Infinitely many radial positive solutions for nonlocal problems with lack of compactness

Infinitely many radial positive solutions for nonlocal problems with lack of compactness

Fen Zhou

, Zifei Shen

and Vicent

iu D. R˘adulescu

1Department of Mathematics and Computer Science, Zhejiang Normal University, Jinhua 321004, P. R. China

2Department of Mathematics, Yunnan Normal University, Kunming 650500, P. R. China

3Faculty of Applied Mathematics, AGH University of Science and Technology, 30-059 Kraków, Poland

4Department of Mathematics, University of Craiova, 200585 Craiova, Romania

Received 14 January 2021, appeared 11 April 2021 Communicated by Dimitri Mugnai

Abstract. We are concerned with the qualitative and asymptotic analysis of solutions to the nonlocal equation

(−∆)^su+V(|z|)u=Q(|z|)u^p inR^N,

where N≥3, 0<s<1, and 1< p< _N−2s^2N . Asr→∞, we assume that the potentials V(r)andQ(r)behave as

V(r) =V0+^a1 r^α +O

1

r^α+θ1

Q(r) =Q0+^a2 r^β +O

1

r^β+θ2

where a₁, a₂∈R,α, β> _N+2s+1^N+2s , andθ₁, θ₂>0, V₀, Q₀>0. Under various hypothe- ses on a₁,a₂,α,β, we establish the existence of infinitely many radial solutions. A key role in our arguments is played by the Lyapunov–Schmidt reduction method.

Keywords: fractional Laplacian, radial solution, lack of compactness, Lyapunov–

Schmidt reduction method.

2020 Mathematics Subject Classification: 35R11, 35A15, 35B40, 47G20.

1 Introduction and the main result

We consider the following nonlocal equation driven by the fractional Laplace operator

(−_∆)^su+V(|z|)u=Q(|z|)u^p, inR^N. (1.1)

BCorresponding author.

Emails: zhoufen_85@163.com (F. Zhou), szf@zjnu.edu.cn (Z. Shen), radulescu@inf.ucv.ro (V. D. R˘adulescu).

Fractional powers of the Laplacian arise in various equations in mathematical physics and related fields; see, e.g., [1], [9], and [14]. Numerous results related to equations with fractional Laplace operator sprout in literature. A characterization of the fractional Laplacian through Dirichlet–Neumann maps was given in [3]. Regularity for fractional elliptic equations was investigated in [4] and [17]. Existence of solutions was studied in many papers; see, e.g., [2,7,12].

Along with different results, there are various enlightening approaches. In [10], the author obtained some symmetry results for equations involving the fractional Laplacian inR^Nby the method of moving planes. In [2], symmetry results for nonlinear equations with fractional Laplacian were achieved by the sliding method. Geometric inequality was applied to inves- tigate symmetry properties for a boundary reaction problem in [18]. The method of moving planes and ABP (Aleksandrov–Bakelman–Pucci) estimates for fractional Laplacian were em- ployed in [6] to study radial symmetry and monotonicity properties for positive solutions of fractional Laplacian. We refer the readers to [7] and [11] for very recent new approaches dealing with fractional Laplacian equations, and to [15] for a comprehensive overview of variational methods for nonlocal fractional problems.

Inspired by [19], we obtain the existence of radial positive solutions to (1.1) by Lyapunov–

Schmidt reduction. To the best of our knowledge, this method has never been employed in investigating radial solutions to equations as (1.1).

We will use the radial solution of

(−_∆)^su+u=u^p in R^N (1.2)

to build up the approximate solutions of problem (1.1). The uniqueness and nondegeneracy of the radial positive solution to problem (1.2) are established in [8].

Our result is based on the following growth assumptions forV(|z|)andQ(|z|)near infin- ity:

(V): there exist constants a₁ ∈ R, α> _{1, and}θ₁ > 0, such that V(r) = V₀+ ^a_rα¹ +O( ¹

r^α+θ1) _as r→_∞;

(Q): there exist constants a₂ ∈ R, β> _{1, and} θ₂ > 0, such that Q(r) =Q₀+ ^a2

r^β +O( ¹

r^β+θ2)_as r→_∞.

We assume throughout this paper thatV₀ =_{1 and}Q₀ =_1.

Let

x_j =

rcos2(j−1)π

k ,rsin2(j−1)π k , 0

, j=1, . . . ,k, where 0 is the zero vector inR^N−2,r ∈ r₀k^{NN++2s2s−τ},r₁k^{NN++2s2s−τ}

,τ=min{α,β}, 0<r₀ <r₁, and k is the number of the bumps of the solution.

Setz= (z⁰,z⁰⁰),z⁰ ∈_R², z⁰⁰ ∈_R^N−2 and define H_rs=

u:u∈ H^s(_R^N),uis even inz_h, h=2, . . . ,N, u(rcosθ,rsinθ,z⁰⁰) =u

rcos

θ+^2πj k

,rsin

θ+ ^2πj k

,z⁰⁰

, whereH^s(_R^N)represents the fractional Sobolev space

H^s(_R^N):= (

u∈ L²(_R^N): u(x)−u(y)

|x−y|^N2+s ∈ L²(_R^N×_R^N) )

, 0<s<1.

Let W be the unique nondegenerate radial positive solution of problem (1.2), Then the result in [8] shows that there exist constants B₁ >B2 >0, such that

B₂

1+|z−x_j|^N+2s ≤W_xj(z)≤ ^B1

1+|z−x_j|^N+2s, whereW_xj(z) =W(z−x_j)_.

Set

Ur(z) =

∑

Wx_j(z).

The main result of this paper establishes the following multiplicity property.

Theorem 1.1. Assume that V(r), Q(r) satisfy (V) and (Q), while a₁, a2, α, β satisfy one of the following conditions:

(i) a₁ >0, a₂ =0, α< N+2s, and α≤ β;

(ii) a₁ >0, a2 >0, α< N+2s, and β≥ N+2s;

(iii) a₁ >0, a₂ <0, α< N+2s, and α> β;

(iv) a₁ =0, a2 <0, α≥ β, andβ< N+2s;

(v) a₁ <0, a₂ <0, α≥ N+2s, and β< N+2s.

Then there exists a positive integer k₀such that for any k ≥k₀, problem(1.1)has a solution U_k of the form

U_k(z) =Ur_k(z) +w_k, where w_k ∈ H_rs, r_k ∈ r₀k^{NN++2s2s−τ},r₁k^{NN++2s2s−τ}

, and as k→+_∞,

Z

R^N |(−_∆)^2sw_k|²+w²_k

→0.

For some of the abstract methods used in this paper, we refer to the monographs by Molica Bisci and Pucci [13] and Papageorgiou, R˘adulescu and Repovš [16].

2 Reduction

Let

P_j = ^∂Wxj

∂r , j=1, . . . ,k, where

x_j =

rcos2(j−1)π

k ,rsin2(j−1)π k , 0

, j=1, . . . ,k and

r∈ S:=

"

N+2s τ −e

_N+¹_2s−τ

k^{NN++2s2s−τ},

N+2s τ +e

_N+2s¹_−τ k^{NN++2s2s−τ}

# .

We have denotedτ:=min{α,β}, whereαandβare the constants in the expansions ofVand Q, ande>0 is a small constant.

Define

H:=

u:u∈ Hrs, Z

R^NW_x^P_j⁻¹P_ju=0, j=1, . . . ,k.

. The norm and the inner product inH^s(_R^N)are defined as

kuk= hu,ui¹², u∈ H^s(_R^N), hu,vi=

Z

R^N (−_∆)^2su(−_∆)^s2v+V(|z|)uv

, u,v∈ H^s(_R^N). We can easily check that

Z

R^N (−_∆)^s2u(−_∆)^2sv+V(|z|)uv−pQ(|z|)U_r^p−1uv

, u,v∈ H

is a bounded bilinear functional inH. Thus, there exists a bounded linear operator Mfrom H toH satisfying

hMu,vi=

Z

R^N (−_∆)^2su(−_∆)^s2v+V(|z|)uv−pQ(|z|)U_r^p−1uv

, u,v∈ H. (2.1) We now establish that M is invertible in H.

Lemma 2.1. There exists a constantρ >0, independent of k, such that for any r∈ S, kMuk ≥ρkuk, u∈ H.

Proof. We argue by contradiction. If the thesis does not hold, then for anyρ_k = ¹_k(k → +_∞), there existsr_k ∈S,u_k ∈ H, such that

kMu_kk<ρ_kku_kk. It follows that

kMu_kk= o(1)ku_kk. Then,

hMu_k,ϕi=o(1)ku_kkkϕk, ∀ϕ∈H. (2.2) We can assumeku_kk²=k.

Let

Ωj =

z = (z⁰,z⁰⁰)∈_R²×_R^N−2: z⁰

|z⁰|^, x_j

|x_j|

≥cosπ k

. By symmetry and the definition ofM, we conclude from (2.2) that for allϕ∈ H,

Z

Ω1

(−_∆)^s2u_k(−_∆)^s2ϕ+V(|z|)u_kϕ−pQ(|z|)U_r^p_k⁻¹u_kϕ

= ¹

khMu_k,ϕi=o 1

√ k

kϕk. (2.3) Particularly,

Z

Ω1

|(−_∆)^s2u_k|²+V(|z|)u_k²−pQ(|z|)U_r^p_k⁻¹u_k²

=o(1) and

Z

Ω1

|(−_∆)^s2u_k|²+V(|z|)u_k²

=_{1. (2.4)}

Letue_k(z) =u_k(z+x₁). Since

|x₂−x₁|=2rsinπ

k ≥2rπ 2k ≥

N+2s 2τ

_N+¹_2s−

τ

k^N+τ2s−τπ,

it follows that for any R>0,B_R(x₁)⊂_Ω1. Then from (2.4), we have for all R>0, Z

B_R(₀)

|(−_∆)^2su_ek|²+V(|z|)u_ek²

≤1.

So, we can assume that there existsu∈ H^s(_R^N), such that ask →+_∞, ue_k *u, in H_loc^s (_R^N),

and

ue_k →u, in L²_loc(_R^N).

Sinceue_k is even inz_h, h=2, . . . .,N, thenuis even inz_h, h=2, . . . ,N.

Besides, by

Z

R^NW_x^P₁⁻¹P₁u_k =0, we know that

Z

R^NW^P−1∂W

∂x₁ue_k =0.

So,usatisfies

Z

R^NW^P−1∂W

∂x₁u=0. (2.5)

We prove in what follows thatusatisfies

(−_∆)^s_u+_u−pW^p−1u=_{0, in R}^N_{. (2.6)} Define

He =

ϕ:ϕ∈ H^s(_R^N), Z

R^NW^P−1∂W

∂x₁ϕ=0

.

For any R > 0, let ϕ ∈ C₀^∞(B_R(0))∩He be any function which is even in z_h, h = 2, . . . ,N.

Then ϕ_k(z):= ϕ(z−x₁) ∈ C^∞₀ (B_R(x₁)). Substituting ϕin (2.3) with ϕ_k, then by Lemma A.1, we get

Z

R^N (−_∆)^s2u(−_∆)^2sϕ+uϕ−pW^p−1uϕ

=0. (2.7)

In addition, since u is even in z_h, h = 2, . . . ,N, relation (2.7) holds for any ϕ ∈ C₀^∞(_R^N), which is odd in z_h,h = 2, . . . ,N. Thus, relation (2.7) is true for any ϕ ∈ C₀^∞(B_R(0))∩H. Bye the density ofC₀^∞(_R^N)_in H^s(_R^N)_{, we have}

Z

R^N (−_∆)^2su(−_∆)^s2ϕ+uϕ−pW^p−1uϕ

=_0, ∀ϕ∈ He (2.8)

Meanwhile, relation (2.8) holds for ϕ = ^∂W_∂x

1. Therefore, (2.8) holds for any ϕ ∈ H^s(_R^N). Substituting ϕin (2.8) with uyields (2.6).

Since W is non-degenerate, we have u = C^∂W_∂x

1 because u is even in z_h, h = 2, . . . .,N. By (2.5), we know that

u=0, which implies

Z

B_R(x₁)u_k² =o(1), ∀R>0.

Besides, it follows from LemmaA.1that there existsC⁰ >0 such that Ur_k(x)≤C⁰, for all x∈ _Ω1.

It follows that

o(1) =

Z

Ω1

|(−_∆)^s2u_k|²+V(|z|)u_k²−pQ(|z|)U_r^p_k⁻¹u_k²

=

Z

Ω1

|(−_∆)^s2u_k|²+V(|z|)u_k²

+o(1)−C Z

Ω1

u_k²

≥ ¹ 2

Z

Ω1

|(−_∆)^2su_k|²+V(|z|)u_k²

+o(1), which contradicts (2.4). The proof is now complete.

Define

I(u) = ¹ 2

Z

R^N |(−_∆)^s2u|²+V(|z|)u²

− ¹ p+1

Z

R^NQ(|z|)|u|^p+1. (2.9) Let

J(φ) = I(Ur+φ), φ∈ H.

Then, J(₀) = ¹

2 Z

R^N |(−_∆)^2s_Ur|²+_V(|z|)Ur2

− ¹ p+1

Z

R^NQ(|z|)|Ur|^p+1.

= ¹ 2

Z

R^NU_r

∑

W_x^p_j+¹ 2

Z

R^N(V(|z|)−1)U_r²− ¹ p+₁

Z

R^N(Q(|z|)−1)U_r^p+1− ¹ p+₁

Z

R^NU_r^p+1 becauseW_xj solves (1.2).

Lemma 2.2. There exists a positive integer k₀ such that for each k ≥ k₀, there is a C¹ map from S to H_rs: φ_k =φ_k(r), r=|x₁|, satisfyingφ_k ∈ H, and

J⁰(φ_k)

H =0.

Moreover, there exists a constant C>0, independent of k, such that kφ_kk ≤ ^C

k

(N+2s)(τ−1)+τ 2(N+2s−τ) +δ

, (2.10)

whereδ >₀is a small constant.

Proof. We expand J(φ_k)as

J(φ_k) = J(0) +l(φ_k) + ¹

2hMφ_k,φ_ki+R(φ_k), φ_k ∈ H, where

l(φ_k) =hI⁰(U_r),φ_ki

=

Z

R^N(V(|z|)−1)Urφ_k+

Z

R^N

_k

j

∑

W_x^p_j−U_r^p

φ_k−

Z

R^N(Q(|z|)−1)U_r^pφ_k. Mis the bounded linear map defined in (2.1) and

R(φ_k) =− ¹ p+1

Z

R^NQ(|z|)

|U_r+φ_k|^p+1−U_r^p+1−(p+1)U_r^Pφ_k−¹

2p(p+1)U_r^p−1φ_k²

. Sincel(φ_k)is a bounded linear functional in H, there existsl_k ∈ H, such that

l(φ_k) =hl_k,φ_ki. Then,φ_k being a critical point of J is equivalent to

l_k+Mφ_k+R⁰(φ_k) =_{0. (2.11)}

Since Mis invertible, we can infer from (2.11) that

φ_k = T(φ_k):=−M⁻¹(l_k+R⁰(φ_k)). Define

E= (

φ_k :φ_k ∈H, kφ_kk ≤ ¹ k

(N+2s)(τ−1)+τ 2(N+2s−τ)

) . Next, we check thatTis a contraction map fromEto E.

Case 1: p≤2. It is easy to verify that

kR⁰(φ_k)k ≤Ckφ_kk|^p. In fact,

|hR⁰(φ_k),vi|= Z

R^NQ(|z|)(pU_r^p−1φ_kv+U_r^pv− |U_r+φ_k|^p−1(U_r+φ_k)v)

= Z

R^NQ(|z|)p(|U_r+θφ_k|^p−1−U_r^p−1)φ_kv

≤C Z

R^N|(|U_r+θφ_k|^p−1−U_r^p−1)| |φ_k| |v|

≤C Z

R^N|θφ_k|^p−1|φ_k| |v|

≤Ckφ_kk^pkvk where 0<θ <_1.

Then, by the boundedness of Mand Lemma 2.3, kT(φ_k)k ≤C(kl_kk+kφ_kk^p)≤ ^C

k

(N+2s)(τ−1)+τ 2(N+2s−τ) +δ

+ ^C

k

(N+2s)(τ−1)+τ

2(N+2s−τ) p ≤ ¹

k

(N+2s)(τ−1)+τ 2(N+2s−τ)

(2.12) which implies thatTmaps EtoE.

In addition,

kR⁰⁰(φ_k)k ≤Ckφ_kk^p−1. In fact,

|hR⁰⁰(φ_k)v,hi|= Z

R^NQ(|z|)(pU_r^p−1vh−p|Ur+φ_k|^p−1vh)

= p Z

R^NQ(|z|)(U_r^p−1vh− |Ur+φ_k|^p−1)vh

≤C Z

R^N|φ_k|^p−1|v||h|

≤Ckφ_kk^p−1kvk khk_. Thus, forφ_k1, φ_k2 ∈E,

kT(φ_k1)−T(φ_k2)k= kM⁻¹R⁰(φ_k1)−M⁻¹R⁰(φ_k2)k

≤ kM⁻¹k kR⁰(φ_k1)−R⁰(φ_k2)k ≤ kM⁻¹k kR⁰⁰(φ_k1 +θ(φ_k2−φ_k1)k kφ_k2 −φ_k1k

≤C(kφ_k1k^p−1+kφ_k2k^p−1k)kφ_k1−φ_k2k ≤ ¹

2kφ_k1 −φ_k2k.

Note that the last inequality holds only whenk is large enough, which implies the existence of k0 in Lemma 2.2. Therefore, T is a contraction map from E to E. Then the contraction mapping theorem implies the existence ofφ_k as a critical point of J restricted to H.

Case 2: p>2.

Settingh(t) =|U_r+tφ_k|^p−1(U_r+tφ_k)v, then by Taylor’s formula,

|hR⁰(φ_k)_,vi|= Z

R^NQ(|z|)(pU_r^p−1φ_kv+U_r^pv− |U_r+φ_k|^p−1(U_r+φ_k)v)

= Z

R^NQ(|z|)(−¹ 2h⁰⁰(θ))

≤C Z

R^N p(p−1)|U_r+θφ_k|^{p−2 Ur}+θφ_k

|Ur+θφ_k|^φ

2kv

≤C Z

R^N(kφ_kk²+kφ_kk^p)kvk

≤Ckφ_kk²kvk which implies thatkR⁰(φ_k)k ≤Ckφ_kk².

By the mean value theorem we obtain

|hR⁰⁰(φ_k)v,hi|= p Z

R^NQ(|z|)(U_r^p−1vh− |Ur+φ_k|^p−1)vh

= p(p−1) Z

R^NQ(|z|)|Ur+θφ_k|^p−3(Ur+θφ_k)φ_kvh

≤C Z

R^N(U_r^p−2+|φ_k|^p−2)|φ_k||v||h|

where 0<θ <1.

By Hölder’s inequality, Z

R^NU_r^p−2|φ_k||v||h| ≤C Z

R^NU_r^p+1

^p_p⁻₊²₁ Z

R^N|φ_k|^p+1

_p+¹₁ Z

R^N|v|^p+1

_p+¹₁ Z

R^N|h|^p+1 _p+¹₁

≤Ckφ_kk kvk khk and

Z

R^N|φ_k|^p−1|v||h| ≤ _Z

R^N|φ_k|^p+1

^p_p⁻₊¹_{1 Z}

R^N|v|^p+1

_p+¹_{1 Z}

R^N|h|^p+1 _p+¹₁

≤Ckφ_kk^p−1kvkkhk. Therefore,kR⁰⁰(φ_k)k ≤Ckφ_kk.

Arguing similarly as in case 1, we have,

kT(φ_k)k ≤C(kl_kk+kφ_kk²)≤ ¹ k

(N+2s)(τ−1)+τ 2(N+2s−τ)

. (2.13)

and T is a contraction map from E to E. The existence of φ_k follows from the contraction mapping theorem, and (2.10) follows from (2.12) and (2.13).

Following the argument employed in [5] to prove Lemma 4.4, we conclude that φ_k(r) is continuously differentiable in r.

Lemma 2.3. Ifτ=min{α,β}< N+2s, there exists a small constantδ>0such that kl_kk ≤ ^C

k

(N+2s)(τ−1)+τ 2(N+2s−τ) +δ

. Proof. We have

hl_k,φ_ki= l(φ_k)

=

Z

R^N(V(|z|)−1)U_rφ_k+

Z

R^N

_k

j

∑

W_x^p_j−U_r^p

φ_k−

Z

R^N(Q(|z| −1)U_r^pφ_k. (2.14) By symmetry,

Z

R^N(V(|z|)−1)U_rφ_k =

Z

R^N(V(|z|)−1) _k

j

∑

W_xj

φ_k

=

∑

Z

R^N(V(|z|)−1)W_xjφ_k =k Z

R^N(V(|z|)−1)W_x1φ_k (2.15) and

Z

R^N(V(|z|)−₁)W_x1φ_k =

Z

Br 2(₀)

(V(|z|)−₁)W_x1φ_k+

Z

R^N\Br 2(₀)

(V(|z|)−₁)W_x1φ_k. For z∈_R^N\B^r

2(0),

V(|z|)−1= ^a1

|z|^α +O 1

|z|^α+θ1

≤ ^C

|z|^α ≤ ²

αC

|r|^α,

Z

R^N\Br 2(0)

(V(|z|)−1)Wx₁φ_k = ²

αC

|r|^α

Z

R^N\Br

2(0)Wx₁φ_k

≤ ²

αC

|r|^α Z

R^N\Br 2(0)W_x²₁

¹₂Z

R^N\Br 2(0)

φ_k^{2 1}₂

=O 1

r^α

kφ_kk, Z

Br 2(0)

(V(|z|)−1)Wx₁φ_k ≤ C Z

Br 2(0)W_x²₁

¹₂Z

Br 2(0)

φ²_k ¹₂

≤C 1

r^N2+2skφ_kk. Then we conclude that

Z

R^N(V(|z|)−1)Wx₁φ_k ≤O 1

r^α

kφ_kk+C 1

r^N2+2skφ_kk. (2.16) By the mean value theorem and LemmaA.1,

Z

R^N

_k

∑

W_x^p_j −U_r^p

φ_k

=k Z

Ω1

_k

j

∑

W_x^p_j−U_r^p

φ_k

≤Ck Z

Ω1

W_x^p₁⁻¹ _k

j

∑

W_xj

φ_k

≤Ck 1 (k⁻¹r)^N+2s

Z

Ω1

W_x^p₁ ^p

−1

p Z

Ω1

|φ_k|^{p 1}_p

≤Ck 1

(k⁻¹r)^N+2skφ_kk. (2.17)

By the boundedness ofU_rwe have Z

R^N(Q(|z| −1)U_r^pφ_k =

Z

R^N(Q(|z| −1)U_r^p−1Urφ_k

≤C Z

R^N(Q(|z| −1)U_rφ_k ≤Ck Z

R^N(Q(|z| −1)W_x1φ_k

≤Ck O 1

r^β

+ ¹ r^N2+2s

!

kφ_kk. (2.18)

Combining relations (2.14)–(2.18), we obtain hl_k,φ_ki=

Z

R^N(V(|z|)−1)U_rφ_k+

Z

R^N

∑

W_x^p_j−U_r^p

! φ_k−

Z

R^N(Q(|z| −1)U_r^pφ_k

≤ k O 1

r^α

+O 1

r^β

+C 1 r^N2+2s

+ ^C

(k⁻¹r)^N+2s

! kφ_kk

≤ k O 1

r^τ

+ ^C

r^N2+2s + ^C (k⁻¹r)^N+2s

!

kφ_kk. (2.19)

Byr∈S, it holds thatr ∼k^{NN++2s2s−τ}, ^C

(k⁻¹r)^N+2s ∼O _r¹τ

,

kO 1

r^τ

<C 1

k^{(NN++2s2s−)ττ−1}

≤ ^C

k

(N+2s)(τ−1)+τ 2(N+2s−τ) +δ

, andk ¹

r^N2+2s < ¹

k

(N+2s)(τ−1)+τ 2(N+2s−τ)

, forτ< N+2s.

We conclude that ifτ<N+2s, then

kl_kk ≤ ^C k

(N+2s)(τ−1)+τ 2(N+2s−τ) +δ

. The proof is now complete.

3 Proof of the main result

Define

G(r) =I(U_r+φ_k), ∀r ∈S, whereφ_k =φ_k(r)is the map obtained in Lemma2.2.

According to Lemma 6.1 in [5], ifris a critical point of G(r), thenU_r+φ_k(r)is a solution of (1.1).

From the energy expansion in the Appendix, we have J(0) = I(Ur)

=k D+ ^a1A1

r^α − ^a2A2

r^β − ^B

(k⁻¹r)^N+2s +O 1

r^α+τ1

+O 1

r^β+τ2

+O

1 (k⁻¹r)^N+2s+σ

! . Set

H(r) = ^a1A1

r^α − ^a2A2

r^β − ^B

(k⁻¹r)^N+2s.

We prove in what follows that in any of the cases in Theorem 1.1, H(r) has a maximum pointr_k.

For case (i): ifa₁ >0, a₂=0, α<N+2s, andα≤ βthen H⁰(r) =−^αa1A1

r^α+1 + ^B(N+2s)k^N+2s r^N+2s+1 andr_k satisfies

αa₁A₁

r^α_k⁺¹ = ^B(N+2s)k^N+2s r_k^N+2s+1 .

Actually, calculating the maximum points in these cases can be summed up as τC

r^τ_k⁺¹ = ^C

0(N+2s)k^N+2s r^N_k^+2s+1 , whereτ=min{α,β}. ThenH(r)has a maximum point

r_k =

(N+2s)C⁰ τC

_N+¹_2s−τ

k^{NN++2s2s−τ},

which is an interior point ofS. Then, there exists a small constant δsuch that J(0) =k D+ ^a1A1

r^α − ^a2A2

r^β − ^B

(k⁻¹r)^N+2s +O 1

k^{(NN++2s2s−)ττ+δ} !

.

Consequently,

G(_r) =_I(_Wr+φ_k) = _I(_Wr) +_l(φ_k) + ¹

2hMφ_k,φ_ki+_R(φ_k)

= J(0) +O(kl_kkkφ_kk+kφ_kk²)

= J(0) +O

1 k^{(NN++2s2s−)ττ−1+δ}

=k D+ ^a1A1

r^α − ^a2A2

r^β − ^B

(k⁻¹r)^N+2s+O 1

k^{(NN++2s2s−)ττ+δ} !

.

Since H(r)has a maximum point r_k which is an interior point ofS in any of the cases listed, thenG(r)has a critical pointre_k in the interior ofS. This means that the function

Urek+φ_k(r_ek)

is a solution of problem (1.1). The proof is now complete.

A Appendix. Energy expansions

In this section, we obtain some energy estimates for the approximate solutions. Recall that Ωj =

z = (z⁰,z⁰⁰)∈_R²×_R^N−2: z⁰

|z⁰|^, x_j

|x_j|

≥cosπ k

,

x_j =

rcos2(j−1)π

k ,rsin2(j−1)π k , 0

, j=1, . . . ,k,

r∈ S:=

"

N+2s τ

−e _N+¹_2s−τ

k^{NN++2s2s−τ},

N+2s τ

+e _N+2s¹_−τ

k^{NN++2s2s−τ}

# ,

I(u) = ¹ 2

Z

R^N |(−_∆)^s2u|²+V(|z|)u²

− ¹ p+₁

Z

R^NQ(|z|)|u|^p+1, and

I(Ur) = ¹ 2

Z

R^N |(−_∆)^2sUr|²+V(|z|)Ur2

− ¹ p+1

Z

R^NQ(|z|)|Ur|^p+1

= ¹ 2

Z

R^NUr

∑

Wx^p_j +¹ 2

Z

R^N(_V(|z|)−1)_Ur²

− ¹ p+1

Z

R^N(Q(|z|)−1)U_r^p+1− ¹ p+1

Z

R^NU_r^p+1. Lemma A.1. For any z∈ _Ω1, there exists C >0, such that

∑

W_xj(z)≤ ^C (k⁻¹r)^N+2s