |x|α
2
1
−
R3
b
| un |p dx 2
|∇ u|2 dx
4
R3
∗ | un − u|
= Φµ (un ) − Φµ (u) +
|un − u|p dx + on (1)
p
1 D(un ) − D(u) − D(un − u) + on (1). 2p
(3.12)
From Lemma 3.3(i), D(un ) − D(u) − D(un − u) → 0 as n → ∞. Thus from (3.12) we obtain (3.10). In order to show (3.11), let v ∈ H , it is easy to see that
⟨Φµ′ ( un ), v⟩
=
⟨Φµ′ (un ), v⟩
−
1
R3
|x|α
+
⟨Φµ′ (u), v⟩
+ on (1) −
p
∗ | un |
| un |
p−2
un v dx −
1
R3
|x|α
1
R3
|x|α
| un |p−2 un v dx
p
∗ | un |
p
∗ | u|
|u|p−2 uv dx.
By Lemma 3.3(ii), we have that
lim
sup
n→∞ ∥v∥ ≤1 µ
1
p
|x|α
R3
∗ | un |
p−2
| un |
un −
1
|x|α
p
∗ | un |
p−2
|un |
un +
1
|x|α
p
∗ | u|
p−2
|u|
u v dx = 0.
Thus we have lim ⟨Φµ′ ( un ), v⟩ = 0,
n→∞
∀v ∈ H ,
which implies (3.11) and this completes the proof of Lemma 3.4.
Lemma 3.5. Let α ∈ (0, 3), p ∈ ( 6−α , 6 − α) and C ∗ be fixed. Given ε > 0 there exist µε = µ(ε, C ∗ ) > 0 and 3 ∗ Rε = R(ε, C ) > 0 such that, if {un } is a (PS )c -sequence of Φµ (u) with c ≤ C ∗ , µ ≥ µε , then
lim sup n→∞
R3 \BRε
1
|x|α
p
∗ | un |
|un |p dx ≤ ε.
(3.13)
D. Lü / Nonlinear Analysis 99 (2014) 35–48
43
Proof. For R > 0, we set
ΩR+ := {x ∈ R3 : |x| ≥ R, g (x) ≥ M },
ΩR− := {x ∈ R3 : |x| ≥ R, g (x) < M },
then
ΩR+
|un |2 dx ≤ ≤ ≤
≤
1 1 + µM
R3
(1 + µg (x))|un |2 dx
1 1 + µM 1
(a|∇ un |2 + (1 + µg (x))|un |2 )dx R3
2pc
1 + µM
1
+ on (∥un ∥µ )
p−1
1 + µM
(by (3.1))
2pC ∗
+ on (1)
p−1
→ 0 as µ → +∞.
(3.14)
Using the Hölder inequality, (2.1) and Lemma 3.1(ii), for 1 < q < 3 we have
ΩR−
|un |2 dx ≤
|un |2q dx
dx ΩR−
R3
≤ C ∥un ∥2µ · L(ΩR− ) ≤ C
q−q 1
1q
2pC ∗ p−1
· L(ΩR− )
q−1 q q−1 q
→ 0 as R → ∞,
(3.15)
where C = C (q) is a positive constant. Using the Hardy–Littlewood–Sobolev inequality, we have
R3 \ΩR−
Setting ℓ =
1
|x|α
∗ |un |p |un |p dx ≤ C
3(p−2)+α , 2p
R3 \ΩR−
|un |
6p 6−α
R3 \ΩR−
|un |
6p 6−α
6−α 3 .
dx
(3.16)
by using the Gagliardo–Nirenberg inequality, we get
dx ≤ C R3 \ΩR−
63p−αℓ |∇ un |2 dx · 63p−αℓ |∇ un | dx ·
1−ℓ) 3p6(−α
R3 \ΩR−
2
≤ C R3 \ΩR− 6pℓ 6−α
≤ C ∥ un ∥ µ
· ΩR+
|un |2 dx +
|un |2 dx
ΩR+
ΩR−
1−ℓ) 3p6(−α
2
|un | dx +
2
ΩR−
|un | dx
1−ℓ) 3p6(−α
|un |2 dx
→ 0 as µ, R → ∞ (by (3.14) and (3.15)). Combining (3.16) and (3.17), we conclude the proof of Lemma 3.5.
(3.17)
Then we have the following compactness result. Lemma 3.6. Let α ∈ (0, 3), p ∈ (2, 6 − α) and (g1 )–(g3 ) hold. Then for any C0 > 0, there exists µ0 > 0 such that Φµ satisfies the (PS )c -condition for all c ≤ C0 and µ ≥ µ0 . Proof. Let c0 > 0 be given by Lemma 3.1(ii) and choose ε > 0 such that ε < p−01 . Then for given C0 > 0, we choose µε > 0 and Rε > 0 as in Lemma 3.5. We claim that µ0 = µε is required in Lemma 3.6. Let {un } ⊂ H be a (PS )c -sequence of Φµ (u) with µ ≥ µ0 and c ≤ C0 . By Lemma 3.4, we may suppose that un ⇀ u in H and un = un − u is a (PS )c ′ -sequence of Φµ with c ′ = c − Φµ (u). Now we claim c ′ = 0. In fact, if c ′ ̸= 0, then by Lemma 3.1(ii) we obtain that c ′ ≥ c0 > 0. Since { un } is a (PS )c ′ -sequence of Φµ , we have pc
Φµ ( un ) = c ′ + on (1) and Φµ′ ( un ) = on (1).
44
D. Lü / Nonlinear Analysis 99 (2014) 35–48
Then we get, c ′ + on (1) −
1 2
on (∥ un ∥µ ) = Φµ ( un ) − b
=− ≤
2
⟨Φµ′ ( un ), un ⟩ 2 2
|∇ un | dx
4
R3
1
1
2
1
−
2p
+
−
2
1 2p
R3
1
|x|α
p
∗ | un |
| un |p dx
1
| un |p dx,
p
∗ | un |
|x|α
R3
1
(3.18)
from (3.18) we deduce that
1
R3
|x|α
lim
n→∞
p
p
∗ | un |
| un | dx ≥ c
′
1
−
2
−1
1
≥
2p
2pc0 p−1
.
On the other hand, by Lemma 3.5 we have that
lim sup n→∞
1
∗ | un |
|x|α
R3 \BRε
| un |p dx ≤ ε <
p
pc0 p−1
.
This implies un ⇀ u in H with u ̸= 0, which is a contradiction. Therefore c ′ = 0 and it follows from (3.1) that lim ∥ un ∥2µ ≤
n→∞
2pc ′
= 0,
p−1
hence un → 0 in H , that is, un → u in H . This completes the proof of Lemma 3.6.
In the following, we will show that there exists uµ ∈ Mµ with Φµ (uµ ) = cµ , that is, uµ is a ground state solution of (Pµ ).
Theorem 3.1. Assume that α ∈ (0, 3), p ∈ (2, 6 − α) and (g1 )–(g3 ) are satisfied. Then there exists µ∗ > 0 such that cµ is achieved for all µ ≥ µ∗ at some uµ which is a ground state solution of (Pµ ). Proof. By Lemma 2.1, Φµ satisfies the mountain-pass geometry, then using a version of the mountain-pass theorem without the (PS) condition, there exists {un } ⊂ H such that Φµ (un ) → cµ and Φµ′ (un ) → 0. Moreover, by Lemma 3.1(i) {un } is
bounded in H . Then, up to a subsequence, we may assume that un ⇀ u0 in H and un → u0 for a.e. in R3 . By Lemma 3.6, there exists µ∗ > 0, such that for µ ≥ µ∗ , un → u0 in H . By Lemma 3.4 we have that Φµ′ (u0 ) = 0. In addition, by the definition of cµ and Lemma 2.3, we have cµ > 0, this implies that u0 ̸= 0. Then u0 ∈ Mµ . Using Fatou’s lemma we have that
Φµ (u0 ) = Φµ (u0 ) −
=
1 2
−
1
1 2p
≤ lim inf
2
∥ u0 ∥ µ +
2p
1 2
n→∞
⟨Φµ′ (u0 ), u0 ⟩
−
1
4
−
b
2p
= lim inf Φµ (un ) −
2
∥ un ∥ µ + 1 2p
2
2
|∇ u0 | dx
2p
n→∞
b
R3
b 4
−
b 2p
2
2
|∇ un | dx R3
⟨Φµ′ (un ), un ⟩
= cµ . Hence, Φµ (u0 ) ≤ cµ . On the other hand, by the definition of cµ , we have cµ ≤ Φµ (u0 ). So, Φµ (u0 ) = cµ , take uµ = u0 , then uµ is a ground state solution of problem (Pµ ). 4. Proof of the main results In this section we give the proofs of our main results. First, we consider the following limit problem:
1 − a + b |∇ u|2 dx 1u + u = ∗ |u|p |u|p−2 u, |x|α Ω u ∈ H01 (Ω ), where Ω = int g −1 (0) is defined in assumption (g2 ).
(P∞ )
D. Lü / Nonlinear Analysis 99 (2014) 35–48
45
The energy functional associated with (P∞ ) is defined by
Φ∞ (u) =
1
2
Ω
b
(a|∇ u| + |u| )dx + 2
2
2
2
|∇ u| dx
4
−
Ω
1
1
2p
Ω
|x|α
p
∗ | u|
|u|p dx.
Let ′ M∞ = {u ∈ H01 (Ω ) \ {0} : ⟨Φ∞ (u), u⟩ = 0}
be the Nehari manifold and set c∞ = inf Φ∞ (u). u∈M∞
Then we have the following lemma. Lemma 4.1. limµ→+∞ cµ = c∞ , where cµ is defined in (2.5). Proof. It is easy to see that cµ ≤ c∞ for all µ ≥ 0. Assume that limn→∞ cµn = k < c∞ for a sequence µn → ∞ as n → ∞. Lemma 3.1(ii) implies k > 0 and by Theorem 3.1, for µ large enough, there exists a sequence un ∈ Mµn , a solution of (Pµn ), such that Φµn (un ) = cµn . Since {un } is bounded in H , we may assume that
un ⇀ u, un → u, un → u,
in H , a.e. in R3 , in Lsloc (R3 ), 2 ≤ s < 6.
(4.1)
We claim that u|Ω c = 0, where Ω c = R3 \ Ω . In fact, if u|Ω c ̸= 0, then there exists a compact subset Σ ⊂ Ω c with dist(Σ , ∂ Ω ) > 0 such that u|Σ ̸= 0. Then by (4.1)
2
Σ
|un | dx →
Σ
|u|2 dx > 0.
Moreover, there exists ε0 > 0 such that g (x) ≥ ε0 for any x ∈ Σ . We also note that un ∈ Mµn , then we get that 1
Φµn (un ) =
2
=
R3
1 2
≥
1 2
≥
1 2
(a|∇ un | + (1 + µn g (x))|un | )dx + 2
−
−
−
1
1
R3
2p
2
2
|∇ un | dx
4
Σ
−
R3
(a|∇ un | + (1 + µn g (x))|un | )dx + 2
R3
2p
b
2p 1
2
2
b 4
−
b
1 2p
D(un )
2
2
|∇ un | dx
2p
R3
(1 + µn g (x))|un |2 dx
(1 + µn ε0 )|un |2 dx
→ +∞ as n → ∞. This contradiction shows that u|Ω c = 0. Next we claim that un → u in Ls (R3 ) for 2 < s < 6. Otherwise, by the concentration compactness principle of Lions [31], there exist δ > 0, ρ > 0 and xn ∈ R3 with |xn | → +∞ such that
lim inf n→∞
Bρ (xn )
|un − u|2 dx ≥ δ > 0.
On the other hand, we notice that un ∈ Mµn and L(Bρ (xn ) ∩ {x|g (x) ≤ M }) → 0 as n → ∞, then we have
Φµn (un ) ≥
2
≥
1 2
=
1
1 2
−
−
−
→ +∞,
1
2p 1
Bρ (xn )∩{x|g (x)≥M }
2p 1 2p
µn M
µn M
(a|∇ un |2 + (1 + µn g (x))|un |2 )dx 2
Bρ (xn )
|un − u| dx −
2
Bρ (xn )∩{x|g (x)≤M }
|un − u| dx − on (1) 2
Bρ (xn )
as n → ∞.
This contradiction implies un → u in Ls (R3 ) for 2 < s < 6.
|un − u| dx
46
D. Lü / Nonlinear Analysis 99 (2014) 35–48
Since {un } is bounded, by Fatou’s lemma, we obtain that
n→∞
|∇ un |2 dx ≥
lim inf R3
R3
|∇ u|2 dx.
On the other hand, by the choice of {un }, we have
(a|∇ un | + (1 + µn g (x))|un | )dx + b 2
R3
2
2
2
|∇ un | dx
1
R3
|x|α
=
R3
p
∗ | un |
|un |p dx.
Consequently
(a|∇ u| + |u| )dx + b 2
Ω
2
2
2
|∇ u| dx
≤ lim
n→∞
Ω
(a|∇ un | + |un | )dx + b 2
R3
1
R3
|x|α
≤ lim
n→∞
2
2
2
|∇ un | dx R3
p
∗ | un |
|un |p dx.
(4.2)
In the following we first show that
1
R3
|x|α
lim
n→∞
∗ |un |p |un |p dx =
1
R3
|x|α
∗ |u|p |u|p dx.
(4.3)
Indeed, for given ε > 0, we can argue as in the proof of Lemma 3.5 to conclude that, for some ρ > 0 large, there holds
lim sup n→∞
1
|x|α
ρ (0)
R3 \B
|un |p dx ≤ ε.
p
∗ | un |
By taking ρ larger if necessary, we may suppose that
ρ (0)
R 3 \B
∗ |u|p |u|p dx ≤ ε . Moreover, the local convergence in
1 |x|α
(4.1) and the Lebesgue dominated convergence theorem imply that
Bρ (0)
1
|x|α
p
∗ | un |
p
|un | dx →
1
p
|x|α
Bρ (0)
∗ | u|
|u|p dx
as n → ∞. Since
1 1 p p p ∗ |un | |un | − ∗ |u| |u|p dx α α R3 |x| |x| 1 1 p p p ∗ |un | |un | dx + ∗ |u| |u|p dx ≤ α α R3 \Bρ (0) |x| R3 \Bρ (0) |x| 1 1 p p p p ∗ | u | | u | − ∗ | u | | u | dx + , n n Bρ (0) |x|α |x|α it follows from the above estimates and convergences that
lim sup n→∞
R3
1
|x|α
p
∗ | un |
1
p
| un | −
|x|α
p
∗ | u|
|u|
p
dx ≤ 2ε,
and therefore (4.3) holds. By (4.2) and (4.3) we have
(a|∇ u| + |u| )dx + b 2
Ω
2
2
2
|∇ u| dx
1
R3
|x|α
1
Ω
|x|α
≤
Ω
∗ | u|
p
|u|p dx.
Since u|Ω c = 0, then we get
(a|∇ u| + |u| )dx + b 2
Ω
2
2
2
|∇ u| dx Ω
≤
∗ | u|
p
|u|p dx.
D. Lü / Nonlinear Analysis 99 (2014) 35–48
47
Thus similar to Lemma 2.4, there exists t1 ∈ (0, 1] such that t1 u ∈ M∞ and
Φ∞ (t1 u) ≤ Φ∞ (u), hence c∞ ≤ Φ∞ (t1 u) ≤ Φ∞ (u) ≤ limn→∞ Φµn (un ) = k < c∞ , which is a contradiction and this completes the proof.
Now we give the proofs of Theorems 1.1 and 1.2. Proof of Theorem 1.1. By Theorem 3.1, the first part of Theorem 1.1 has been proved. Now we consider the concentration behavior of the solutions. Let {un } ⊂ H 1 (R3 ) with un ∈ Mµn , Φµn (un ) = cµn and µn → ∞. Since such a sequence must be bounded in H 1 (R3 ), we may assume that un ⇀ u in H 1 (R3 ) and un → u in Lsloc (R3 ) for 2 ≤ s < 6. Next we shall show that u ∈ H01 (Ω ) is a ground state solution of (P∞ ), that is Φ∞ (u) = c∞ , and that un → u in H 1 (R3 ). Using the argument in the proof of Lemma 4.1 we can prove that u|Ω c = 0 and un → u in Ls (R3 ) for 2 ≤ s < 6. Moreover, Φ∞ (u) ≤ limn→∞ Φµn (un ) ≤ c∞ . If u ∈ M∞ , then Φ∞ (u) = c∞ . In order to show u ∈ M∞ , it suffices to show that
|∇ un |2 dx =
lim
n→∞
R3
R3
|∇ u|2 dx,
lim µn
n→∞
R3
g (x)|un |2 dx = 0.
In fact, if one of the above limits does not hold, by Fatou’s lemma, we have
(|∇ u| + |u| )dx + b 2
R3
2
2
2
|∇ u| dx R3
<
1
R3
|x|α
p
∗ | u|
|u|p dx,
similar to the proof of Lemma 4.1, there exists σ0 ∈ (0, 1) such that σ0 u ∈ M∞ and c∞ ≤ Φ∞ (σ0 u) < Φ∞ (u) ≤ lim Φµn (un ) ≤ c∞ , n→∞
which is a contradiction. This completes the proof of Theorem 1.1.
Proof of Theorem 1.2. For any sequence µn → +∞, let un := uµn be the solution of (Pµn ). Since lim infn→∞ Φµn (uµn ) < ∞, it is easy to see that {un } must be bounded in H 1 (R3 ). Therefore, we may assume that un ⇀ u¯ in H 1 (R3 ) and un → u¯ in Lsloc (R3 ) for 2 ≤ s < 6. As in the proof of Lemma 4.1, we can prove that u¯ |Ω c = 0 and un → u¯ in Ls (R3 ) for 2 ≤ s < 6. Furthermore, as in the proof of Theorem 1.1, we can prove that u¯ ∈ H01 (Ω ) is a ground state solution of (P∞ ). To complete the proof it suffices to show that un → u¯ in H 1 (R3 ). In fact, we use the weak convergence of {un }, the fact that un ∈ H 1 (R3 ) is the solution of (Pµn ) and u¯ ∈ M∞ to get that
(a|∇(un − u¯ )|2 + Vµn (x)|un − u¯ |2 )dx 2 2 = (a|∇ un | + Vµn (x)|un | )dx − (a|∇ u¯ |2 + Vµn (x)|¯u|2 )dx + on (1) R3 R3
∥un − u¯ ∥2µn =
R3
1
= R3
|x|α
R3
2
2
|∇ un | dx
−b
1
∗ |un |p |un |p dx −
∗ |¯u|p |¯u|p dx 2
2
|∇ u¯ | dx
+b
R3
|x|α
R3
+ on (1)
= on (1) (by (4.3) and (3.7)), which implies that un → u¯ in H 1 (R3 ). This completes the proof of Theorem 1.2.
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A note on Kirchhoff-type equations with Hartree-type nonlinearities Dengfeng Lü ∗ School of Mathematics and Statistics, Hubei Engineering University, Hubei 432000, PR China
article
info
abstract
Article history: Received 15 October 2013 Accepted 24 December 2013 Communicated by S. Carl
We consider the following Kirchhoff-type equation in R3
MSC: 35J60 35Q55 35J10
where a > 0, b ≥ 0 are constants, α ∈ (0, 3), p ∈ (2, 6 − α), µ > 0 is a parameter and g (x) is a nonnegative continuous potential satisfying some conditions. By using the Nehari manifold and the concentration compactness principle, we establish the existence of ground state solutions for the equation if the parameter µ is large enough. Moreover, some concentration behaviors of these solutions as µ → +∞ are discussed. © 2014 Elsevier Ltd. All rights reserved.
− a+b
Keywords: Kirchhoff-type equation Hartree-type nonlinearity Variational methods Ground state solution
1 p ∗ | u | |u|p−2 u, |∇ u|2 dx ∆u + (1 + µg (x))u = |x|α R3
1. Introduction and main results In this paper, we are concerned with the existence and concentration of ground state solutions for the following Kirchhoff-type equation in R3 :
− a + b
R3
|∇ u| dx 1u + Vµ (x)u = 2
1
|x|α
∗ | u|
p
|u|p−2 u,
(Pµ )
u ∈ H (R ), 1
3
where a > 0, b ≥ 0 are constants, Vµ (x) = 1 + µg (x), µ > 0 is a parameter and g (x) is a continuous potential function on R3 , α ∈ (0, 3), p ∈ (2, 6 − α) and ∗ is a notation for the convolution of two functions in R3 . It is pointed out in [1] that such equations arise in various models of physical and biological systems, for example, problem (Pµ ) is related to the stationary analogue of the equation
∂ 2u ρ 2 − ∂t
L 2 2 ∂ u ∂u + = 0, dx h 2L 0 ∂ x ∂ x2
P0
E
(1.1)
which was proposed by Kirchhoff in [2] as an extension of the classical D’Alembert wave equation for free vibrations of elastic strings. Kirchhoff’s model considers the changes in the length of the string produced by transverse vibrations. The
∗
Fax: +86 712 2345434. E-mail address: [email protected].
0362-546X/$ – see front matter © 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.12.022
36
D. Lü / Nonlinear Analysis 99 (2014) 35–48
parameters in Eq. (1.1) have the following meanings: ρ is the mass density, P0 is the initial tension, h is the area of the crosssection, E is the Young modulus of the material, and L is the length of the string. After the pioneer work of Lions [3], where a functional analysis approach was proposed, problem (1.1) began to call attention of several researchers. In recent years, many authors studied the following Kirchhoff-type equation
2 − a+b |∇ u| dx 1u + V (x)u = f (x, u) R3 u ∈ H 1 (R3 ), u > 0
in R3 ,
(1.2)
in R , 3
1
where f is a C function and satisfies certain conditions. Some interesting results were obtained. For example, in [4], Wu obtained existence results for nontrivial solutions and a sequence of high energy solutions for problem (1.2) by applying the symmetric mountain pass theorem. Subsequently, Liu and He [5] proved the existence of infinitely many high energy solutions for (1.2) when f is a subcritical nonlinearity which needs not satisfy the usual Ambrosetti–Rabinowitz-type growth conditions. We also note that several existence results have been obtained for (1.2) on a bounded domain Ω ⊂ R3 , see [1,6–8] and we also refer the reader to [9–13] for more results. On the other hand, in many physical applications, the Hartree-type nonlinearities appear naturally, that is f (x, u) = (K (x) ∗ G(u))g (u), where G ∈ C 1 (R, R) and g = G′ . The function K (x) here is usually called the response function. In (Pµ ), if we set a = 1, b = 0, µ = 0, α = 1 and p = 2, it reduces to the following equation
− 1u + u =
1
|x|
u,
2
∗ | u|
u ∈ H 1 (R3 ).
(1.3)
Eq. (1.3) is usually called the Choquard equation which arises in various branches of mathematical physics, such as the quantum theory of large systems of nonrelativistic bosonic atoms and molecules, physics of multiple-particle systems, etc., see for example [14]. Eq. (1.3) was proposed by Choquard in 1976 as an approximation to Hartree–Fock theory for one component plasma [15]. It was also proposed by Penrose [16] as a model for the self-gravitational collapse of a quantum mechanical wave function. Lieb [15] and Lions [17] obtained the existence of solutions for (1.3) by using variational methods. Further results for related problems may be found in [18–22] and the references therein. Recently, Ma and Zhao [23] considered the generalized Choquard equation
− 1u + u =
1
|x|α
p
∗ | u|
|u|p−2 u,
u ∈ H 1 (RN ),
(1.4)
where p ≥ 2. Under some assumptions on N , α and p, they proved that every positive solution of (1.4) is radially symmetric and monotone decreasing about some point. More recently, Clapp and Salazar [24] gave the existence of positive and sign changing solutions of (1.4) when RN and u are replaced by bounded domains Ω and W (x)u respectively. Moroz and Schaftingen [25] showed the regularity, positivity and radial symmetry of the ground state solutions for the optimal range of parameters, and also obtained decay asymptotics at infinity for them. Motivated by the works we mentioned above, in this paper, we study the existence and concentration of ground state solutions for a class of Kirchhoff-type equations involving Such problems are often referred to as Hartree-type nonlinearities. being nonlocal because of the appearance of the terms ( R3 |∇ u|2 dx)1u and R3 |x1|α ∗ |u|p |u|p dx which imply that problem (Pµ ) is no longer a pointwise identity. This phenomenon provokes some mathematical difficulties, which make the study of such problems particularly interesting. To the best of our knowledge, it seems that there is almost no work on the existence of ground state solutions to Eq. (Pµ ), which is just our aim. The main difficulties when dealing with this problem lie in the presence of the nonlocal terms and the lack of compactness due to the unboundedness of the domain R3 . Different from the results mentioned above, by exploiting the Nehari manifold method and the concentration compactness principle, we get the ground state solutions to problem (Pµ ) and the asymptotic behavior of these solutions as µ → +∞. Before stating our main results, we need to introduce some hypotheses on the potential function g (x): (g1 ) g (x) ∈ C (R3 , R) and g (x) ≥ 0 for all x ∈ R3 ; (g2 ) Ω = int g −1 (0) is nonempty with smooth boundary and Ω = g −1 (0); (g3 ) there exists M > 0 such that L({x ∈ R3 |g (x) ≤ M }) < ∞, where L denotes the Lebesgue measure in R3 . This kind of hypotheses was first introduced by Bartsch and Wang [26] in the study of a nonlinear Schrödinger equation. The hypotheses (g1 )–(g3 ) imply that Vµ (x) represents a potential well whose depth is controlled by µ and Vµ (x) is called a steep potential well if µ is large. It is worth mentioning that we do not impose any other hypotheses on the behavior of g (x) for |x| → ∞. The main results in this paper show that the following Kirchhoff-type equation
1 2 p − a+b |∇ u| dx 1u + u = ∗ |u| |u|p−2 u, α | x | Ω u ∈ H01 (Ω ) plays a special role: it can be seen as the limit problem for (Pµ ) as µ → +∞, where Ω = int g −1 (0).
(P∞ )
D. Lü / Nonlinear Analysis 99 (2014) 35–48
37
Our main results for (Pµ ) are the following: Theorem 1.1. Assume that α ∈ (0, 3), p ∈ (2, 6 − α) and (g1 )–(g3 ) are satisfied. Then there exists µ∗ > 0 such that, for all µ ≥ µ∗ , the problem (Pµ ) has at least one ground state solution uµ in H 1 (R3 ). Furthermore, for any sequence µn → +∞, uµn converges in H 1 (R3 ) along a subsequence to a ground state solution of (P∞ ). Theorem 1.2. Let α ∈ (0, 3), p ∈ (2, 6 − α) and (g1 )–(g3 ) hold. Assume that uµn be a sequence of solutions of (Pµn ) with µn → +∞ and such that lim infn→∞ Φµn (uµn ) < ∞, here Φµ is defined in (2.3). Then uµn → u¯ in H 1 (R3 ) such that u¯ is a nontrivial solution of (P∞ ). Remark 1.1. If we let a = 1 and b = 0, (Pµ ) reduces to the Choquard equation, thus our results cover the case for the Choquard equation. The rest of this paper is organized as follows. In Section 2, we introduce some notations, set the variational framework for problem (Pµ ) and present some preliminary results about the Nehari manifold. In Section 3, we prove some important lemmas that will be used for the proofs of the main results. In Section 4, we prove the main results. 2. Variational setting and preliminaries In this paper we will use the following notations:
• • • • • •
C , C1 , C2 , . . . denote various positive constants whose exact values are not important. → (respectively ⇀) denotes strong (respectively weak) convergence. on (1) denotes a quantity such that on (1) → 0 as n → ∞. Br denotes a ball centered at the origin with radius r > 0. The dual space of a Banach space H will be denoted by H −1 . ⟨·, ·⟩ denotes the duality pairing between H −1 and H .
Throughout this paper we suppose that (g1 )–(g3 ) are satisfied. H 1 (R3 ) is the usual Sobolev space endowed with the standard inner product
⟨u, v⟩ =
∇ u∇v + uv dx, R3
and the associated norm ∥u∥2 = ⟨u, u⟩. Let H := {u ∈ H 1 (R3 ) : the inner product
(u, v)H =
R3
g (x)|u|2 dx < +∞} be the Hilbert space equipped with
∇ u∇v + g (x)uv dx R3
and endowed with the equivalent norm ∥u∥2µ = R3 (a|∇ u|2 + Vµ (x)|u|2 )dx. It is clear that, for each q ∈ [2, 2∗ ], there exists aq > 0 (independent of µ) such that if µ ≥ 1 then
|u|q ≤ aq ∥u∥µ ,
∀ u ∈ H,
(2.1)
1 = 6 and |u|q = ( R3 |u|q dx) q is the usual norm in Lq (R3 ). Set where 2∗ = N2N −2 1 |u(x)|p |u(y)|p p p D(u) = ∗ | u | | u | dx = dxdy. α |x − y|α R3 |x| R3 R3 Using the Hardy–Littlewood–Sobolev inequality (see [27, Theorem 4.3])
R3
R3
φ(x)ψ(y) dxdy ≤ C (κ)|φ|r |ψ|s , |x − y|κ
where 0 < κ < 3, 1 < r , s < ∞ and
|D(u)| ≤ C (α)
|u|
6p 6−α
1 r
+
1 s
+
κ 3
∀ φ ∈ Lr (R3 ), ∀ ψ ∈ Ls (R3 ), = 2, we have the estimate of the D(u) as
6−α 3 = C (α)|u|2p pr
dx
R3
(2.2)
6 for every u ∈ H 1 (R3 ), where C (α) is a positive constant and r = 6−α . For using the Sobolev embedding, we let
6p 6−α
∈ (2, 2 ), ∗
that is, p ∈
6−α 3
,6 − α .
38
D. Lü / Nonlinear Analysis 99 (2014) 35–48
By (2.2) we know that D(u) is well-defined in H . Moreover, similar to the proof of Lemma 2.5 in [22], we can get that D(u) ∈ C 1 (H , R). Let
Φµ (u) =
1 2
b
2
∥ u∥ µ +
2
2
|∇ u| dx
4
−
R3
1
1
2p
R3
|x|α
∗ |u|
|u|p dx.
p
(2.3)
In view of the hypotheses (g1 )–(g3 ), (2.2) and D(u) ∈ C 1 (H , R), the functional Φµ (u) is well-defined and Φµ (u) ∈ C 1 (H , R). Hence, the solutions of problem (Pµ ) are the critical points of the energy functional Φµ (u). Moreover, the functional Φµ (u) satisfies the mountain-pass geometry. Lemma 2.1. Let α ∈ (0, 3), p ∈ (2, 6 − α) and (g1 )–(g3 ) hold. Then the functional Φµ (u) satisfies the following conditions. (i) There exist θ , ρ > 0 such that Φµ (u) ≥ θ > 0 for all ∥u∥µ = ρ . (ii) There exists e ∈ H with ∥e∥µ > ρ such that Φµ (e) ≤ 0. Proof. (i) By (2.2) and (2.1), we have that
Φµ (u) =
≥ ≥
1 2 1 2 1 2
2
b
2
2
|∇ u| dx
∥ u∥ µ +
4
∥u∥2µ −
2p
−
R3
C0
1
1
2p
R3
|x|α
p
|u|p dx
∗ | u|
|u|2p pr
∥u∥2µ − C1 ∥u∥2p µ,
(2.4)
where C1 > 0 is independent of µ. Since p > 2, thus we can choose some θ , ρ > 0 such that Φµ (u) ≥ θ > 0 for all
∥u∥µ = ρ .
(ii) First we note that, for each µ > 0, Φµ (0) = 0. Furthermore, since p > 2, we have that
lim Φµ (tu) = lim
t →+∞
t →+∞
t2 2
∥u∥2µ +
bt 4
2
4
|∇ u|2 dx
−
R3
t 2p
1
2p
R3
|x|α
∗ |u|p |u|p dx = −∞.
Hence, we can choose t0 > 0 large enough such that ∥t0 u∥µ > ρ and Φµ (t0 u) < 0. Let e = t0 u, then (ii) holds.
In order to find the weak solutions of problem (Pµ ), we consider the Nehari manifold
Mµ = u ∈ H \ {0} : γ (u) = 0 ,
where
γ ( u) =
⟨Φµ′ (u), u⟩
2
2
= ∥u∥µ + b
2
|∇ u| dx
1
R3
|x|α
−
R3
p
∗ | u|
|u|p dx.
We define the minimax cµ as cµ = inf Φµ (u).
(2.5)
u∈Mµ
Now we are ready to prove some results concerning the Nehari manifold Mµ . Lemma 2.2. For any u ∈ Mµ there exist σ , δ > 0 such that ∥u∥µ ≥ σ and ⟨γ ′ (u), u⟩ ≤ −δ . Proof. For any u ∈ Mµ , by (2.2) and (2.1), we have 0 = ⟨Φµ′ (u), u⟩ 2
= ∥u∥µ + b
2
2
|∇ u| dx R3
1
R3
|x|α
−
2 2p ≥ ∥u∥2µ − C0 |u|2p pr ≥ ∥u∥µ − C2 ∥u∥µ .
p
∗ | u|
|u|p dx
D. Lü / Nonlinear Analysis 99 (2014) 35–48
39
Note that p > 2, thus there exists σ > 0 such that ∥u∥µ ≥ σ . Furthermore,
⟨γ (u), u⟩ = 2∥u∥µ + 4b ′
2
2
2
|∇ u| dx
− 2p
R3
= (2 − 2p)∥u∥µ + (4b − 2bp)
p
|x|α 2
R3
2
1
∗ | u|
|u|p dx
|∇ u|2 dx R3
≤ −(2p − 2)σ 2 < 0. By Lemma 2.2, Mµ is a smooth manifold in H . It is easy to see that Φµ is well-defined and smooth on Mµ . Moreover, we can proceed analogously to the proof of Theorem 4.3 in [28] to show that if u is a critical point of Φµ constrained to Mµ , then u is a nontrivial solution for (Pµ ). Lemma 2.3. Φµ is bounded from below by a positive constant for all u ∈ Mµ . Proof. Let u ∈ Mµ , by the definition of Mµ and Lemma 2.2 we obtain
Φµ (u) =
1 2
=
∥ u∥ µ + 1 2
>
b
2
1 2
−
−
1
4
2
|∇ u| dx
−
R3
2
∥ u∥ µ +
2p 1
2
b
1
2p
R3
|x|α
∗ | u|
p
|u|p dx
2
b
−
4
1
2
|∇ u| dx
2p
R3
σ 2 > 0.
2p
Lemma 2.4. For any 0 ̸= u ∈ H , there is a unique tu > 0 such that tu u ∈ Mµ . Moreover, Φµ (tu u) = maxt >0 Φµ (tu). Proof. Let u ∈ H \ {0} be fixed and for t > 0, we consider the fibering maps f : t → Φµ (tu) defined by f (t ) := Φµ (tu) =
t2 2
2
∥ u∥ µ +
bt 4
2
2
|∇ u| dx
4
−
t 2p
1
2p
R3
|x|α
R3
p
∗ | u|
|u|p dx.
First we claim that f (t ) > 0 for t > 0 small. Indeed, by (2.4), we have that f (t ) ≥
t2 2
∥u∥2µ − C1 t 2p ∥u∥2p µ,
so f (t ) > 0 if t > 0 is small enough. Using the similar argument as in the proof of Lemma 2.1(ii), we see that f (t ) → −∞ as t → +∞. Hence, there exists tu > 0 such that f (t ) has a positive maximum and f ′ (tu ) = 0. On the other hand, we notice that if tu ∈ Mµ , then t must be such that 0 = ⟨Φµ′ (tu), tu⟩ = t 2 ∥u∥2µ + bt 4
2
|∇ u|2 dx
− t 2p
1
R3
|x|α
R3
∗ |u|p |u|p dx
and setting
β = ∥u∥µ , 2
η=
2
2
|∇ u| dx R3
,
ϑ=
1
R3
|x|α
p
∗ | u|
|u|p dx,
we are led to find a positive solution of t 2 (β + bηt 2 − ϑ t 2p−2 ) = 0 with β, η, ϑ > 0 and b ≥ 0. Obviously, since p > 2, the equation β + bηt 2 − ϑ t 2p−2 = 0 has a unique solution t = tu > 0. We observe that f ′ (t ) = 1t ⟨Φµ′ (tu), tu⟩. So tu u ∈ Mµ and Φµ (tu u) = maxt >0 Φµ (tu). The proof is completed. 3. The (PS )c condition In this section, the main goal is to show that functional Φµ (u) satisfies the (PS )c condition. Recall that, for a given functional I ∈ C 1 (H , R), we say that a sequence {zn } ⊂ H is a (PS )c sequence if it satisfies I (zn ) → c and I ′ (zn ) → 0
40
D. Lü / Nonlinear Analysis 99 (2014) 35–48
as n → ∞. Moreover, if any (PS )c sequence has a convergent subsequence, then we say that I satisfies the (PS )c condition. In this paper, we will take I = Φµ (u). Lemma 3.1. Under the assumptions of Theorem 1.1 and letting {un } be a (PS )c sequence for Φµ (u), we have (i) {un } is bounded in H ; (ii) either c ≥ c0 for some c0 > 0 independent of µ or c = 0. Proof. (i) Let {un } be a (PS )c sequence for Φµ (u), i.e.,
Φµ (un ) = c + on (1) and Φµ′ (un ) = on (1). Note that p > 2 and b ≥ 0, then we have, c + on (1) −
1 2p
on (∥un ∥µ ) = Φµ (un ) −
=
2
≥
1
1 2
−
−
1 2p
⟨Φµ′ (un ), un ⟩
1
∥un ∥2µ +
2p
b 4
−
b
2
2p
|∇ un |2 dx R3
1 2p
∥un ∥2µ .
Thus
2
∥un ∥µ ≤ c
1 2
−
1
−1 ,
2p
(3.1)
for n large enough and therefore (i) holds. (ii) Since Φµ′ (un ) = on (1), we have that on (∥un ∥µ ) = ⟨Φµ′ (un ), un ⟩ 2
2
= ∥un ∥µ + b
2
|∇ un | dx
1
R3
|x|α
−
R3
p
∗ | un |
|un |p dx
≥ ∥un ∥2µ − C2 ∥un ∥2p µ, since p > 2, there exists σ1 > 0(σ1 < 1) such that
⟨Φµ′ (un ), un ⟩ ≥ Now, if c <
(p−1)σ12 2p
∥un ∥2µ ,
for ∥un ∥µ < σ1 .
(3.2)
and {un } is a (PS )c -sequence of Φµ , then by (3.1), we get that
lim ∥un ∥2µ ≤
n→∞
1 4
2pc p−1
< σ12 .
Hence, ∥un ∥µ < σ1 for n large, then by (3.2), we have 1 4
∥un ∥2µ ≤ ⟨Φµ′ (un ), un ⟩ = on (1)∥un ∥µ ,
which implies ∥un ∥µ → 0 as n → ∞ and c = 0, it follows that (ii) holds for c0 =
(p−1)σ12 2p
.
Next, we recall that pointwise convergence of a bounded sequence implies weak convergence (see [29, Proposition 5.4.7]). Lemma 3.2. Let N ≥ 3, q ∈ (1, +∞) and {un } be a bounded sequence in Lq (RN ). If un → u almost everywhere in RN as n → ∞, then un ⇀ u weakly in Lq (RN ). Then we have the following Brézis–Lieb type lemma for the nonlocal term D(u). Lemma 3.3. Let α ∈ (0, 3), p ∈ ( 6−α , 6 − α). If {un } ⊂ H is a bounded sequence such that un → u almost everywhere in R3 3 as n → ∞, then the following hold. (i) D(un ) − D(un − u) → D(u) as n → ∞. (ii) D′ (un ) − D′ (un − u) → D′ (u) in H −1 as n → ∞.
D. Lü / Nonlinear Analysis 99 (2014) 35–48
41
Proof. The proof is analogous to that of Lemma 3.5 in [18], but we exhibit it here for completeness. First, similarly to the proof of the Brézis–Lieb Lemma [30], we know that 6
|un − u|p − |un |p → |u|p in L 6−α (R3 ) as n → ∞.
(3.3)
By the Hardy–Littlewood–Sobolev inequality, we can deduce that 1
∗ (|un − u|p − |un |p ) →
|x|α
1
6
∗ |u|p in L α (R3 ) as n → ∞.
|x|α
(3.4)
We notice that
1
|x|α
R3
=
∗ | un |
p
|un | dx −
1
1
p
|x|α
R3
|x|α
R3
p
∗ | un − u|
|un − u|p dx
∗ (|un | − |un − u| ) (|un | − |un − u| )dx + 2 p
p
p
p
1
R3
|x|α
∗ (|un | − |un − u| ) |un − u|p dx. p
p
(3.5)
On the other hand, by Lemma 3.2, we have that 6
|un − u|p ⇀ 0 in L 6−α (R3 ) as n → ∞.
(3.6)
Combining (3.3)–(3.6), conclusion (i) holds. The proof of conclusion (ii) is similar to (i), and is omitted here.
Lemma 3.4. Under the assumptions of Theorem 1.1, let µ > 0 be fixed and {un } be a (PS )c -sequence of Φµ . Then up to a subsequence un ⇀ u in H with u being a weak solution of (Pµ ). Moreover, Φµ (un − u) → c − Φµ (u) and Φµ′ (un − u) → 0 as n → ∞. Proof. By Lemma 3.1(i), we know that {un } is bounded in H . Therefore, there is a subsequence of {un } such that un ⇀ u in H as n → ∞. In order to see that u is a critical point of Φµ we recall that un ⇀ u in H , un → u, a.e. in R3 , un → u in Lsloc (R3 ), 2 ≤ s < 6. Moreover, there exists A ∈ R, such that
|∇ un |2 dx → A as n → ∞. R3
Then by Fatou’s lemma we get that
R3
|∇ u|2 dx ≤ A.
We claim that R3 |∇ u|2 dx = A. Arguing by contradiction, we assume that H , for any ϕ ∈ H , we have
a∇ u∇ϕ dx + R3
R3
(1 + µg (x))uϕ dx + Ab
1
R3
|x|α
∇ u∇ϕ dx − R3
R3
|∇ u|2 dx < A. By Φµ′ (un ) → 0 and un ⇀ u in p
∗ | u|
|u|p−2 uϕ dx = 0.
Then ⟨Φµ′ (u), u⟩ < 0. On the other hand, by Lemma 2.4 it is easy to get that ⟨Φµ′ (tu), tu⟩ > 0 for t > 0 is small enough. Hence, there exists t0 ∈ (0, 1) satisfying ⟨Φµ′ (t0 u), t0 u⟩ = 0. Moreover, Φµ (t0 u) = max0≤t ≤1 Φµ (tu). So 1
cµ ≤ Φµ (t0 u) = Φµ (t0 u) −
= <
t02 4 1 4
2
∥ u∥ µ + 2
∥ u∥ µ +
= lim inf
1 4
n→∞
1 4
1 4
−
−
1
2p
R3
2
∥un ∥µ +
1 4
1 4
−
|x|α
R3
1
= lim inf Φµ (un ) −
1
2p
t0
2p
n→∞
⟨Φµ′ (t0 u), t0 u⟩ 4
∗ |u|p |u|p dx
1
p
∗ | u| |x|α
1
2p
R3
|u|p dx 1
|x|α
∗ | un |
⟨Φµ′ (un ), un ⟩
p
= cµ ,
p
|un | dx
42
D. Lü / Nonlinear Analysis 99 (2014) 35–48
which is a contradiction. Then
|∇ u|2 dx = A = lim
n→∞
R3
|∇ un |2 dx.
R3
(3.7)
Thus for any ϕ ∈ H , we have
⟨Φµ′ (u), ϕ⟩ = lim ⟨Φµ′ (un ), ϕ⟩ = 0, n→∞
which shows that u is a weak solution of (Pµ ). Now we consider a new sequence un = un − u. By the Brézis–Lieb lemma [30], we have that
∥ un ∥2µ = ∥un ∥2µ − ∥u∥2µ + on (1), 2 2 |∇ un |2 dx
(3.8)
|∇ un |2 dx
=
R3
2
+ on (1).
|∇ u|2 dx
−
R3
R3
(3.9)
Next we prove that
Φµ ( un ) = c − Φµ (u) as n → ∞
(3.10)
Φµ′ ( un ) → 0 as n → ∞.
(3.11)
and
Using (3.8) and (3.9) we get that
Φµ ( un ) =
=
1 2 1 2
b
2
∥ un ∥ µ +
2
2
|∇ un | dx
4 1
b
2
4
∥un ∥2µ − ∥u∥2µ +
−
−
R3
1
1
2p
R3
|x|α
1
2p
R3
|∇ un |2 dx
p
∗ | un | |x|α
2
1
−
R3
b
| un |p dx 2
|∇ u|2 dx
4
R3
∗ | un − u|
= Φµ (un ) − Φµ (u) +
|un − u|p dx + on (1)
p
1 D(un ) − D(u) − D(un − u) + on (1). 2p
(3.12)
From Lemma 3.3(i), D(un ) − D(u) − D(un − u) → 0 as n → ∞. Thus from (3.12) we obtain (3.10). In order to show (3.11), let v ∈ H , it is easy to see that
⟨Φµ′ ( un ), v⟩
=
⟨Φµ′ (un ), v⟩
−
1
R3
|x|α
+
⟨Φµ′ (u), v⟩
+ on (1) −
p
∗ | un |
| un |
p−2
un v dx −
1
R3
|x|α
1
R3
|x|α
| un |p−2 un v dx
p
∗ | un |
p
∗ | u|
|u|p−2 uv dx.
By Lemma 3.3(ii), we have that
lim
sup
n→∞ ∥v∥ ≤1 µ
1
p
|x|α
R3
∗ | un |
p−2
| un |
un −
1
|x|α
p
∗ | un |
p−2
|un |
un +
1
|x|α
p
∗ | u|
p−2
|u|
u v dx = 0.
Thus we have lim ⟨Φµ′ ( un ), v⟩ = 0,
n→∞
∀v ∈ H ,
which implies (3.11) and this completes the proof of Lemma 3.4.
Lemma 3.5. Let α ∈ (0, 3), p ∈ ( 6−α , 6 − α) and C ∗ be fixed. Given ε > 0 there exist µε = µ(ε, C ∗ ) > 0 and 3 ∗ Rε = R(ε, C ) > 0 such that, if {un } is a (PS )c -sequence of Φµ (u) with c ≤ C ∗ , µ ≥ µε , then
lim sup n→∞
R3 \BRε
1
|x|α
p
∗ | un |
|un |p dx ≤ ε.
(3.13)
D. Lü / Nonlinear Analysis 99 (2014) 35–48
43
Proof. For R > 0, we set
ΩR+ := {x ∈ R3 : |x| ≥ R, g (x) ≥ M },
ΩR− := {x ∈ R3 : |x| ≥ R, g (x) < M },
then
ΩR+
|un |2 dx ≤ ≤ ≤
≤
1 1 + µM
R3
(1 + µg (x))|un |2 dx
1 1 + µM 1
(a|∇ un |2 + (1 + µg (x))|un |2 )dx R3
2pc
1 + µM
1
+ on (∥un ∥µ )
p−1
1 + µM
(by (3.1))
2pC ∗
+ on (1)
p−1
→ 0 as µ → +∞.
(3.14)
Using the Hölder inequality, (2.1) and Lemma 3.1(ii), for 1 < q < 3 we have
ΩR−
|un |2 dx ≤
|un |2q dx
dx ΩR−
R3
≤ C ∥un ∥2µ · L(ΩR− ) ≤ C
q−q 1
1q
2pC ∗ p−1
· L(ΩR− )
q−1 q q−1 q
→ 0 as R → ∞,
(3.15)
where C = C (q) is a positive constant. Using the Hardy–Littlewood–Sobolev inequality, we have
R3 \ΩR−
Setting ℓ =
1
|x|α
∗ |un |p |un |p dx ≤ C
3(p−2)+α , 2p
R3 \ΩR−
|un |
6p 6−α
R3 \ΩR−
|un |
6p 6−α
6−α 3 .
dx
(3.16)
by using the Gagliardo–Nirenberg inequality, we get
dx ≤ C R3 \ΩR−
63p−αℓ |∇ un |2 dx · 63p−αℓ |∇ un | dx ·
1−ℓ) 3p6(−α
R3 \ΩR−
2
≤ C R3 \ΩR− 6pℓ 6−α
≤ C ∥ un ∥ µ
· ΩR+
|un |2 dx +
|un |2 dx
ΩR+
ΩR−
1−ℓ) 3p6(−α
2
|un | dx +
2
ΩR−
|un | dx
1−ℓ) 3p6(−α
|un |2 dx
→ 0 as µ, R → ∞ (by (3.14) and (3.15)). Combining (3.16) and (3.17), we conclude the proof of Lemma 3.5.
(3.17)
Then we have the following compactness result. Lemma 3.6. Let α ∈ (0, 3), p ∈ (2, 6 − α) and (g1 )–(g3 ) hold. Then for any C0 > 0, there exists µ0 > 0 such that Φµ satisfies the (PS )c -condition for all c ≤ C0 and µ ≥ µ0 . Proof. Let c0 > 0 be given by Lemma 3.1(ii) and choose ε > 0 such that ε < p−01 . Then for given C0 > 0, we choose µε > 0 and Rε > 0 as in Lemma 3.5. We claim that µ0 = µε is required in Lemma 3.6. Let {un } ⊂ H be a (PS )c -sequence of Φµ (u) with µ ≥ µ0 and c ≤ C0 . By Lemma 3.4, we may suppose that un ⇀ u in H and un = un − u is a (PS )c ′ -sequence of Φµ with c ′ = c − Φµ (u). Now we claim c ′ = 0. In fact, if c ′ ̸= 0, then by Lemma 3.1(ii) we obtain that c ′ ≥ c0 > 0. Since { un } is a (PS )c ′ -sequence of Φµ , we have pc
Φµ ( un ) = c ′ + on (1) and Φµ′ ( un ) = on (1).
44
D. Lü / Nonlinear Analysis 99 (2014) 35–48
Then we get, c ′ + on (1) −
1 2
on (∥ un ∥µ ) = Φµ ( un ) − b
=− ≤
2
⟨Φµ′ ( un ), un ⟩ 2 2
|∇ un | dx
4
R3
1
1
2
1
−
2p
+
−
2
1 2p
R3
1
|x|α
p
∗ | un |
| un |p dx
1
| un |p dx,
p
∗ | un |
|x|α
R3
1
(3.18)
from (3.18) we deduce that
1
R3
|x|α
lim
n→∞
p
p
∗ | un |
| un | dx ≥ c
′
1
−
2
−1
1
≥
2p
2pc0 p−1
.
On the other hand, by Lemma 3.5 we have that
lim sup n→∞
1
∗ | un |
|x|α
R3 \BRε
| un |p dx ≤ ε <
p
pc0 p−1
.
This implies un ⇀ u in H with u ̸= 0, which is a contradiction. Therefore c ′ = 0 and it follows from (3.1) that lim ∥ un ∥2µ ≤
n→∞
2pc ′
= 0,
p−1
hence un → 0 in H , that is, un → u in H . This completes the proof of Lemma 3.6.
In the following, we will show that there exists uµ ∈ Mµ with Φµ (uµ ) = cµ , that is, uµ is a ground state solution of (Pµ ).
Theorem 3.1. Assume that α ∈ (0, 3), p ∈ (2, 6 − α) and (g1 )–(g3 ) are satisfied. Then there exists µ∗ > 0 such that cµ is achieved for all µ ≥ µ∗ at some uµ which is a ground state solution of (Pµ ). Proof. By Lemma 2.1, Φµ satisfies the mountain-pass geometry, then using a version of the mountain-pass theorem without the (PS) condition, there exists {un } ⊂ H such that Φµ (un ) → cµ and Φµ′ (un ) → 0. Moreover, by Lemma 3.1(i) {un } is
bounded in H . Then, up to a subsequence, we may assume that un ⇀ u0 in H and un → u0 for a.e. in R3 . By Lemma 3.6, there exists µ∗ > 0, such that for µ ≥ µ∗ , un → u0 in H . By Lemma 3.4 we have that Φµ′ (u0 ) = 0. In addition, by the definition of cµ and Lemma 2.3, we have cµ > 0, this implies that u0 ̸= 0. Then u0 ∈ Mµ . Using Fatou’s lemma we have that
Φµ (u0 ) = Φµ (u0 ) −
=
1 2
−
1
1 2p
≤ lim inf
2
∥ u0 ∥ µ +
2p
1 2
n→∞
⟨Φµ′ (u0 ), u0 ⟩
−
1
4
−
b
2p
= lim inf Φµ (un ) −
2
∥ un ∥ µ + 1 2p
2
2
|∇ u0 | dx
2p
n→∞
b
R3
b 4
−
b 2p
2
2
|∇ un | dx R3
⟨Φµ′ (un ), un ⟩
= cµ . Hence, Φµ (u0 ) ≤ cµ . On the other hand, by the definition of cµ , we have cµ ≤ Φµ (u0 ). So, Φµ (u0 ) = cµ , take uµ = u0 , then uµ is a ground state solution of problem (Pµ ). 4. Proof of the main results In this section we give the proofs of our main results. First, we consider the following limit problem:
1 − a + b |∇ u|2 dx 1u + u = ∗ |u|p |u|p−2 u, |x|α Ω u ∈ H01 (Ω ), where Ω = int g −1 (0) is defined in assumption (g2 ).
(P∞ )
D. Lü / Nonlinear Analysis 99 (2014) 35–48
45
The energy functional associated with (P∞ ) is defined by
Φ∞ (u) =
1
2
Ω
b
(a|∇ u| + |u| )dx + 2
2
2
2
|∇ u| dx
4
−
Ω
1
1
2p
Ω
|x|α
p
∗ | u|
|u|p dx.
Let ′ M∞ = {u ∈ H01 (Ω ) \ {0} : ⟨Φ∞ (u), u⟩ = 0}
be the Nehari manifold and set c∞ = inf Φ∞ (u). u∈M∞
Then we have the following lemma. Lemma 4.1. limµ→+∞ cµ = c∞ , where cµ is defined in (2.5). Proof. It is easy to see that cµ ≤ c∞ for all µ ≥ 0. Assume that limn→∞ cµn = k < c∞ for a sequence µn → ∞ as n → ∞. Lemma 3.1(ii) implies k > 0 and by Theorem 3.1, for µ large enough, there exists a sequence un ∈ Mµn , a solution of (Pµn ), such that Φµn (un ) = cµn . Since {un } is bounded in H , we may assume that
un ⇀ u, un → u, un → u,
in H , a.e. in R3 , in Lsloc (R3 ), 2 ≤ s < 6.
(4.1)
We claim that u|Ω c = 0, where Ω c = R3 \ Ω . In fact, if u|Ω c ̸= 0, then there exists a compact subset Σ ⊂ Ω c with dist(Σ , ∂ Ω ) > 0 such that u|Σ ̸= 0. Then by (4.1)
2
Σ
|un | dx →
Σ
|u|2 dx > 0.
Moreover, there exists ε0 > 0 such that g (x) ≥ ε0 for any x ∈ Σ . We also note that un ∈ Mµn , then we get that 1
Φµn (un ) =
2
=
R3
1 2
≥
1 2
≥
1 2
(a|∇ un | + (1 + µn g (x))|un | )dx + 2
−
−
−
1
1
R3
2p
2
2
|∇ un | dx
4
Σ
−
R3
(a|∇ un | + (1 + µn g (x))|un | )dx + 2
R3
2p
b
2p 1
2
2
b 4
−
b
1 2p
D(un )
2
2
|∇ un | dx
2p
R3
(1 + µn g (x))|un |2 dx
(1 + µn ε0 )|un |2 dx
→ +∞ as n → ∞. This contradiction shows that u|Ω c = 0. Next we claim that un → u in Ls (R3 ) for 2 < s < 6. Otherwise, by the concentration compactness principle of Lions [31], there exist δ > 0, ρ > 0 and xn ∈ R3 with |xn | → +∞ such that
lim inf n→∞
Bρ (xn )
|un − u|2 dx ≥ δ > 0.
On the other hand, we notice that un ∈ Mµn and L(Bρ (xn ) ∩ {x|g (x) ≤ M }) → 0 as n → ∞, then we have
Φµn (un ) ≥
2
≥
1 2
=
1
1 2
−
−
−
→ +∞,
1
2p 1
Bρ (xn )∩{x|g (x)≥M }
2p 1 2p
µn M
µn M
(a|∇ un |2 + (1 + µn g (x))|un |2 )dx 2
Bρ (xn )
|un − u| dx −
2
Bρ (xn )∩{x|g (x)≤M }
|un − u| dx − on (1) 2
Bρ (xn )
as n → ∞.
This contradiction implies un → u in Ls (R3 ) for 2 < s < 6.
|un − u| dx
46
D. Lü / Nonlinear Analysis 99 (2014) 35–48
Since {un } is bounded, by Fatou’s lemma, we obtain that
n→∞
|∇ un |2 dx ≥
lim inf R3
R3
|∇ u|2 dx.
On the other hand, by the choice of {un }, we have
(a|∇ un | + (1 + µn g (x))|un | )dx + b 2
R3
2
2
2
|∇ un | dx
1
R3
|x|α
=
R3
p
∗ | un |
|un |p dx.
Consequently
(a|∇ u| + |u| )dx + b 2
Ω
2
2
2
|∇ u| dx
≤ lim
n→∞
Ω
(a|∇ un | + |un | )dx + b 2
R3
1
R3
|x|α
≤ lim
n→∞
2
2
2
|∇ un | dx R3
p
∗ | un |
|un |p dx.
(4.2)
In the following we first show that
1
R3
|x|α
lim
n→∞
∗ |un |p |un |p dx =
1
R3
|x|α
∗ |u|p |u|p dx.
(4.3)
Indeed, for given ε > 0, we can argue as in the proof of Lemma 3.5 to conclude that, for some ρ > 0 large, there holds
lim sup n→∞
1
|x|α
ρ (0)
R3 \B
|un |p dx ≤ ε.
p
∗ | un |
By taking ρ larger if necessary, we may suppose that
ρ (0)
R 3 \B
∗ |u|p |u|p dx ≤ ε . Moreover, the local convergence in
1 |x|α
(4.1) and the Lebesgue dominated convergence theorem imply that
Bρ (0)
1
|x|α
p
∗ | un |
p
|un | dx →
1
p
|x|α
Bρ (0)
∗ | u|
|u|p dx
as n → ∞. Since
1 1 p p p ∗ |un | |un | − ∗ |u| |u|p dx α α R3 |x| |x| 1 1 p p p ∗ |un | |un | dx + ∗ |u| |u|p dx ≤ α α R3 \Bρ (0) |x| R3 \Bρ (0) |x| 1 1 p p p p ∗ | u | | u | − ∗ | u | | u | dx + , n n Bρ (0) |x|α |x|α it follows from the above estimates and convergences that
lim sup n→∞
R3
1
|x|α
p
∗ | un |
1
p
| un | −
|x|α
p
∗ | u|
|u|
p
dx ≤ 2ε,
and therefore (4.3) holds. By (4.2) and (4.3) we have
(a|∇ u| + |u| )dx + b 2
Ω
2
2
2
|∇ u| dx
1
R3
|x|α
1
Ω
|x|α
≤
Ω
∗ | u|
p
|u|p dx.
Since u|Ω c = 0, then we get
(a|∇ u| + |u| )dx + b 2
Ω
2
2
2
|∇ u| dx Ω
≤
∗ | u|
p
|u|p dx.
D. Lü / Nonlinear Analysis 99 (2014) 35–48
47
Thus similar to Lemma 2.4, there exists t1 ∈ (0, 1] such that t1 u ∈ M∞ and
Φ∞ (t1 u) ≤ Φ∞ (u), hence c∞ ≤ Φ∞ (t1 u) ≤ Φ∞ (u) ≤ limn→∞ Φµn (un ) = k < c∞ , which is a contradiction and this completes the proof.
Now we give the proofs of Theorems 1.1 and 1.2. Proof of Theorem 1.1. By Theorem 3.1, the first part of Theorem 1.1 has been proved. Now we consider the concentration behavior of the solutions. Let {un } ⊂ H 1 (R3 ) with un ∈ Mµn , Φµn (un ) = cµn and µn → ∞. Since such a sequence must be bounded in H 1 (R3 ), we may assume that un ⇀ u in H 1 (R3 ) and un → u in Lsloc (R3 ) for 2 ≤ s < 6. Next we shall show that u ∈ H01 (Ω ) is a ground state solution of (P∞ ), that is Φ∞ (u) = c∞ , and that un → u in H 1 (R3 ). Using the argument in the proof of Lemma 4.1 we can prove that u|Ω c = 0 and un → u in Ls (R3 ) for 2 ≤ s < 6. Moreover, Φ∞ (u) ≤ limn→∞ Φµn (un ) ≤ c∞ . If u ∈ M∞ , then Φ∞ (u) = c∞ . In order to show u ∈ M∞ , it suffices to show that
|∇ un |2 dx =
lim
n→∞
R3
R3
|∇ u|2 dx,
lim µn
n→∞
R3
g (x)|un |2 dx = 0.
In fact, if one of the above limits does not hold, by Fatou’s lemma, we have
(|∇ u| + |u| )dx + b 2
R3
2
2
2
|∇ u| dx R3
<
1
R3
|x|α
p
∗ | u|
|u|p dx,
similar to the proof of Lemma 4.1, there exists σ0 ∈ (0, 1) such that σ0 u ∈ M∞ and c∞ ≤ Φ∞ (σ0 u) < Φ∞ (u) ≤ lim Φµn (un ) ≤ c∞ , n→∞
which is a contradiction. This completes the proof of Theorem 1.1.
Proof of Theorem 1.2. For any sequence µn → +∞, let un := uµn be the solution of (Pµn ). Since lim infn→∞ Φµn (uµn ) < ∞, it is easy to see that {un } must be bounded in H 1 (R3 ). Therefore, we may assume that un ⇀ u¯ in H 1 (R3 ) and un → u¯ in Lsloc (R3 ) for 2 ≤ s < 6. As in the proof of Lemma 4.1, we can prove that u¯ |Ω c = 0 and un → u¯ in Ls (R3 ) for 2 ≤ s < 6. Furthermore, as in the proof of Theorem 1.1, we can prove that u¯ ∈ H01 (Ω ) is a ground state solution of (P∞ ). To complete the proof it suffices to show that un → u¯ in H 1 (R3 ). In fact, we use the weak convergence of {un }, the fact that un ∈ H 1 (R3 ) is the solution of (Pµn ) and u¯ ∈ M∞ to get that
(a|∇(un − u¯ )|2 + Vµn (x)|un − u¯ |2 )dx 2 2 = (a|∇ un | + Vµn (x)|un | )dx − (a|∇ u¯ |2 + Vµn (x)|¯u|2 )dx + on (1) R3 R3
∥un − u¯ ∥2µn =
R3
1
= R3
|x|α
R3
2
2
|∇ un | dx
−b
1
∗ |un |p |un |p dx −
∗ |¯u|p |¯u|p dx 2
2
|∇ u¯ | dx
+b
R3
|x|α
R3
+ on (1)
= on (1) (by (4.3) and (3.7)), which implies that un → u¯ in H 1 (R3 ). This completes the proof of Theorem 1.2.
Acknowledgments The author would like to thank the referees for valuable comments and helpful suggestions. References [1] C.O. Alves, F.J.S.A. Correa, T.F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005) 85–93. [2] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [3] J.L. Lions, On some questions in boundary value problems of mathematical physics, in: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proceedings of International Symposium, Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977, in: North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, 1978, pp. 284–346. [4] X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in RN , Nonlinear Anal. RWA 12 (2011) 1278–1287. [5] W. Liu, X.M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput. 39 (2012) 473–487. [6] X.M. He, W.M. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal. 70 (2009) 1407–1414.
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