Concentrating solutions of nonlinear fractional Schrödinger equation with potentials

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ScienceDirect J. Differential Equations ••• (••••) •••–••• www.elsevier.com/locate/jde

Concentrating solutions of nonlinear fractional Schrödinger equation with potentials Xudong Shang a,b , Jihui Zhang a,∗ a Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023,

PR China b School of Mathematics, Nanjing Normal University Taizhou College, Taizhou 225300, PR China

Received 24 April 2014; revised 17 October 2014

Abstract In this paper we study the concentration phenomenon of solutions for the nonlinear fractional Schrödinger equation ε2s (−)s u + V (x)u = K(x)|u|p−1 u,

x ∈ RN ,

where ε is a positive parameter, s ∈ (0, 1), N ≥ 2 and 1 < p < N+2s N−2s , V (x) and K(x) are positive smooth p+1

−N

−

2

functions. Let Γ (x) = [V (x)] p−1 2s [K(x)] p−1 . Under certain assumptions on V (x) and K(x), we show existence and multiplicity of solutions which concentrate near some critical points of Γ (x) by a perturbative variational method. © 2014 Elsevier Inc. All rights reserved. Keywords: Fractional Schrödinger equations; Concentrating solutions; Variational method

* Corresponding author.

E-mail addresses: [email protected] (X. Shang), [email protected] (J. Zhang). http://dx.doi.org/10.1016/j.jde.2014.10.012 0022-0396/© 2014 Elsevier Inc. All rights reserved.

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1. Introduction and main results This paper is devoted to the existence and concentration behavior of semiclassical standing wave solutions for the Schrödinger equations with fractional Laplacian. More precisely, we are concerned with the following equation ε 2s (−)s u + V (x)u = K(x)|u|p−1 u,

x ∈ RN ,

(1.1)

where ε > 0 is a positive parameter, N ≥ 2, 0 < s < 1 and (−)s is defined as  (−)s u(x) = kN,s P.V. RN

u(x) − u(y) dy. |x − y|N+2s

(1.2)

The symbol P.V. stands for the Cauchy principal value, and kN,s is a dimensional constant that  ζ1 depends on N and s, precisely given by kN,s = ( RN 1−cos dζ )−1 . |ζ |N+2s The study of elliptic equations involving fractional powers of the Laplacian appears to be important in many physical situations such as combustion, and in dislocations in mechanical systems or in crystals. A basic motivation for the study of Eq. (1.1) arises in the study of standing wave solutions of the type ψ(x, t) = e−iλt/ε u(x) for the following time-dependent fractional Schrödinger equation iε

  ∂ψ = ε 2s (−)s ψ + V (x) + λ ψ − K(x)|ψ|p−1 ψ, ∂t

x ∈ RN ,

(1.3)

where ε is the Planck constant. Such a ψ solves (1.3) if the standing wave u(x) satisfies (1.1). Eq. (1.3) was introduced by Laskin [19], it is a fundamental equation of fractional quantum mechanics in the study of particles on stochastic fields modeled by Lévy processes. We refer to [9,20] for more physical background. Very recently, the study on problems of fractional Schrödinger equations has attracted much attention from many mathematicians. Coti Zelati and Nolasco [27] proved the existence of a 1 ground state solution of some fractional Schrödinger equation involving the operator (− +d 2 ) 2 with d > 0. Cheng [5] obtained the existence of ground state solution of the following equation (−)s u + V (x)u − |u|p−1 u = 0,

x ∈ RN ,

(1.4)

with unbounded potential V . In (1.4), when V (x) ≡ 1, Dipierro et al. [14] proved existence and symmetry results for the solutions, and in [17], Felmer et al. studied the same equation with a more general nonlinearity f (x, u), they obtained the existence, regularity and qualitative properties of ground states. Secchi [25] obtained positive solutions of a more general fractional Schrödinger equation by variational method. For other related investigations, one can see [10,26] and references therein. An important feature of semiclassical states uε of (1.1) is that they can concentrate as ε → 0. We say uε concentrates at a point x0 ∈ RN in the following sense: for any σ > 0, there exist

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positive constants ε0 and ρ such that uε (x) ≤ σ,

for |x − x0 | ≥ ερ, ∀ε < ε0 .

As for the classical case s = 1, we rewrite (1.1) as follows −ε 2 u + V (x)u = K(x)|u|p−1 u,

x ∈ RN .

(1.5)

There are many works focusing on Eq. (1.5). When K(x) ≡ 1, in the pioneering work [18], Floer and Weinstein studied the case N = 1 and p = 3, they constructed a positive solution uε which concentrates around the critical point of potential V (x), by using the Lyapunov–Schmidt reduction. This method and results have been generalized by [23] to the higher dimensional case. Ambrosetti et al. in [1] proved existence of standing wave solutions by assuming that the potential V (x) has a degenerate local minimum or maximum. In [4], Ambrosetti et al. obtained multiplicity results for some classes of potentials V (x) and K(x) in Eq. (1.5), and proved these solutions concentrations near several non-degenerate critical points of the auxiliary func2p+2+N−Np

−

2

tion A(x) = [V (x)] 2p−2 [K(x)] p−1 . Existence of solutions concentrating at one or several points to Eq. (1.5) with potentials vanishing at infinity has been obtained in [2,7,28] and references therein. In the fractional case 0 < s < 1, much less is known. When K(x) = 1 in (1.1), Chen and Zheng [11] considered the existence and concentration phenomenon under further constraints in the space dimension N and the values of s, by using the Lyapunov–Schmidt reduction method; Dávila et al. [15] proved that if V (x) satisfies     V ∈ C 1,α RN ∩ L∞ RN

and

inf V (x) > 0,

x∈RN

then (1.1) has multi-peak solutions; in [12], Dávila et al. considered the fractional Schrödinger equation in a bounded domain with zero Dirichlet datum, and built a family of solutions that concentrate at an interior point of the domain. The aim of this paper is to extend the existence and multiplicity results in [4] for the nonlocal problem (1.1). We obtain the existence and some multiplicity results of (1.1) for all 1 < p < N+2s N−2s under the following assumptions on potentials V and K: (V1) V (x) ∈ C 2 (RN , R), V and its derivatives are uniformly bounded; (V2) infx∈RN V (x) ≥ τ > 0; (K1) K(x) ∈ C 2 (RN , R), K(x) > 0, K and its derivatives are uniformly bounded. Set − 2   p+1 − N  Γ (x) = V (x) p−1 2s K(x) p−1 . By (V1) and (K1), Γ (x) is a C 2 -smooth function. We say that x0 is an isolated stable stationary point of Γ (x) if the Leray–Schauder index ind(∇Γ, x0 , 0) = 0. The index ind(∇Γ, x0 , 0) = limr→0 deg(∇Γ, Br (x0 ), 0). It is easy to see that local isolated maxima and minima as well as non-degenerate stationary points are stable.

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Now we are ready to state our main results. Theorem 1.1. Let (V1), (V2) and (K1) hold. Suppose that x0 is an isolated stable stationary point of Γ (x). Then, for ε > 0 small, Eq. (1.1) has a solution uε ∈ H s (RN ) that concentrates at x0 . We recall that the category cat(Σ, X) of a subset Σ of a topological space X is defined as the minimal k ∈ N such that Σ is covered by k closed subsets of X which are contractible in X. The cup long l(Σ) of Σ is defined by  l(Σ) = 1 + sup k ∈ N : ∃α1 , · · · , αk ∈ H˘ ∗ (Σ) \ 1, α1 ∪ · · · ∪ αk = 0 . If no such class exists, we set l(Σ) = 1. Where H˘ ∗ (Σ) is the Alexander cohomology of Σ with real coefficients and ∪ denotes the cup product. In general, one has that l(Σ) ≤ catΣ (Σδ ), where Σδ denotes its δ neighborhood. Let Σ be a smooth compact manifold of critical points of Γ (x), which is non-degenerate in the sense that for every x ∈ Σ one has that Tx Σ = ker[Γ  (x)]. Theorem 1.2. Let (V1), (V2) and (K1) hold. Suppose that Γ (x) has a non-degenerate smooth compact manifold of critical points Σ . Then, for ε > 0 small, Eq. (1.1) has at least l(Σ) solutions that concentrate near points of Σ . Theorem 1.3. Let (V1), (V2) and (K1) hold. Suppose that there is a compact set Σ where Γ (x) achieves an isolated strict local minimum, or maximum. Then, there exists εδ > 0, Eq. (1.1) has at least cat(Σ, Σδ ) solutions that concentrate near points of Σ for ε ∈ (0, εδ ). It is worth noting that, a common approach to deal with the fractional nonlocal problem, which was given by Caffarelli and Silvestre [8], is via the Dirichlet–Neumann map transforming (1.1) into a local problem. In this work, we prefer to analyze the problem directly in H s (RN ). The proofs of our main results are based on the perturbation method, variational in nature, see [3,4]. The basic idea is to use the non-degeneracy result in [16] to construct solutions of (1.1). To the best of our knowledge, there is no result on the multiplicity and concentration of solutions for fractional nonlinear Schrödinger equation with potentials. At present paper, we are first devoted to the proof of the existence and concentration of solutions for Eq. (1.1), and then study the multiplicity and concentration of solutions for (1.1). This is the first result for fractional nonlinear Schrödinger equation with potentials. We complement and improve the main results in [11,15], in the sense that we are considering the multiplicity results. Our results are in clear accordance with those for the classical local counterpart, while s = 1. We organize this paper as follows. Section 2 contains some known results. In Section 3, we solve the auxiliary equation which plays a key role in the proofs of our main theorems. In Section 4 the problem is reduced to a finite dimensional variational problem. Finally, in Section 5, we prove our main results. 2. Preliminary results In this section, we recall some preliminary results which will be useful along the paper. First, we will give some useful facts of the fractional order Sobolev spaces.

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For any 0 < s < 1, the fractional Sobolev space H s (RN ) is defined by

     |u(x) − u(y)|  N  2 N R , H s RN = u ∈ L 2 RN : ∈ L × R N+2s |x − y| 2 endowed with the natural norm  uH s (RN ) =

 |u| dx + 2

RN

R2N

|u(x) − u(y)|2 dxdy |x − y|N+2s

1

2

,

where the term  [u]H s (RN ) = R2N

|u(x) − u(y)|2 dxdy |x − y|N+2s

1

2

is the so-called Gagliardo semi-norm of u. Let S be the Schwartz space of rapidly decaying C ∞ functions in RN . Indeed, the fractional Laplacian (−)s can be viewed as a pseudo-differential operator of symbol |ξ |2s , as stated in the following Lemma 2.1. (See [21].) Let s ∈ (0, 1) and let (−)s : S → L2 (RN ) be the fractional Laplacian operator defined by (1.2). Then, for any u ∈ S ,   (−)s u(x) = F −1 |ξ |2s (F u) ,

∀ξ ∈ RN .

Now, one can see that an alternative definition of the fractional Sobolev space H s (RN ) via the Fourier transform is as follows

   N  N   s 2 2s 2 1 + |ξ | |F u| dξ < +∞ . H R = u∈L R : RN

It can be proved (Propositions 3.4 and 3.6 of [13]) that −1 2kN,s



 s 2 −1  |ξ |2s |F u|2 dξ = 2kN,s (−) 2 uL2 (RN ) = [u]2H s (RN ) ,

RN

where F denotes the Fourier transform. As a consequence, the norms on H s (RN ) u → uH s (RN ) ,  1  s 2 u → uL2 (RN ) + (−) 2 uL2 (RN ) 2 ,

1



u → uL2 (RN ) + RN

|ξ | |F u| dξ 2s

2

2

,

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are all equivalent. For the reader’s convenience, we review the main embedding result for this class of fractional Sobolev spaces. Lemma 2.2. (See [22,13].) Let N ≥ 1, 0 < s < 1 such that 2s < N . Then there exists a constant C = C(N, s) > 0, such that uL2∗s (RN ) ≤ CuH s (RN ) 2N for every u ∈ H s (RN ), where 2∗s = N−2s is the fractional critical exponent. Moreover, the embedding H s (RN ) ⊂ Lp (RN ) is continuous for any p ∈ [2, 2∗s ], and is locally compact whenever p ∈ [2, 2∗s ).

We need some tools to handle the nonlocality of the fractional Laplacian. The next lemma below provides a way to manipulate smooth truncations for the fractional Laplacian. First we give the homogeneous Sobolev space       s ∗ H0s RN = u ∈ L2s RN : |ξ | 2 F (u) ∈ L2 RN . This space can be equivalently defined as the completion of C0∞(RN ) under the norm  u2H s (RN ) = 0

|ξ |2s |F u|2 dξ.

RN

Lemma 2.3. (See [24].) Suppose that 0 < 2s < N and u ∈ H0s (RN ). Let ϕ ∈ C0∞ (RN ) and for each r > 0, ϕr (x) = ϕ(x/r). Then   uϕr → 0 in H0s RN as r → 0. If, in addition, ϕ ≡ 1 in a neighborhood of the origin, then   uϕr → u in H0s RN as r → ∞. From [16], we know that problem (−)s u + u = |u|p−1 u,

  u ∈ H s RN ,

(2.1)

has a unique radial positive ground state solution Q(x). The solution Q(x) is smooth and satisfies C C¯ ≤ Q(x) ≤ , 1 + |x|N+2s 1 + |x|N+2s

x ∈ RN

(2.2)

with some constants C  ≥ C¯ > 0. Moreover, Q(x) is non-degenerate, that is, the kernel of the ∂Q , i = 1, · · · , N }. Then linearized operator (−)s + 1 − pQp−1 is spanned by { ∂x i (−)s

∂Q ∂Q ∂Q − pQp−1 =− , ∂xi ∂xi ∂xi

i = 1, · · · , N.

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We apply Lemma C.2 of [16] to find the following decay estimate |∂xi Q| ≤

C , 1 + |x|N+2s

i = 1, · · · , N.

(2.3)

Making the change of variables, we are led to study the following equation (−)s u + V (εx)u = K(εx)|u|p−1 u,

  u ∈ H s RN .

(2.4)

If uε is a solution of (2.4) then uε ( xε ) solves (1.1). Solutions of (2.4) are the critical points of Iε (u) =



1 2

   (−) 2s u2 + V (εx)|u|2 dx −

RN

1 p+1

 K(εx)|u|p+1 dx. RN

Then Iε is well defined on H s (RN ) and belongs to C 2 (H s (RN ), R) under our assumptions. The solutions of (2.4) will be found near solutions of (−)s u + V (εξ )u = K(εξ )|u|p−1 u,

  u ∈ H s RN ,

(2.5)

where ξ ∈ RN is regarded as a parameter instead of an independent variable. The functional corresponding to problem (2.5) is 1 Jε (u) = 2



   (−) 2s u2 + V (εξ )|u|2 dx −

RN

1 p+1

 K(εξ )|u|p+1 dx. RN

Define 

V (εξ ) α(εξ ) = K(εξ )



1 p−1

,

 1 β(εξ ) = V (εξ ) 2s ,

and

  U (x) = α(εξ )Q β(εξ )x .

Then, it is easy to check that U (x) satisfies (2.5). Since (2.5) is translation invariant, it follows that any Uξ (x) = U (x − ξ ) is also a solution of (2.5). Let us introduce the critical manifold of Jε  M = Uξ (x) : ξ ∈ RN . Letting TUξ M denote the tangent space to M at Uξ , we know that TUξ M = span{∂ξ1 Uξ , · · · , ∂ξN Uξ }. We observe that   ∂ξi Uξ (x) = −∂xi Uξ (x) + ε∂xi α(εξ )Q β(εξ )(x − ξ )   + εα(εξ )∂xi β(εξ )∇Q β(εξ )(x − ξ ) · (x − ξ ).

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By the definition of α(x), β(x) and (2.3), we obtain ∂ξi Uξ (x) = −∂xi Uξ (x) + O(ε).

(2.6)

3. Solving the auxiliary equation πIε (Uξ + φ) = 0 In this section, we will find a solution φ ∈ (TUξ M)⊥ which satisfies the auxiliary equation πIε (Uξ + φ) = 0, where π denotes the orthogonal projection onto (TUξ M)⊥ . First of all, we show that the manifold M is a manifold of approximate critical points for Iε for ε > 0 small. We denote C and C0 , C1 , C2 , · · · are positive (possibly different) constants. Lemma 3.1. There exists C0 > 0 such that for all ξ ∈ RN and ε > 0 small, we have    I (Uξ ) ≤ C0 ε.

(3.1)

ε

Proof. Let us estimate Iε (Uξ )[v] for any v ∈ H s (RN ). Since Jε (Uξ ) = 0, we obtain Iε (Uξ )[v] =





 V (εx) − V (εξ ) Uξ vdx −

RN





 p K(εx) − K(εξ ) Uξ vdx.

RN

Using Hölder’s inequality, we infer that    I (Uξ )[v] ≤ v2 ·



ε

  V (εx) − V (εξ )2 U 2 dx

1

2

ξ

RN



  p+1 K(εx) − K(εξ ) p |Uξ |p+1 dx

+ vp+1 ·

p p+1

.

RN

By the conditions (V1) and (K1), there exist positive constant V1 and K1 , and one finds   V (εx) − V (εξ ) ≤ εV1 |x − ξ |,

  K(εx) − K(εξ ) ≤ εK1 |x − ξ |

for any x, ξ ∈ RN . It follows from the definition of Uξ that 

  V (εx) − V (εξ )2 U 2 dx ≤ ε 2 V 2 α 2 β −N−2 ξ



1

RN

|z|2 Q2 (z)dz,

RN

and   p+1    |K(εx) − K(εξ )| p |Uξ |p+1 dx    RN

≤ε

p+1 p

p+1

K1 p α p+1 β

− p+1+Np p

 |z| RN

p+1 p

Qp+1 (z)dz.

(3.2)

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By (2.2), we see that 

 |z| Q (z)dz ≤ C, 2

|z|

2

RN

p+1 p

Qp+1 (z)dz ≤ C.

RN

Hence, using Lemma 2.2, we obtain (3.1). This completes the proof. 2 We now show that Iε (Uξ ) is invertible on (TUξ M)⊥ , which plays a key role in solving the auxiliary equation. Lemma 3.2. There exist C1 > 0 and T > 0 such that ε > 0 small and ξ ∈ RN with |ξ | ≤ T . Then πIε (Uξ ) is invertible and [πIε (Uξ )]−1  ≤ C1 . Proof. Let us set Lε,ξ : (TUξ M)⊥ → (TUξ M)⊥ : (Lε,ξ u, v) = πIε (Uξ )[u, v],

for all u, v ∈ (TUξ M)⊥ .

So we only need to show that there exists k > 0 such that the interval (−k, k) does not have any eigenvalue of Lε,ξ on (TUξ M)⊥ . We decompose (TUξ M)⊥ = X1 ⊕ X2⊥ , where X1 is the space spanned by πUξ , X2 = Uξ ⊕ TUξ M. Since J  (Uξ ) = 0, we deduce that Iε (Uξ )[Uξ , Uξ ] =



RN



=

   (−) 2s Uξ 2 + V (εx)|Uξ |2 dx − p

 K(εx)|Uξ |p+1 dx

RN

    V (εx) − V (εξ ) |Uξ |2 dx − p K(εx) − K(εξ )|Uξ |p+1 dx

RN



+ (1 − p)

RN

 K(εξ ) |Uξ |p+1 dx.

RN

So, following the proof of Lemma 3.1, one finds Iε (Uξ )[Uξ , Uξ ] ≤ Cε + (1 − p)



   (−) 2s Uξ 2 + V (εξ )|Uξ |2 dx

RN

≤ Cε + C(1 − p)Uξ 2 . Hence, for ε > 0 small, we have Iε (Uξ )[Uξ , Uξ ] ≤ −CUξ 2 . Next, we will prove the inequality Iε (Uξ )[u, u] ≥ C4 u2 ,

∀u ∈ X2⊥ .

(3.3)

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Define a smooth cut-off function η(x) ∈ C0∞ (RN , [0, 1]) such that η(x) = 1, |x| ≤ 1; η(x) = 0, |x| ≥ 2. For R > 0, set ηR (x) = η( Rx ). We decompose u as u = u1 + u2 , where u1 = ηR (x − ξ )u(x), u2 = (1 − ηR (x − ξ ))u(x). It is easy to check that Iε (Uξ )[u, u] =



   (−) 2s u2 + V (εx)|u|2 dx − p

RN

 K(εx)|Uξ |p−1 u2 dx

RN

= Iε (Uξ )[u1 , u1 ] + Iε (Uξ )[u2 , u2 ] + 2Iε (Uξ )[u1 , u2 ].

(3.4)

Let us estimate Iε (Uξ )[u1 , u2 ]. We recall that Iε (Uξ )[u1 , u2 ] =



s



s

(−) 2 u1 (−) 2 u2 dx +

RN

RN



−p

V (εx)u1 u2 dx

K(εx)|Uξ |p−1 u1 u2 dx.

RN

Using the boundedness of K(x), Hölder’s inequality and Sobolev embeddings, one derives      K(εx)|Uξ |p−1 u1 u2 dx  ≤ C   RN

 |U (y)|p−1 u2 (y + ξ )dy

R≤|y|≤2R

≤ Cu



2

  U (y)p+1 dy

p−1 p+1

.

R≤|y|≤2R

It follows from (2.2) that      K(εx)|Uξ |p−1 u1 u2 dx  = oR (1)u2 .   RN

Thus, Iε (Uξ )[u1 , u2 ] ≥ C5





 s s (−) 2 u1 (−) 2 u2 + u1 u2 dx + oR (1)u2 .

(3.5)

RN

Furthermore, in a similar way we infer that Iε (Uξ )[u2 , u2 ] =



   (−) 2s u2 2 + V (εx)|u2 |2 dx − p

RN

≥ C5 u2 2 + oR (1)u2 .

 K(εx)|Uξ |p−1 u22 dx

RN

(3.6)

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Finally, we estimate the term Iε (Uξ )[u1 , u1 ]. Set w = u1 − v, where v is the projection of u1 onto X2 . It follows that  Uξ ∂ξi Uξ + u1 , ∂ξi Uξ  . 2 Uξ  ∂ξi Uξ 2 N

v = u1 , Uξ 

(3.7)

i=1

It is easy to see that Iε (Uξ )[u1 , u1 ] =



   (−) 2s u1 2 + V (εx)|u1 |2 dx − p

RN

= Jε (Uξ )[u1 , u1 ] +  −p



 K(εx)|Uξ |p−1 u21 dx

RN



 V (εx) − V (εξ ) |u1 |2 dx

RN



 K(εx) − K(εξ ) |Uξ |p−1 |u1 |2 dx.

(3.8)

RN

By (3.2), the definition of u1 , we can easily get        2   V (εx) − V (εξ ) |u | dx ≤ εV 1 1  

|y|u2 (y + ξ )dy ≤ εC6 u2 ,

|y|≤2R

RN

and        p−1 2  ≤ εK1 max U p−1  K(εx) − K(εξ ) |U | |u | dx ξ 1   x∈RN

|y|u2 (y + ξ )dy

|y|≤2R

RN

≤ εC7 u2 . From (3.8), we have then that Iε (Uξ )[u1 , u1 ] ≥ Jε (Uξ )[u1 , u1 ] − εCu2 .

(3.9)

Jε (Uξ )[u1 , u1 ] = Jε (Uξ )[w, w] + Jε (Uξ )[v, v] + 2Jε (Uξ )[w, v].

(3.10)

On the other hand,

Note that the ground state of (2.1) is non-degenerate, and using the same indirect argument as in [11], we deduce that Jε (Uξ )[w, w] ≥ cw2 .

(3.11)

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Recalling that u = u1 + u2 and u ∈ X2⊥ , one finds           s s u1 , Uξ  = u2 , Uξ  =  2 u2 (−) 2 Uξ + u2 Uξ dx . (−)   RN

From Lemma 2.3 and the definition of u2 , we infer 

s



s

s

(−) 2 u2 (−) 2 Uξ dx = RN

s

(−) 2 u(−) 2 Uξ dx

RN



−

 s s (−) 2 Uξ (−) 2 uηR (x − ξ ) dx

RN

= oR (1). By Hölder’s inequality, Lemma 2.2 and (2.2), we obtain         u2 Uξ dx  ≤    

  u(y + ξ )U (y)dy 

|y|≥R

RN

 ≤ Cu ·

  U (y)2 dy

1

2

= oR (1)u.

|y|≥R

Thus, u1 , Uξ  = oR (1)u.

(3.12)

From (2.3) and (2.6), using the same arguments we get u1 , ∂xi Uξ  = oR (1)u,

i = 1, · · · , N.

(3.13)

Putting together (3.7), (3.12) and (3.13), we have v = oR (1)u.

(3.14)

This implies that  K(εξ )|Uξ |p−1 |v|2 dx = oR (1)u2 . RN

It follows from (3.14) and the boundedness of V (x) that Jε (Uξ )[v, v] = oR (1)u2 .

(3.15)

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13

We can easily obtain Jε (Uξ )[w, v] = oR (1)u2 .

(3.16)

Putting together (3.11), (3.15), (3.16) and (3.10), we have Jε (Uξ )[u1 , u1 ] ≥ cw2 + oR (1)u2 . It follows from the definition of w, (3.14) and (3.9) that Iε (Uξ )[u1 , u1 ] ≥ cu1 2 + oR (1)u2 − Cεu2 .

(3.17)

The combination of (3.4)–(3.6) and (3.17) implies Iε (Uξ )[u, u] ≥ Cu2 + oR (1)u2 − Cεu2 . Letting ε → 0 and R → ∞, (3.3) follows. This completes the proof.

2

Lemma 3.3. For ε > 0 small enough and |ξ | ≤ T , there exists a unique function φε,ξ ∈ (TUξ M)⊥ such that πIε (Uξ + φ) = 0. Moreover, φε,ξ is of class C 1 with respect to ξ , and for some C > 0 there holds ∇ξ φε,ξ  ≤ Cε  ,

 = min{1, p − 1}.

(3.18)

Proof. For each φ ∈ (TUξ M)⊥ , we apply Taylor’s expansion Iε (Uξ + φ) = Iε (Uξ ) + Iε (Uξ )[φ] + S(Uξ , φ), where  S(Uξ , φ)[v] = −

 p p−1  K(εx) (Uξ + φ)p − Uξ − pUξ φ vdx,

RN

for v ∈ H s (RN ). By Lemma 3.2, solving the auxiliary equation πIε (Uξ + φ) = 0 is equivalent to find a fixed point of φ = Fε,ξ (φ),  where Fε,ξ (φ) = −L−1 ε,ξ π(Iε (Uξ ) + S(Uξ , φ)). For γ = 2C0 C1 , we set

 Bε = u ∈ (TUξ M)⊥ : u ≤ γ ε . We only need to show that Fε,ξ is a contraction on Bε .

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Using the Hölder inequality, the Sobolev embeddings and the boundedness of K(x), we observe that     2  S(Uξ , φ)[v] ≤ C |φ| + |φ|p |v|dx RN

  ≤ C  φ2 + φp v.

(3.19)

Similarly, we obtain   S(Uξ , φ1 )[v] − S(Uξ , φ2 )[v]   ≤ C  φ1 p−1 + φ2 p−1 + φ1  + φ2  φ1 − φ2 v.

(3.20)

For any φ ∈ Bε , by (3.19), Lemmas 3.1 and 3.2, we conclude       Fε,ξ (φ) ≤ C1 I  (Uξ ) + S(Uξ , φ) ε   ≤ C1 C0 + C1 C  γ 2 ε + C1 C  γ p ε p−1 ε. This implies that Fε,ξ (φ) ∈ Bε if ε is small enough. From Lemma 3.2 and (3.20), for φ1 , φ2 ∈ Bε , we find      Fε,ξ (φ1 ) − Fε,ξ (φ2 ) = L−1 π S(Uξ , φ1 ) − S(Uξ , φ2 )  ε,ξ   ≤ C1 C  φ1 p−1 + φ2 p−1 + φ1  + φ2  φ1 − φ2    ≤ 2C1 C  γ p−1 ε p−1 + γ ε φ1 − φ2  1 ≤ φ1 − φ2  2 for ε > 0 small enough. Thus Fε,ξ is a contraction mapping in Bε , and hence has a unique solution φε,ξ ∈ (TUξ M)⊥ of πIε (Uξ + φ) = 0 and satisfying φε,ξ  ≤ γ ε. We prove next that ξ → φε,ξ is of class C 1 . For fixed ε > 0 small, we set Hε (ξ, ω) = ω − Fε,ξ (ω),

ω ∈ (TUξ M)⊥ ∩ Bε .

Then, Hε (ξ, φε,ξ ) = 0. On the other hand,   Dω Hε (ξ, ω)[ϕ] = ϕ + L−1 ε,ξ π ∂ω S(Uξ , ω)[ϕ] where  ∂ω S(Uξ , ω)[ϕ, v] = −p RN

 p−1  ϕvdx. K(εξ ) (Uξ + ω)p−1 − Uξ

(3.21)

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Similarly, by (3.21), we get     ∂ω S(Uξ , ω) ≤ C γ ε + γ p−1 ε p−1 . Then, Dω Hε (ξ, ω) is an invertible operator if ε is sufficiently small; we also note Dω Hε (ξ, ω) and Dξ Hε (ξ, ω) are continuous. Thus, the implicit function theorem implies that the unique fixed point φε,ξ of Fε,ξ is of class C 1 with respect to ξ . Finally, we estimate the gradient ∇ξ φε,ξ . We recall that p−1

Lε,ξ φε,ξ = (−)s φε,ξ + V (εx)φε,ξ − pK(εx)Uξ

φε,ξ .

For all v ∈ (TUξ M)⊥ , we observe that 

 vLε,ξ φε,ξ dx +

RN



 V (εx) − V (εξ ) Uξ vdx

RN





 p K(εξ ) − K(εx) Uξ vdx −

+ RN

 K(εx)f (Uξ , φε,ξ )vdx = 0,

(3.22)

RN

where p

p−1

f (Uξ , φε,ξ ) = (Uξ + φε,ξ )p − Uξ − pUξ

φε,ξ .

By differentiation of ξ in (3.22), we have 



p−2

vLε,ξ ∂ξi φε,ξ dx − p(p − 1) RN

 +p

RN

∂ξi Uξ φε,ξ vdx

RN

 − ε∂xi V (εξ )

K(εx)Uξ

 Uξ vdx +



 V (εx) − V (εξ ) ∂ξi Uξ vdx

RN



 p−1 K(εξ ) − K(εx) Uξ ∂ξi Uξ vdx + ε∂xi K(εξ )

RN



p

Uξ vdx

RN



K(εx)(fUξ · ∂ξi Uξ + fφε,ξ · ∂ξi φε,ξ )vdx = 0,

−

(3.23)

RN

where p−1

fUξ · ∂ξi Uξ = p(Uξ + φε,ξ )p−1 ∂ξi Uξ − pUξ

p−2

∂ξi Uξ − p(p − 1)Uξ

and p−1

fφε,ξ · ∂ξi φε,ξ = p(Uξ + φε,ξ )p−1 ∂ξi φε,ξ − pUξ

∂ξi φε,ξ .

∂ξi Uξ φε,ξ

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 Denote L = Lε,ξ − Gφε,ξ , where Gφ is defined as (Gφ u, v) = RN K(εx)fφ uvdx. We can rewrite (3.23) as follows      p−2 L ∂ξi φε,ξ , v = p(p − 1) K(εx)Uξ ∂ξi Uξ φε,ξ vdx + ε∂xi V (εξ ) Uξ vdx  −

RN



 V (εx) − V (εξ ) ∂ξi Uξ vdx − p

RN



p

RN



 p−1 K(εξ ) − K(εx) Uξ ∂ξi Uξ vdx

RN



Uξ vdx +

− ε∂xi K(εξ )

RN



K(εx)fUξ ∂ξi Uξ vdx.

RN

Using the estimates (2.6) and the arguments already carried out before, one can easily prove that       L ∂ξ φε,ξ , v  ≤ C ε + φε,ξ  v. i Let us point out that fφ → 0 as φ → 0. Hence, the operator L is invertible for ε > 0 small by Lemma 3.2. Thus, from (3.21) we obtain ∂ξi φε,ξ  ≤ Cε  . 2

This completes the proof.

4. The finite-dimensional variational reduction In this section, we look for critical points of Iε with the form u = Uξ + φε,ξ , where φε,ξ is the function obtained in Lemma 3.3. Define  M = Uξ + φε,ξ : ξ ∈ RN . We show that any constrained critical point u of Iε on M is a stationary point of Iε , namely Iε (u) = 0. Define the reduced functional Φε (ξ ) = Iε (Uξ + φε,ξ ). Lemma 4.1. Assume ξ0 ∈ RN . If Φε (ξ0 ) = 0, then Iε (Uξ0 + φε,ξ0 ) = 0. Proof. Let us set uξ = Uξ + φε,ξ . We observe that 

s 2

∂ξi Φε (ξ ) =



s 2

(−) uξ (−) ∂ξi uξ dx +

RN



− 

RN

RN

K(εx)|uξ |p−1 uξ ∂ξi uξ dx

 = I  (uξ ), ∂ξi uξ .

V (εx)uξ ∂ξi uξ dx

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It follows that 

 I  (uξ ), ∂ξi uξ ξ =ξ = 0.

(4.1)

0

By (3.18), we have ∂ξi φε,ξ → 0 as ε → 0, for i = 1, · · · , N . Then, TUξ M is close to TUξ M for ε > 0 small enough. By (4.1), we have Iε (uξ0 ) is orthogonal to TUξ0 M. On the other hand, Lemma 3.3 implies Iε (uξ0 ) ∈ TUξ0 M. Thus, Iε (Uξ0 + φε,ξ0 ) = 0. This completes the proof. 2 Let us now expand the reduced functional Φε (ξ ) = Iε (Uξ + φε,ξ ). Since Uξ satisfies Eq. (2.5), we have Φε (ξ ) =



1 2

  (−) 2s (Uξ + φε,ξ )2 dx + 1 2

RN

− =

1 p+1

+

1 2

K(εx)|Uξ + φε,ξ |p+1 dx RN

 K(εξ )|Uξ |

p+1



1 dx + 2

RN



 V (εx) − V (εξ ) |Uξ |2 dx

RN

   (−) 2s φε,ξ 2 + V (εx)|φε,ξ |2 dx + I  (Uξ )[φε,ξ ] ε

RN

−

V (εx)|Uξ + φε,ξ |2 dx RN



1 1 − 2 p+1 



1 p+1

1 − p+1



  p+1 p K(εx) |Uξ + φε,ξ |p+1 − Uξ − (p + 1)Uξ φε,ξ dx

RN





 K(εx) − K(εξ ) |Uξ |p+1 dx.

RN

It follows from the definition of Uξ that 

 K(εξ )|Uξ |

p+1

dx = Γ (εξ )

RN



We denote m = ( 12 − 1 Υε (ξ ) = 2

 RN

and



1 p+1 ) RN

Qp+1 dx.

RN

Qp+1 dx,

 V (εx) − V (εξ ) |Uξ |2 dx −

1 p+1

 RN



 K(εx) − K(εξ ) |Uξ |p+1 dx,

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18

1 Hε (ξ ) = 2



   (−) 2s φε,ξ 2 + V (εx)|φε,ξ |2 dx

RN

−

1 p+1



  p+1 p K(εx) |Uξ + φε,ξ |p+1 − Uξ − (p + 1)Uξ φε,ξ dx.

RN

Then, Φε (ξ ) = mΓ (εξ ) + Iε (Uξ )[φε,ξ ] + Υε (ξ ) + Hε (ξ ). Lemma 4.2. The following estimates hold: Φε (ξ ) = mΓ (εξ ) + o(ε),

(4.2)

∇ξ Φε (ξ ) = εm∇Γ (εξ ) + o(ε).

(4.3)

and

Proof. By Lemma 3.1 and (3.21), we get    I (Uξ )[φε,ξ ] ≤ C0 εφε,ξ  ≤ C0 γ ε 2 . ε

Using (3.2) and the polynomial decay of Q, we infer   Υε (ξ ) = o(ε). Using the boundedness of V (x) and (3.21), we obtain 

   (−) 2s φε,ξ 2 + V (εx)|φε,ξ |2 dx ≤ Cφε,ξ 2 ≤ Cγ 2 ε 2 .

RN

Furthermore, by the boundedness of K(x), the Sobolev embeddings and (3.21), we get          K(εx) |Uξ + φε,ξ |p+1 − U p+1 − (p + 1)U p φε,ξ dx  ≤ C γ 2 ε 2 + γ p+1 ε p+1 . ξ ξ   RN

Hence, we observe that   Hε (ξ ) = o(ε). Then, (4.2) holds. Next, we estimate the derivatives of Φε with respect to ξ . We compute   ∂ξi Φε (ξ ) = mε∂xi Γ (εξ ) + ∂ξi Iε (Uξ )[φε,ξ ] + ∂ξi Υε (ξ ) + ∂ξi Hε (ξ ).

(4.4)

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By a direct calculation, we find   ∂ξi Iε (Uξ )[φε,ξ ] = Iε (Uξ )[∂ξi φε,ξ ] + Iε (Uξ )[∂ξi Uξ , φε,ξ ].

(4.5)

From Lemma 3.1 and (3.18), we obtain    I (Uξ )[∂ξ φε,ξ ] ≤ Cε +1 . ε i

(4.6)

Since Uξ satisfies (2.5), by the non-degeneracy of Q, we have now that Iε (Uξ )[∂ξi Uξ , φε,ξ ] = Iε (Uξ )[∂ξi Uξ , φε,ξ ] + Jε (Uξ )[∂xi Uξ , φε,ξ ]  = φε,ξ (−)s (∂ξi Uξ + ∂xi Uξ )dx RN



+ RN



 V (εx)∂ξi Uξ + V (εξ )∂xi Uξ φε,ξ dx



−p



 p−1 K(εx)∂ξi Uξ + K(εξ )∂xi Uξ Uξ φε,ξ dx.

RN

Using (3.21) and (2.6), we find 

 φε,ξ (−) (∂ξi Uξ + ∂xi Uξ )dx = s

RN

(∂ξi Uξ + ∂xi Uξ )(−)s φε,ξ dx

RN



= O(ε)

(−)s φε,ξ dx

RN

≤ O(ε)φε,ξ 2 ≤ Cε 3 . Moreover, by (3.21), (2.6) and arguing as in the proof of Lemma 3.1, one easily obtains        = o(ε),  V (εx)∂ φ U + V (εξ )∂ U dx ξi ξ xi ξ ε,ξ   RN

      p−1  K(εx)∂ξi Uξ + K(εξ )∂xi Uξ Uξ φε,ξ dx  = o(ε).  RN

Hence, we have Iε (Uξ )[∂ξi Uξ , φε,ξ ] = o(ε). It follows from (4.5) and (4.6) that     ∂ξ I (Uξ )[φε,ξ ]  = o(ε). ε i

(4.7)

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Let us compute 



 ε V (εx) − V (εξ ) Uξ ∂ξi Uξ dx − ∂xi V (εξ ) 2

∂ξi Υε (ξ ) = RN

+

ε ∂x K(εξ ) p+1 i



p+1

Uξ

 dx −

RN

 Uξ2 dx

RN



 p K(εx) − K(εξ ) Uξ ∂ξi Uξ dx.

RN

By the boundedness of ∇V and ∇K, we have     ε  ∂x V (εξ ) U 2 dx  = o(ε), ξ  2 i

    ε  p+1    p + 1 ∂xi K(εξ ) Uξ dx  = o(ε).

RN

RN

It follows from the similar proof of Lemma 3.1 that   ∂ξ Υε (ξ ) = o(ε). i We observe that 



 s s (−) 2 φε,ξ (−) 2 ∂ξi φε,ξ + V (εx)φε,ξ ∂ξi φε,ξ dx

∂ξi Hε (ξ ) = RN



−

 K(εx) |Uξ + φε,ξ |p−1 (Uξ + φε,ξ )(∂ξi Uξ + ∂ξi φε,ξ )

RN p

p−1

− Uξ ∂ξi Uξ − pUξ

 p ∂ξi Uξ φε,ξ − Uξ ∂ξi φε,ξ dx.

From the boundedness of V (x), (3.18) and (3.21), we deduce that       s s  2 2 (−) φε,ξ (−) ∂ξi φε,ξ + V (εx)φε,ξ ∂ξi φε,ξ dx   RN

    ≤ C φε,ξ 2 + ∂ξi φε,ξ 2 ≤ C ε2 + ε 2 .

Using the Hölder inequality, Sobolev embeddings, (3.21) and (2.6), we infer        K(εx) |Uξ + φε,ξ |p−1 (Uξ + φε,ξ ) − U p − pU p−1 φε,ξ ∂ξ Uξ dx  i ξ ξ   RN

    ≤ C φε,ξ 2 + φε,ξ p ≤ C ε 2 + ε p ,

and          K(εx) |Uξ + φε,ξ |p−1 (Uξ + φε,ξ ) − U p ∂ξ φε,ξ dx  ≤ C ε 1+ + ε p+ . i ξ   RN

(4.8)

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Thus,   ∂ξ Hε (ξ ) = o(ε). i

(4.9)

Putting together (4.7)–(4.9) with (4.4), yields the estimate (4.3). This concludes the proof. 2 5. The proof of main results In this section, we will show our main results of the present paper. First, we give the proof of the existence of solution of (1.1) ε (ξ ) = Φε ( ξ ). It follows from (4.2) that Proof of Theorem 1.1. Set Φ ε ε (ξ ) = mΓ (ξ ) + o(ε). Φ Let x0 ∈ RN be an isolated minimum of Γ (x), that is, for some δ > 0, Γ (x) > Γ (x0 )

for all 0 < |x − x0 | < δ.

(5.1)

It is easy to see that   ε (x0 ) = m Γ (ξ ) − Γ (x0 ) + o(ε). ε (ξ ) − Φ Φ By (5.1), we observe that Γ (ξ ) − Γ (x0 ) ≥ σ > 0 for all 0 < |ξ − x0 | = ρ ≤ δ. Hence, inf

|ξ −x0 |=ρ

ε (x0 ) ε (ξ ) > Φ Φ

ε (ξ ) has a critical point ξε satisfying for all 0 < |ξ − x0 | = ρ ≤ δ and ε > 0 small. It follows that Φ ξε → x0 as ε → 0. Then, ξεε is a critical point of Φε . From Lemma 4.1 we see that U (x − ξεε ) + φε,ξε is a critical point of Iε and hence a solution of (2.4). Thus, (1.1) has a solution uε of the form

x − ξε uε (x) = U + φε,ξε ε with ξε → x0 and φε,ξε  → 0 as ε → 0. The case of a maximum requires merely obvious changes. Now, we consider x0 ∈ RN which is an isolated stable critical point of Γ (x). By (4.3), we infer that for ε > 0 small, ε , x0 , 0) = ind(∇ξ Γ, x0 , 0) = 0. ind(∇ξ Φ ε (ξ ) has a critical point ξε satisfying ξε → x0 as ε → 0. Similarly, we deduce This implies that Φ that (1.1) has a solution uε which concentrates at x0 as ε → 0. 2 In order to prove the problem (1.1) has l(Σ) solutions, we recall a result of [6].

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Theorem 5.1. Let f ∈ C 2 (RN , R) and suppose that Σ ⊂ RN is a non-degenerate compact manifold of critical points of f . Let N be a neighborhood of Σ and let g ∈ C 1 (RN , R). If f − gC 1 is sufficiently small, the function g has at least l(Σ) critical points in N . Proof of Theorem 1.2. We let f (ξ ) = mΓ (ξ ), Σ is obviously a non-degenerate smooth comε (ξ ). By Lemma 4.2, we know pact manifold of f . Fix a neighborhood N of Σ, and set g(ξ ) = Φ that f − gC 1 → 0 as ε → 0. From Theorem 5.1, we can deduce the existence of at least l(Σ) critical points of g in N when ε → 0 is small enough. Now let ξε,i ∈ N be any of those critical ξ ξε points. Then, ε,i ε is a critical point of Φε . It follows from Lemma 4.1 that U (x − ε ) + φε,ξε is a critical point of Iε and hence a solution of (2.4). Thus, (1.1) has a solution uε,i of the form

x − ξε,i uε,i (x) = U ε

+ φε,ξε,i

with φε,ξε,i  → 0 and any ξε,i →  ξi ∈ N as ε → 0. Using (4.3), we infer that  ξi is a stationary  point of Γ . Then, taking N possibly smaller, it follows that ξi ∈ Σ , and hence uε,i (x) concentrates near a point of Σ. 2 Proof of Theorem 1.3. We consider the case that Γ (x) has an isolated strict local minimum in compact set Σ , that is, there exists δ > 0 satisfying  θ = inf Γ (x) : x ∈ ∂Σδ > μ = Γ (x)|Σ , ε (ξ ) ≤ where Σδ is the δ-neighborhood of Σ. Set X = {ξ ∈ Σδ : Φ εδ > 0 such that for all ε ∈ (0, εδ ), we have Σ ⊂ X ⊂ Σδ .

(5.2)

m(θ+μ) }. By (5.2), there exists 2

(5.3)

ε (ξ0 ) ≤ Let {ξn } ⊂ X, suppose that there exists ξ0 ∈ RN such that ξn → ξ0 as n → ∞. Then, Φ m(θ+μ) and ξ ∈ Σ . If ξ ∈ ∂Σ , using (5.2) we observe that 0 δ 0 δ 2 ε (ξ0 ) ≥ mθ + o(ε) > Φ

m(θ + μ) , 2

a contradiction. We conclude ξ0 ∈ Σ , and hence X is compact. Then Lusternik–Schnirelman ε (ξ ) has at least cat(X, Σδ ) critical points on Σδ . It follows from (5.3) theorems imply that Φ ε (ξ ) has at and the properties of the category that cat(X, Σδ ) ≥ cat(Σ, Σδ ). Thus, we obtain Φ least cat(Σ, Σδ ) critical points. Using arguments already carried out in the proof of Theorem 1.2, we conclude that Eq. (1.1) has at least cat(Σ, Σδ ) solutions that concentrate near points of Σ for ε ∈ (0, εδ ). Similarly, we deduce the case of a maximum. 2 Acknowledgments The authors would like to express sincere thanks to the anonymous referee for his/her valuable comments and suggestions. This research was supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions (No. 14KJB110017), Project of Graduate Education

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