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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four

Jérôme Vétois
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  • Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec H3A 0B9, Canada
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/ Shaodong Wang
  • Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec H3A 0B9, Canada
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Published Online: 2017-08-18 | DOI: https://doi.org/10.1515/anona-2017-0085

Abstract

We extend Chen, Wei and Yan’s constructions of families of solutions with unbounded energies [5] to the case of cubic nonlinear Schrödinger equations in the optimal dimension four.

Keywords: Nonlinear Schrödinger equations; blowing-up solutions; unbounded energies

MSC 2010: 35J61

1 Introduction and main results

In this note, we consider the cubic nonlinear Schrödinger equation

Δgu+fu=u3in M,(1.1)

where (M,g) is a Riemannian manifold of dimension 4, Δg:=-divg is the Laplace–Beltrami operator and fC0,α(M), α(0,1).

For (M,g)=(𝕊4,g0), where g0 is the standard metric on the sphere 𝕊4, we obtain the following result.

Theorem 1.1.

Assume that (M,g)=(S4,g0) and f>2 is constant. Then there exists a family of positive solutions (uε)ε>0 to (1.1) such that uεL2(S4) as ε0.

Theorem 1.1 extends a result obtained by Chen, Wei and Yan [5], in dimensions n5, for positive solutions of the equation

Δgu+fu=u2*-1in M,(1.2)

where 2*:=2nn-2. The dimension four is optimal for this result since Li and Zhu [11] obtained the existence of a priori bounds on the energy of positive solutions to (1.2) in dimension three.

It is also interesting to mention that in the case where n{3,6} and f>n(n-2)4 on 𝕊n (or, more generally, f>n-24(n-1)Scalg on a general closed manifold, where Scalg is the scalar curvature), Druet [6] obtained a compactness result for families of positive solutions (uε)ε>0 of (1.2) with bounded energies, i.e., such that uεL2(M)<C for some constant C independent of ε. Theorem 1.1, together with the result of Chen, Wei and Yan [5] in dimensions n5, shows that the energy assumption in Druet’s result is necessary at least in the case of the standard sphere.

In the case where fn(n-2)4 and (M,g)=(𝕊n,g0), the positive solutions of (1.2) have been classified by Obata [12] (see also [4]). In this case, the solutions are not bounded in L(𝕊n) but they all have the same energy. We refer to [2, 3, 10] and the references therein for results on the set of solutions of (1.2) for fn-24(n-1)Scalg and (M,g)(𝕊n,g0). On the other hand, for f<n-24(n-1)Scalg on a general closed manifold, Druet [7] obtained pointwise a priori bounds on the set of positive solutions of (1.2). Note that if, moreover, 0<f<n-24(n-1)Scalg is constant, then uf(n-2)/4 is the unique positive solution of (1.2), see [1]. We refer to the books of Druet, Hebey and Robert [8], and Hebey [9] for more results on equations of type (1.1) on a closed manifold.

As in [5], we obtain Theorem 1.1 by proving a more general result for the case (M,g)=(4,δ0), where δ0 is the Euclidean metric on 4. We let D1,2(4) be the completion of the set of smooth functions with compact support in 4 with respect to the norm uD1,2(4)=uL2(4). For simplicity, we will use the notation Δ:=Δδ0, ,:=,δ0 and ||:=||δ0. We say that the operator Δ+f is coercive in D1,2(4) if

4(|u|2+fu2)𝑑xCuD1,2(4)2for all uD1,2(4),

for some constant C>0. We have the following result.

Theorem 1.2.

Assume that (M,g)=(R4,δ0) and that fC0,α(R4)L2(R4) is radially symmetric about the point 0. Assume, moreover, that the operator Δ+f is coercive in D1,2(R4) and the function rr2f(r) has a strict local maximum point r0>0 such that f(r0)>0. Then there exists a family of positive solutions (uε)ε>0 in C2,α(R4)D1,2(R4) of (1.1) such that uεL2(R4) as ε0.

The proof of Theorem 1.2 relies on a Lyapunov–Schmidt-type method, as in [5]. This method for constructing solutions with infinitely many peaks was invented and successfully used in previous works by Wang, Wei and Yan [13, 14] and Wei and Yan [15, 16, 17, 18]. A specificity in our case is that the number of peaks in the construction behaves as a logarithm of the peak’s height, while it behaves as a power of the peak’s height in the higher dimensional case (see [5]). Due to this logarithm behavior, we need to introduce some suitable changes of variables in order to find the critical points of the reduced energy in this case (see the proof of Theorem 1.2 at the end of Section 2).

2 Proof of Theorems 1.1 and 1.2

This section is devoted to the proof of Theorems 1.1 and 1.2. For any integer k1, we let Hk be the set of all functions uD1,2(4) such that u is even in x2,x3,x4 and

u(rcos(θ),rsin(θ),x3,x4)=u(rcos(θ+2π/k),rsin(θ+2π/k),x3,x4)

for all r>0 and θ,x3,x4. Assuming that the operator Δ+f is coercive in D1,2(4), we can equip Hk with the inner product

u,vHk:=4(u,v+fuv)𝑑xfor all u,vHk,

and the norm

uHk:=u,uHkfor all uHk.

For any k1 and r,μ>0, we define

Wk,r,μ:=i=1kUi,k,r,μ,where Ui,k,r,μ(x):=22μ1+μ2|x-xi,k,r|2 for all x4,

with

xi,k,r:=(rcos(2(i-1)π/k),rsin(2(i-1)π/k),0,0).

Moreover, we define

Pk,r,μ:={ϕHk:i=1kϕ,Zi,j,k,r,μHk=0 for all j{1,2}},

where

Zi,1,k,r,μ:=1μddr[Ui,k,r,μ]andZi,2,k,r,μ:=μddμ[Ui,k,r,μ].

First, in Proposition 2.1 below, we solve the equation

Qk,r,μ(Wk,r,μ+ϕ-(Δ+f)-1((Wk,r,μ+ϕ)+3))=0,(2.1)

where ϕPk,r,μ is the unknown function, Qk,r,μ is the orthogonal projection of Hk onto Pk,r,μ and, as usual, u+:=max(u,0) for all u:4.

We will prove the following result in Section 3.

Proposition 2.1.

Let fC0,α(R4)L2(R4) be a radially symmetric function about the point 0 and such that the operator Δ+f is coercive in D1,2(R4). Then, for any a,b,c,d>0, with a<b and c<d, there exist constants k0>0 and C0>0 such that for any kk0, r[a,b] and μ[eck2,edk2], there exists a unique solution ϕk,r,μPk,r,μ of (2.1) satisfying

ϕk,r,μHkC0k/μ.(2.2)

Moreover, the map (r,μ)ϕk,r,μ is continuously differentiable and if there exists a critical point (rk,μk)[a,b]×[eck2,edk2] of the function

(r,μ)k(r,μ):=I(Wk,r,μ+ϕk,r,μ),

where

I(u):=124(|u|+fu2)𝑑x-144u+4𝑑x,

then the function Wk,rk,μk+ϕk,rk,μk is a positive solution in C2,α(R4)Hk of the equation

Δu+fu=u3in 4.(2.3)

Then we will prove the following result in Section 4.

Proposition 2.2.

Let fC0,α(R4)L2(R4) be a radially symmetric function about the point 0 and such that the operator Δ+f is coercive in D1,2(R4). Then there exist constants c0,c1,c2>0 such that for any a,b,c,d>0, with a<b and c<d, we have

I(Wk,r,μ+ϕk,r,μ)=c0k+c1f(r)klnμμ2-c2k3r2μ2+o(k3μ2)as k,(2.4)

uniformly for r[a,b] and μ[eck2,edk2], where ϕk,r,μ is as in Proposition 2.1.

Now, we prove Theorem 1.2, by using Propositions 2.1 and 2.2.

Proof of Theorem 1.2.

Since f(r0)>0 and r0 is a strict local maximum point of the function rr2f(r), we obtain that there exists δ0>0 such that

0<r2f(r)<r02f(r0)for all r[r0-δ0,r0+δ0].(2.5)

For any k1 and s>0, we define μk(s):=esk2. By applying Proposition 2.2, we obtain

k(r,μk(s))=c0k+k3e-2sk2(c1f(r)s-c2r2+o(1))as k,(2.6)

uniformly for (r,s) in compact subsets of (0,)2. Note that the function

se-2sk2(c1f(r)s-c2r2)

attains its maximal value at the point

sk(r):=c2c1f(r)r2+12k2

for all k1 and r[r0-δ0,r0+δ0]. We define

𝒥k(r,t):=k(r,μk(sk(r)+t)).

By using (2.5), we obtain that there exists t0>0 such that

t0<min(sk(r0)2,23(sk(r0+δ0)-sk(r0)),23(sk(r0-δ0)-sk(r0)))(2.7)

for all k1. Since t0<sk(r0)/2, from (2.6) it follows that

𝒥k(r,t)=c0k+k3e-2(sk(r)+t)k2(c1f(r)t+o(1))as k,(2.8)

uniformly for (r,t)[r0-δ0,r0+δ0]×[-t0,t0]. Since sk(r)>sk(r0) and f(r)>0, from (2.8) it follows that

𝒥k(r,t0)<𝒥k(r0,t0/2)(2.9)

and

𝒥k(r,-t0)<𝒥k(r0,t0/2)(2.10)

as k, uniformly for r[r0-δ0,r0+δ0]. Moreover, by using (2.7) and (2.8), we obtain

𝒥k(r0±δ0,t)<𝒥k(r0,t0/2)(2.11)

as k, uniformly for t[-t0,t0]. From (2.9)–(2.11) it follows that the function 𝒥k has a local maximum point (rk,tk)[r0-δ0,r0+δ0]×[-t0,t0] for large k. We then obtain k(rk,μk(sk(rk)+tk))=0, and so, by applying the second part of Proposition 2.1, we obtain that the function Wk,rk,μk(sk(rk)+tk)+ϕk,rk,μk(sk(rk)+tk) is a positive solution of equation (2.3). Moreover, by using (2.2) together with the definition of Wk,rk,μk(sk(rk)+tk), we easily obtain

(Wk,rk,μk(sk(rk)+tk)+ϕk,rk,μk(sk(rk)+tk))L2as k.

This ends the proof of Theorem 1.2. ∎

Finally, we prove Theorem 1.1, by using Theorem 1.2.

Proof of Theorem 1.1.

By using a stereographic projection, we can see that equation (1.1) on (M,g)=(𝕊4,g0) is equivalent to the problem

{Δu+4(f-2)(1+|y|2)2u=u3in 4,uD1,2(4).(2.12)

It is easy to check that if f>2 is a constant, then the potential function in (2.12) satisfies the assumptions of Theorem 1.2. With this remark, Theorem 1.1 becomes a direct corollary of Theorem 1.2. ∎

3 Proof of Proposition 2.1

In this section we prove Proposition 2.1. Throughout this section, we assume that fC0,α(4)L2(4) is radially symmetric about the point 0, and that the operator Δ+f is coercive in D1,2(4).

We rewrite (2.1) as

Lk,r,μ(ϕ)=Qk,r,μ(Nk,r,μ(ϕ)+Rk,r,μ),

where

Lk,r,μ(ϕ):=Qk,r,μ(ϕ-(Δ+f)-1(3Wk,r,μ2ϕ)),Nk,r,μ(ϕ):=(Δ+f)-1((Wk,r,μ+ϕ)+3-Wk,r,μ3-3Wk,r,μ2ϕ),Rk,r,μ:=(Δ+f)-1(Wk,r,μ3)-Wk,r,μ.

First, we obtain the following result.

Lemma 3.1.

For any a,b,c,d>0, with a<b and c<d, there exist constants k1>0 and C1>0 such that for any kk1, r[a,b] and μ[eck2,edk2], Lk,r,μ is an isomorphism from Pk,r,μ to itself and

Lk,r,μ(ϕ)HkC2ϕHkfor all ϕPk,r,μ.

Proof.

The proof of this result follows the same lines as in [5]. ∎

We then estimate the error term Rk,r,μ. We obtain the following result.

Lemma 3.2.

For any a,b,c,d>0, with a<b and c<d, there exist constants k2>0 and C2>0 such that

Rk,r,μHkC2k/μ(3.1)

for all kk2, r[a,b], and μ[eck2,edk2].

Proof.

For any ϕHk, by integrating by parts, we obtain

Rk,r,μ,ϕHk=4(Wk,r,μ3-ΔWk,r,μ-fWk,r,μ)ϕ𝑑x=4(Wk,r,μ3-i=1kUi,k,r,μ3-fWk,r,μ)ϕ𝑑x=i=1k4(3jiUi,k,r,μ2Uj,k,r,μ-fUi,k,r,μ)ϕ𝑑x.(3.2)

By using Hölder’s inequality and Sobolev’s inequality, from (3.2) it follows that

Rk,r,μHk=i=1kO(jiUi,k,r,μ2Uj,k,r,μL4/3+fUi,k,r,μL4/3).(3.3)

We start with estimating the first term in (3.3). For any α{1,,k}, we define

Ωα,k,r:={(y1,y2,y3,y4)4:(y1,y2,0,0),xα,k,rcos(π/k)}.

We then write

4Ui,k,r,μ8/3Uj,k,r,μ4/3𝑑x=α=1kΩα,k,rUi,k,r,μ8/3Uj,k,r,μ4/3𝑑x.(3.4)

We observe that if αj, then

|x-xj,k,r||x-xα,k,r|and|x-xj,k,r|12|xα,k,r-xj,k,r|(3.5)

for all xΩα,k,r. For any i,j,α{1,,k}, with ij, by using (3.4), we obtain

Ui,k,r,μ(x)8/3Uj,k,r,μ(x)4/3{28/3(22)4μ4/3(1+μ2|x-xi,k,r|2)8/3|xi,k,r-xj,k,r|8/3if α=i,28/3(22)4μ4/3(1+μ2|x-xα,k,r|2)8/3|xi,k,r-xα,k,r|8/3if αi(3.6)

for all xΩα,k,r{xα,k,r}. By using (3.4), (3.6) and straightforward estimates, we obtain

4Ui,k,r,μ8/3Uj,k,r,μ4/3𝑑x=O(μ-8/3|xi,k,r-xj,k,r|8/3+αiμ-8/3|xi,k,r-xα,k,r|8/3)=O(μ-8/3|xi,k,r-xj,k,r|8/3+k8/3μ8/3).(3.7)

From (3.7) it follows that

jiUi,k,r,μ2Uj,k,r,μL4/3=O(k(k/μ)2).(3.8)

Now, we estimate the second term in (3.4). Since fL(4)L2(4), by applying Hölder’s inequality and straightforward estimates, we obtain

4B(xi,k,r,1)|fUi,k,r,μ|4/3dx=O((4B(xi,k,r,1)|Ui,k,r,μ|4dx)1/3)=O(μ-4/3)(3.9)

and

B(xi,k,r,1)|fUi,k,r,μ|4/3dx=O(B(xi,k,r,1)|Ui,k,r,μ|4/3dx)=O(μ-4/3).(3.10)

From (3.9) and (3.10), it follows that

fUi,k,r,μL4/3=O(1/μ).(3.11)

Finally, (3.1) follows from (3.8) and (3.11). ∎

We can now prove Proposition 2.1 by using Lemmas 3.1 and 3.2.

Proof of Proposition 2.1.

We define

Tk,r,μ(ϕ):=Lk,r,μ-1(Qk,r,μ(Nk,r,μ(ϕ)+Rk,r,μ))for all ϕPk,r,μ

and

Vk,r,μ:={ϕPk,r,μ:ϕHkC0k/μ},

where C0>0 is a constant to be fixed later on. From Lemmas 3.1 and 3.2, it follows that

Tk,r,μ(ϕ)HkC1(Nk,r,μ(ϕ)Hk+C2k/μ)(3.12)

for all kk2, r[a,b] and μ[eck2,edk2]. By integrating by parts and using Hölder’s inequality, Sobolev’s inequality and straightforward estimates, we obtain

Nk,r,μ(ϕ),ψHk=4((Wk,r,μ+ϕ)+3-Wk,r,μ3-3Wk,r,μ2ϕ)ψ𝑑x=O((Wk,r,μL4ϕHk2+ϕHk3)ψHk)(3.13)

for all ψHk. Proceeding as in (3.4)–(3.8), we obtain

4Wk,r,μ4dx=O(i=1k4(Ui,k,r,μ4+jiUi,k,r,μ2Uj,k,r,μ2)dx)=O(k+k(k/μ)4lnμ).(3.14)

From (3.13) and (3.14), it follows that

Nk,r,μ(ϕ)Hk=O(k1/4ϕHk2+ϕHk3).(3.15)

Letting C0 be large enough so that C0>C1C2, from (3.12) and (3.15), it follows that there exists a constant k3>0 such that

Tk,r,μ(Vk,r,μ)Vk,r,μ(3.16)

for all kk3, r[a,b] and μ[eck2,edk2]. Now, we prove that if k is large enough, then Tk,r,μ is a contraction map from Vk,r,μ to itself, i.e.,

Tk,r,μ(ϕ1)-Tk,r,μ(ϕ2)HkCϕ1-ϕ2Hkfor all ϕ1,ϕ2Vk,r,μ,(3.17)

for some constant C(0,1). From Lemma 3.1 it follows that

Tk,r,μ(ϕ1)-Tk,r,μ(ϕ2)HkC1Nk,r,μ(ϕ1)-Nk,r,μ(ϕ2)Hk.(3.18)

By integrating by parts and using Hölder’s inequality, Sobolev’s inequality and (3.14), we obtain

Nk,r,μ(ϕ1)-Nk,r,μ(ϕ2),ψHk=4((Wk,r,μ+ϕ1)+3-(Wk,r,μ+ϕ2)+3-3Wk,r,μ2(ϕ1-ϕ2))ψ𝑑x=O((Wk,r,μL4+ϕ1Hk+ϕ2Hk)(ϕ1Hk+ϕ2Hk)ϕ1-ϕ2HkψHk)=O((k1/4+ϕ1Hk+ϕ2Hk)(ϕ1Hk+ϕ2Hk)ϕ1-ϕ2HkψHk).(3.19)

From (3.19) it follows that

Nk,r,μ(ϕ1)-Nk,r,μ(ϕ2)Hk=o(ϕ1-ϕ2Hk)as k,(3.20)

uniformly for r[a,b], μ[eck2,edk2] and ϕ1,ϕ2Vk,r,μ. We then obtain (3.17) by putting together (3.18) and (3.20). From (3.16) and (3.17), it follows that there exists a constant k4k3 such that for any kk4, r[a,b] and μ[eck2,edk2], there exists a unique solution ϕk,r,μVk,r,μ of (2.1). The continuous differentiability of (r,μ)ϕk,r,μ is standard.

Now, we prove the last part of Proposition 2.1. We let (rk,μk)[a,b]×[eck2,edk2] be a critical point of k. Since ϕk,r,μ is a solution of (2.1), there exist c1,k and c2,k such that

DI(Wk,rk,μk+ϕk,rk,μk)=j=12cj,ki=1kZi,j,k,rk,μk,Hk.(3.21)

From (3.21) it follows that

0=kr(rk,μk)=j=12cj,ki=1kZi,j,k,rk,μk,ddr[Wk,r,μk+ϕk,r,μk]r=rkHk=j=12cj,ki=1k(μkα=1kZi,j,k,rk,μk,Zα,1,k,rk,μkHk+Zi,j,k,rk,μk,ddr[ϕk,r,μk]r=rkHk)(3.22)

and

0=kμ(rk,μk)=j=12cj,ki=1kZi,j,k,rk,μk,ddμ[Wk,rk,μ+ϕk,rk,μ]μ=μkHk=j=12cj,ki=1k(1μkα=1kZi,j,k,rk,μk,Zα,2,k,rk,μkHk+Zi,j,k,rk,μk,ddμ[ϕk,rk,μ]μ=μkHk).(3.23)

For any i,α{1,,k} and j,β{1,2}, direct calculations yield

Zi,j,k,rk,μk,Zα,β,k,rk,μkHk=Λjkδiαδjβ+o(k)as k,(3.24)

where Λj>0 is a constant and δiα:=1 if α=i, and δiα:=0 if αi. Moreover, since ϕk,r,μPk,r,μ, we obtain

Zi,j,k,rk,μk,ddr[ϕk,r,μk]r=rkHk=-ddr[Zi,j,k,r,μk]r=rk,ϕk,rk,μkHk,

and therefore, by using the Cauchy–Schwarz inequality, Sobolev’s inequality, (2.2) and similar estimates as in (3.14), we obtain

|Zi,j,k,rk,μk,ddr[ϕk,r,μk]r=rkHk|ddr[Zi,j,k,r,μk]r=rkHkϕk,rk,μkHk=O(k3/2).(3.25)

Similarly, we obtain

|Zi,j,k,rk,μk,ddμ[ϕk,rk,μ]μ=μkHk|ddμ[Zi,j,k,rk,μ]μ=μkHkϕk,rk,μkHk=O(k3/2μk-2).(3.26)

From (3.22)–(3.26) it follows that if k is large enough, then c1,k=c2,k=0, i.e., the function Wk,rk,μk+ϕk,rk,μk is a weak solution of the equation

Δu+fu=u+3in 4.

By using the coercivity of the operator Δ+f in D1,2(4), we obtain that u0 a.e. in 4. Then, from standard elliptic regularity theory and the strong maximum principle, it follows that Wk,rk,μk+ϕk,rk,μk is a strong positive solution in C2,α(4) of (2.3). ∎

4 Proof of Proposition 2.2

In this section we prove Proposition 2.2. Throughout this section we assume that fC0,α(4)L2(4) is radially symmetric about the point 0, and that the operator Δ+f is coercive in D1,2(4). First, we obtain the following result.

Lemma 4.1.

There exist constants c0,c1,c2>0 such that for any a,b,c,d>0, with a<b and c<d, we have

I(Wk,r,μ)=c0k+c1f(r)klnμμ2-c2k3r2μ2+o(k3μ2)as k,(4.1)

uniformly for r[a,b] and μ[eck2,edk2].

Proof.

By integrating by parts, we obtain

I(Wk,r,μ)=124(ΔWk,r,μ+fWk,r,μ)Wk,r,μ𝑑x-144Wk,r,μ4𝑑x=124(i,j=1kUi,k,r,μ3Uj,k,r,μ+fWk,r,μ2-12Wk,r,μ4)dx=12i=1k4(fUi,k,r,μ2+12Ui,k,r,μ4-jiUi,k,r,μ3Uj,k,r,μ-3jiUi,k,r,μ2Uj,k,r,μ2+fjiUi,k,r,μUj,k,r,μ)𝑑x.(4.2)

Direct calculations yield

4Ui,k,r,μ4𝑑x=(22)44dx(1+|x|2)4(4.3)

and

4fUi,k,r,μ2𝑑x=16π2f(r)lnμμ2+o(lnμμ2)as k,(4.4)

uniformly for r[a,b] and μ[eck2,edk2]. Proceeding as in (3.4)–(3.8), we obtain

ji4Ui,k,r,μ3Uj,k,r,μdx=jiΩi,k,r64μ2(1+μ2|x-xi,k,r|2)3(1+O(𝟏Ωi,k,rB(xi,|x1,k,r-x2,k,r|/2))|xi,k,r-xj,k,r|2+O(μ-2+|x-xi,k,r||xi,k,r-xj,k,r||xi,k,r-xj,k,r|4𝟏B(xi,|x1,k,r-x2,k,r|/2)))dx+O(αikμ|xi,k,r-xα,k,r|3Ωα,k,rdx(1+μ2|x-xα,k,r|2)5/2)=ji(64μ-2|xi,k,r-xj,k,r|24dx(1+|x|2)3+O(kμ-3|xi,k,r-xj,k,r|3))=32k2π2r2μ24dx(1+|x|2)3j=11j2+o(k2μ2)(4.5)

uniformly for r[a,b] and μ[eck2,edk2]. Moreover, straightforward estimates give

ji4(B(xi,k,r,μ,|xi,k,r,μ-xj,k,r,μ|/2)B(xj,k,r,μ,|xi,k,r,μ-xj,k,r,μ|/2))Ui,k,r,μ2Uj,k,r,μ2dx=O(μ-4ji4(B(xi,k,r,μ,|xi,k,r,μ-xj,k,r,μ|/2)B(xj,k,r,μ,|xi,k,r,μ-xj,k,r,μ|/2))|x-xi,k,r,μ|-4|x-xj,k,r,μ|-4dx)=O(jiμ-4|xi,k,r,μ-xj,k,r,μ|4)=O((k/μ)4),(4.6)jiB(xi,k,r,μ,|xi,k,r,μ-xj,k,r,μ|/2)B(xj,k,r,μ,|xi,k,r,μ-xj,k,r,μ|/2)Ui,k,r,μ2Uj,k,r,μ2𝑑x=O(jiμ-4|xi,k,r,μ-xj,k,r,μ|4B(0,μ|xi,k,r,μ-xj,k,r,μ|/2)dx(1+|x|2)2)=O(jiμ-4lnμ|xi,k,r,μ-xj,k,r,μ|4)=O((k/μ)4lnμ),(4.7)ji4(B(xi,k,r,μ,1)B(xj,k,r,μ,1))fUi,k,r,μUj,k,r,μdx=O(ji(4B(xj,k,r,μ,1)Uj,k,r,μ4dx)1/2)=O(k(4B(0,μ)dx(1+|x|2)4)1/2)=O(k/μ2),(4.8)jiB(xi,k,r,μ,1)B(xj,k,r,μ,1)fUi,k,r,μUj,k,r,μdx=O(jiB(xi,k,r,μ,1)Ui,k,r,μUj,k,r,μdx)=O(jiB(xi,k,r,μ,1)μ-2dx|x-xi,k,r,μ|2|x-xj,k,r,μ|2)=O(μ-2jiln1|xi,k,r,μ-xj,k,r,μ|)=O(klnkμ2)(4.9)

as k, uniformly for r[a,b] and μ[eck2,edk2]. Finally, (4.1) follows from (4.2)–(4.9). ∎

We can now prove Proposition 2.2, by using Lemma 4.1.

Proof of Proposition 2.2.

By integrating by parts, we obtain

I(Wk,r,μ+ϕk,r,μ)=I(Wk,r,μ)-Rk,r,μ,ϕk,r,μHk+12ϕk,r,μHk2-144((Wk,r,μ+ϕk,r,μ)+4-Wk,r,μ4-4Wk,r,μ3ϕk,r,μ)𝑑x.(4.10)

By using the Cauchy–Schwarz inequality, Lemma 3.1 and Proposition 2.1, we obtain

-Rk,r,μ,ϕk,r,μHk+12ϕk,r,μHk2=O((k/μ)2).(4.11)

Moreover, by using Hölder’s inequality, Sobolev’s inequality, (3.14) and Lemma 3.1, we obtain

4((Wk,r,μ+ϕk,r,μ)+4-Wk,r,μ4-4Wk,r,μ3ϕk,r,μ)𝑑x=O(4(Wk,r,μ2+ϕk,r,μ2)ϕk,r,μ2𝑑x)=O(Wk,r,μL42ϕk,r,μHk2+ϕk,r,μHk4)=O(k(k/μ)2+(k/μ)4).(4.12)

Finally, (2.4) follows from (4.10)–(4.12). ∎

Acknowledgements

This paper was written during the second author’s Ph.D. He wishes to express his gratitude to his supervisors Pengfei Guan and Jérôme Vétois for their support and guidance.

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About the article

Received: 2017-04-05

Accepted: 2017-07-09

Published Online: 2017-08-18


The first author was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. The second author was supported by the China Scholarship Council.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 715–724, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0085.

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