# Existence of Multispike Positive Solutions for a Nonlocal Problem inℝ3

Authors: Jing Yang; Qiuxiang Bian; Na Zhao
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this paper, we study the following nonlinear Choquard equation−ϵ2Δu+Kxu=1/8πϵ2∫ℝ3u2y/x−ydyu,x∈ℝ3, where ϵ>0 and Kx is a positive bounded continuous potential on ℝ3. By applying the reduction method, we proved that for any positive integer k, the above equation has a positive solution with k spikes near the local maximum point of Kx if ϵ>0 is sufficiently small under some suitable conditions on Kx.


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## Body

## 1. Introduction and Main Results

In this paper, we consider the following nonlinear Choquard equations(1)−ϵ2Δu+Kxu=φu,x∈ℝ3,−ϵ2Δφ=u22,x∈ℝ3,where ϵ>0 and Kx is a positive bounded continuous potential. The Choquard equation first appeared as early as in 1954, in a work by Pekar describing the quantum mechanics of a polaron at rest [1]. In 1976, Choquard used it to describe an electron trapped in its own hole in a certain approximation to the Hartree-Fock theory of one component plasma in [2]. Penrose [3] also proposed it as a model of self-gravitating matter, in a programme in which quantum state reduction is understood as a gravitational phenomenon. Moreover, the Choquard equation is also known as the Schrödinger-Newton equation in models coupling the Schrödinger equation of quantum physics together with nonrelativistic Newtonian gravity.Note that the second equation of (1) can be explicitly solved with respect to φ and then (1) reduces to the following single nonlocal equation
(2)−ϵ2Δu+Kxu=18πϵ2∫ℝ3u2yx−ydyu.Equation (2) has attracted considerable attention in recent period and part of the motivation is due to looking for standing waves for the following nonlinear Hartree equations
(3)iϵ∂ψ∂t+ϵ2Δψ+Kx+hψ=18πϵ2∫ℝ3ψ2yx−ydyψ,x,t∈ℝ3×ℝ+,with the form ψx,t=e−iht/ϵux, where i is the imaginary unit, h∈ℝ and ϵ is the Planck constant. The above Hartree equations also appear in quantum mechanics models (see [4–6]) and in the semiconductor theory (see [7–9]).Also, the Choquard equation (2) is a special type of the following generalized Choquard equation
(4)−ϵ2Δu+Kxu=18πϵ2∫ℝ3upyx−yn−αdyup−2u,x∈ℝn,where α∈0,n and p>1. The symmetry and the regularity of solutions of (4) have been established by Ma and Zhao [10] and by Cingolani et al. [11], respectively, under the suitable assumptions on p when ϵ=1. Later, in [12] Moroz and Van Schaftingen derived the regularity, positivity, radial symmetry, and sharp asymptotics of ground state solutions of (4) for the optimal range of parameters (see also [13]).In particular, takingn=3, p=2, and α=2 in (4), we get (2). In [14], Lions derived the existence of ground state solutions of (2) under some suitable conditions on Kx if ϵ>0 is small enough. For any positive integer k>0, Wei and Winter [15] proved that there exist a solution of (2) concentrating at k points which are all local minimums or local maximums or non-degenerate critical points of Kx under the conditions that infℝ3K>0,K∈C2ℝ3 provided ϵ is sufficiently small. Recently, Luo, Peng and Wang [16] showed the uniqueness of positive solutions for (2) concentrating at the non-degenerate critical points of Kx by using a local Pohozaev type identity for ϵ>0 small enough.But, whenϵ=1 and Kx=1, (2) is written as
(5)−Δu+u=18π∫ℝ3u2yx−ydyu,x∈ℝ3.In [17], Lieb obtained the existence and uniqueness of ground state solutions of (5) by using variational method (see also [18, 19]). Later, Tod and Moroz and Tod [20] and Wei and Winter [15] proved the nondegeneracy of the ground state solutions of (5).Applying the existence and the nondegeneracy of ground state solutions for (5), inspired by [21, 22], we want to exploit the finite dimensional reduction method to investigate the existence of positive multi-spikes solutions for (2) under the conditions imposed on Kx as follows:(K1) K has a strict local maximum at some point y0∈ℝ3, that is, there is δ>0 such that Kx<Ky0 for all x∈Bδy0\y0.(K2) infℝ3K≥b>0 and there exist constants L,θ>0 with θ<1 such that
(6)Kx−Ky≤Lx−yθfor all x,y∈B2δy0.We state our main result as follows:Theorem 1.
Assume thatK1, K2 hold, then for any positive integer k, problem (2) has a k-spike solution for sufficiently small ϵ>0.As in [21–23], we mainly use the finite-dimensional reduction to prove our result. Here, our purpose is to verify that if ϵ is small enough, then for any positive integer k, (2) has a solution with k-spikes concentrating near y0 corresponding to any strict local maximum y0 of Kx, namely, a solution with k maximum points converging to y0 as ϵ⟶0.In the end of this part, let us outline the sketch of our proof of Theorem1. Denoted by wx, the unique radially positive solution of the following problem
(7)−Δu+Ky0u=18π∫ℝ3u2yx−ydyu,inℝ3,ux>0inℝ3,u0=maxℝ3ux.It follows from [2, 15] that wx is strictly decreasing and satisfies
(8)limx→+∞wxxex=c0,limx→+∞wxw′x=−1,for some constant c0>0. Also, wx is nondegenerate, namely, if ψx∈H1ℝ3 solves the lineared equation
(9)−Δψ+Ky0ψ=18π∫ℝ3w2yx−ydyψx+14π∫ℝ3wyψyx−ydywx,then ψx is a linear combination of ∂w/∂xj,j=1,2,3.We will use the unique solutionw of (7) to establish the solutions of (2). In what follows, without loss of generality, we assume that y0=0 and K0=1. Let Br0=x∈ℝ3:x<r and denote its closure by B¯r0. For any positive integer k and large R, we define
(10)Dkϵ,δ=y=y1,⋯,yk∈ℝ3k:yj∈Bδ/2y0,yi−yjϵ≥R,i≠j,i,j=1,2⋯,k.Furthermore, sinceinfℝ3K>0, we can define the following Soblev space
(11)Hϵ≔u∈H1ℝ3:∫ℝ3ϵ2∇u2+Kxu2dx<∞,with the corresponding norm uϵ2=u,uϵ, where
(12)u,vϵ=∫ℝ3ϵ2∇u∇v+Kxuvdx,and, in this sequel, we denote by ⋅p the usual norm of Lpℝ3 and let uD≔∫ℝ3∇u2dx1/2, uH1≔∫ℝ3∇u2+u2dx1/2 be the norms of D1,2ℝ3 and H1ℝ3, respectively.Now fixingy∈Dkϵ,δ, set
(13)Wϵ,y=∑j=1kwϵ,yj=∑j=1kwx−yjϵ,Eϵ,k=ϕ∈H1ℝ3:∂wϵ,yj∂ysj,ϕϵ=0,ϕϵ2≤δϵ3,for j=1,⋯,k,s=1,2,3. We will prove Theorem 1 by verifying the following result.Theorem 2.
LetK1, K2 hold. Then, for any positive integer k, there is ϵ0=ϵk such that for ϵ∈0,ϵ0, problem (2) has a solution uϵ of the form
(14)uϵx=∑j=1kwx−yϵjϵ+ϕϵ,for some points yϵj∈ℝ3,j=1,⋯,k and ϕϵ∈H1ℝ3 satisfying as ϵ⟶0,
(15)yϵj⟶y0,yϵi−yϵjϵ⟶+∞i≠j,ϕϵϵ=oϵ3/2.We want to point out that compared with [15], we introduce a little stronger conditions imposed on Kx than that of [15] and the reduction procedure has been modified here to allow for the degenerate of the critical point of Kx. Also, the appearance of nonlocal term forces us to face much difficulties in the reduction process which involves some more delicate estimates.The rest of the paper is organized as follows. In Section2, we will carry out a reduction procedure. We prove our main result in Section 3. Finally, in Appendix, some technical estimates and an energy expansion for the functional corresponding to problem (2) will be established.


## 2. The Finite-Dimensional Reduction

Observe that the variational functional corresponding to (2) is
(16)Iϵu=12∫ℝ3ϵ2∇u2+Kxu2dx−132πϵ2∫ℝ3∫ℝ3u2yx−ydyu2dx.Then by the direct computation, we have for anyϕ∈Eϵ,k,
(17)Jϵy1,⋯,yk≔Jϵy,ϕ=IϵWϵ,y+ϕ=Iϵ∑j=1kwx−yjϵ+ϕ=12∫ℝ3ϵ2∇Wϵ,y2+KxWϵ,y2dx−132πϵ2∫ℝ3∫ℝ3Wϵ,y2yx−ydyWϵ,y2dx+∫ℝ3Kx−1Wϵ,yϕdx−18πϵ2∫ℝ3∫ℝ3∑i≠jwϵ,yjwϵ,yi∣x−y∣dy∑j=1kwϵ,yjϕdx+∑i≠j∫ℝ3∫ℝ3wϵ,yj2∣x−y∣dywϵ,yiϕdx+12∫ℝ3ϵ2∇ϕ2+Kxϕ2dx−116πϵ2∫ℝ3∫ℝ3Wϵ,y2yx−ydyϕ2dx−18πϵ2∫ℝ3∫ℝ3Wϵ,yϕx−ydyWϵ,yϕdx−132πϵ2∫ℝ3∫ℝ3ϕ2yx−ydyϕ2dx+18πϵ2∫ℝ3∫ℝ3Wϵ,yϕx−ydyϕ2dx≕Jϵy,0+ℓϵ,yϕ+12Lϵ,yϕ,ϕ−Rϵ,yϕ,where we use the fact that wϵ,yjj=1,⋯,k solves
(18)−ϵ2Δu+u=18πϵ2∫ℝ3u2yx−ydyu.In order to find a critical point forJϵy,ϕ, we need to discuss each terms in the expansion (17). First, we haveLemma 3.
For anyu∈Lpℝ32≤p≤6, there holds
(19)up≤Cϵ31/p−1/2∥u∥ϵ.Proof.
Takinguϵx=uϵx, then
(20)∫ℝ3updx=ϵ3∫ℝ3uϵxpdx≤Cϵ3∫ℝ3∇uϵ2+Kϵxuϵ2dxp/2=Cϵ3ϵ−3∫ℝ3ϵ2∇u2+Kxu2dxp/2=Cϵ31−p/2uϵp,which implies the conclusion holds.Lemma 4.
There exists a positive constantC independent of ϵ such that
(21)Rϵ,yiϕ≤Cϵ−3ϕϵ+ϵ−3/2ϕϵ3−i,i=0,1,2,where Rϵ,yiϕ denotes the ith derivative of Rϵ,yϕ.Proof.
Note that(22)Rϵ,yϕ=132πϵ2∫ℝ3∫ℝ3ϕ2yx−ydyϕ2dx+18πϵ2∫ℝ3∫ℝ3Wϵ,yϕx−ydyϕ2dx.
So it is easy to check that(23)Rϵ,y′ϕ,ψ=18πϵ2∫ℝ3∫ℝ3ϕ2yx−ydyϕψdx+18πϵ2∫ℝ3∫ℝ3Wϵ,yψx−ydyϕ2dx+14πϵ2∫ℝ3∫ℝ3Wϵ,yϕx−ydyϕψdx,R′ϵ,y′ϕ,ψ,ξ=14πϵ2∫ℝ3∫ℝ3ϕyξyx−ydyϕψdx+18πϵ2∫ℝ3∫ℝ3ϕ2yx−ydyψξdx+14πϵ2∫ℝ3∫ℝ3Wϵ,yψx−ydyϕξdx+14πϵ2∫ℝ3∫ℝ3Wϵ,yξx−ydyϕψdx+14πϵ2∫ℝ3∫ℝ3Wϵ,yϕyx−ydyψξdx.
First, we estimateRϵ,yϕ. Notice that if we denote
(24)φu≔18π∫ℝ3u2yx−ydy,then φu satisfies −Δφu=u2/2 in ℝ3. So,
(25)∫ℝ3∇φu2dx=12∫ℝ3φuu2dx≤cφu6u26/5≤CφuDu12/52,which implies that
(26)φuD≤Cu12/52.
As a result, from Lemma3, we have
(27)∫ℝ3∫ℝ3ϕ2yx−ydyϕ2dx≤Cφϕ6ϕ12/52≤Cϕ12/54≤Cϵ−1ϕϵ4,(28)∫ℝ3∫ℝ3Wϵ,yϕ∣x−y∣dyϕ2dx≤∫ℝ3φWϵ,y1/2φϕ1/2ϕ2dx≤CφWϵ,y61/2φϕ61/2ϕ12/52≤CWϵ,y12/5ϕ12/53≤Cϵ95/12−1/2ϕϵ3ϵ3⋅5/12w12/5≤Cϵ1/2ϕϵ3.
Combining the definition ofRϵ,yϕ and (27), (28), we find
(29)Rϵ,yϕ≤Cϵ−3ϕϵ+ϵ−3/2ϕϵ3.
Now, we discussRϵ,y′ϕ. Similar to (27) and (28), we get
(30)∫ℝ3∫ℝ3ϕ2y∣x−y∣dyϕψdx≤Cφϕ6ϕ12/5ψ12/5≤Cϵ−1ϕϵ3ψϵ,∫ℝ3∫ℝ3Wϵ,yψx−ydyϕ2dx≤Cϵ1/2ϕϵ2ψϵ,∫ℝ3∫ℝ3Wϵ,yϕx−ydyϕψdx≤Cϵ1/2ϕϵ2ψϵ.
So, by the estimates above, we infer that(31)Rϵ,y′ϕ,ψ≤Cϵ−3ϕϵ3+ϵ−3/2ϕϵ2ψϵ,and then
(32)Rϵ,y′ϕ≤Cϵ−3ϕϵ+ϵ−3/2ϕϵ2.
Finally, by the same argument as above, we find(33)∫ℝ3∫ℝ3ϕyξyx−ydyϕψdx≤C∫ℝ3φϕ1/2φξ1/2ϕψdx≤Cϵ−1ϕϵ2ψϵξϵ,∫ℝ3∫ℝ3ϕ2yx−ydyψξdx≤Cϵ−1ϕϵ2ψϵξϵ,∫ℝ3∫ℝ3Wϵ,yψx−ydyϕξdx≤Cϵ1/2ϕϵψϵξϵ,∫ℝ3∫ℝ3Wϵ,yξx−ydyϕψdx≤Cϵ1/2ϕϵψϵξϵ,∫ℝ3∫ℝ3Wϵ,yϕx−ydyξψdx≤Cϵ1/2ϕϵψϵξϵ.
The estimates above imply that(34)R′ϵ,y′ϕ,ψ,ξ≤Cϵ−3ϕϵ2+ϵ−3/2ϕϵψϵξϵ,R′ϵ,y′ϕ≤Cϵ−3ϕϵ+ϵ−3/2ϕϵ.
This completes our proof.Lemma 5.
There holds(35)ℓϵ,y≤Cϵ3/2ϵθ+Σj=1kKyj−1+Σi≠je−yi−yj/2ϵ,where i,j=1,⋯,k,i≠j and C is a constant independent of ϵ.Proof.
Recall that(36)ℓϵ,yϕ=∫ℝ3Kx−1Wϵ,yϕdx−18πϵ2∫ℝ3∫ℝ3∑i≠jwϵ,yjwϵ,yix−ydy∑j=1kwϵ,yjϕdx−18πϵ2∫ℝ3∫ℝ3∑i≠jwϵ,yj2x−ydywϵ,yiϕdx≕ℓ1+ℓ2+ℓ3.
First, we can write(37)ℓ1=∑j=1k∫ℝ3Kx−Kyjwϵ,yjϕdx+∑j=1k∫ℝ3Kyj−1wϵ,yjϕdx≕ℓ1.1+ℓ1.2,and we have
(38)ℓ1,1=∑j=1k∫BδyjKx−Kyjwϵ,yjϕdx+∑j=1k∫ℝ3\BδyjKx−Kyjwϵ,yjϕdx≕ℓ1.1.1+ℓ1.1.2.
Then, by the assumptionK2, Lemma 3 and the decay property of w, we find
(39)ℓ1.1.1≤∑j=1k∫BδyjKx−Kyjwϵ,yjϕdx≤Cϵ3∑j=1k∫Bδ/ϵ0ϵxθwx∣ϕϵx+yj∣dx≤Cϵ3+θ∑j=1k∫Bδ/ϵ0x3/2θw3/2xdx2/3∫Bδ/ϵ0ϕϵx+yj3dx1/3≤Cϵ3+θ∫ℝ3ϕx3ϵ−3dx1/3≤Cϵ3/2+θϕϵ,where we used the fact that
(40)∫Bδ/ϵ0x3/2θw3/2xdx≤∫BR0x3/2θw3/2xdx+∫BRc0x3/2θe−3/2xdx<+∞.
On the other hand, by Hölder inequality, we have(41)ℓ1.1.2≤Cϵ3∫Bδ/ϵc0wxϕϵx+yjdx≤Cϵ3∫Bδ/ϵc0w3/2xdx2/3∫Bδ/ϵc0ϕϵx+yj3dx1/3≤Cϵ3∫δ/ϵ+∞r1/2e−3/2rdr2/3∫ℝ3ϵ−3ϕx3dx1/3≤Cϵ3e−2δ/3ϵϵ−1ϵ−1/2ϕϵ≤Cϵ3/2+θϕϵ,which, together with (41), implies that
(42)ℓ1.1≤Cϵ3/2+θϕϵ.
By the same argument as above, we also deduce that(43)ℓ1.2≤∑j=1kKyj−1∫ℝ3wϵ,yjϕdx≤∑j=1kKyj−1∫ℝ3wxϕϵx+yjϵ3dx≤ϵ3∑j=1kKyj−1∫ℝ3wx2/3dx2/3∫ℝ3ϕϵx+yj3dx1/3≤Cϵ3∑j=1kKyj−1ϵ−1ϵ−1/2ϕϵ=Cϵ3/2∑j=1kKyj−1ϕϵ.
Hence,(44)ℓ1≤Cϵ3/2ϵθ+∑j=1kKyj−1ϕϵ.
Now, in order to estimateℓ2, we recall the Hardy-Littlewood-Sobolev inequality (see [2]): if 1<p,q<∞,0<t<3 and 1/p+1/q+t/3=2, f∈Lpℝ3,g∈Lqℝ3, then
(45)∫ℝ3∫ℝ3fxgyx−ytdxdy≤Cp,q,tfpgq.
Thus, by Hölder inequality and (45), one has
(46)ℓ2=18πϵ2∫ℝ3∫ℝ3∑i≠jwϵ,yjwϵ,yix−ydy∑j=1kwϵ,yjϕdx=ϵ38π∫ℝ3∫ℝ3∑i≠jwywy−yi−yj/ϵx−ydy∑j=1kwxϕϵx+yjdx≤Cϵ3∑i≠jwywy−yi−yjϵ6/5wxϕϵx+yj6/5≤Cϵ3∑i≠je−yi−yj/ϵwx2ϕϵx+yj3≤Cϵ3∑i≠je−yi−yj/2ϵϵ−3/2ϕϵ=Cϵ3/2∑i≠je−yi−yj/2ϵϕϵ.
Finally, using LemmaA.1 and Hölder inequality, we infer that
(47)ℓ3=18πϵ2∫ℝ3∫ℝ3∑i≠jwϵ,yj2∣x−y∣dywϵ,yiϕdx=ϵ38π∫ℝ3∫ℝ3∑i≠jw2y+yi−yj/ϵ∣x−y∣dywxϕϵx+yidx≤Cϵ3∫ℝ3∑i≠je−x−yi−yj/ϵ∑m=−12x−yi−yjϵm+x−yi−yjϵ−12w2x1/2ϕϵx+yi2≤Cϵ3∑i≠j∫ℝ3e−2x−yi−yj/ϵ∑m=−12x−yi−yjϵ2m+x−yi−yjϵ−22w2x1/2ϵ−3/2ϕϵ≤Cϵ3/2∑i≠jϵ−yi−yj/2ϵyi−yjϵ3+yi−yjϵ−1ϕϵ≤Cϵ3/2+θϕϵ,which, together with (44) and (46), concludes this proof.Now, associated to the quadratic formLϵ,yϕ, we define Lϵ,y to be a bounded linear map from Eϵ,k to Eϵ,k as
(48)Lϵ,yv1,v2=∫ℝ3ϵ2∇v1∇v2+Kxv1v2dx−18πϵ2∫ℝ3∫ℝ3Wϵ,y2yx−ydyv1v2dx−14πϵ2∫ℝ3∫ℝ3Wϵ,yv1x−ydyWϵ,yv2dx.Here, we come to show the invertibility ofLϵ,y in Eϵ,k.Proposition 6.
There existϵ0,δ0,ρ,R0>0 such that for R≫R0,ϵ∈0,ϵ0,δ∈0,δ0,
(49)Lϵ,yϕ≥ρϕϵ,∀ϕ∈Eϵ,k.Proof.
We argue by contradiction. Suppose that there existsϵn⟶0, yn=yn,1,⋯,yn,k∈Dkϵn,δ, and ϕn∈Eϵn,k such that
(50)Lϵn,ynϕn,g=on1ϕnϵngϵn,∀g∈Eϵn,k.
Without loss of generality, we can assume thatϕnϵn2=ϵn3. Fix i∈1,⋯,k and let
(51)ϕ¯n,i=ϕnϵnx+yn,i.
So,(52)∫ℝ3ϵn2∇ϕn2+Kxϕn2dx=ϵn3,∫ℝ3∇ϕ¯n,i2+Kϵnx+yn,iϕ¯n,i2dx≤C,which implies that ϕ¯n,i is bounded in H1ℝ3. Thus, up to a subsequence, there exists ϕ∈H1ℝ3 such that as n⟶+∞,
(53)ϕ¯n,i⇀ϕ,inH1ℝ3,ϕ¯n,i⟶ϕ,a.einℝ3,ϕ¯n,i⟶ϕ,ELloctℝ3,2≤t≤2∗.
Next we will proveϕ=0. To this end, from (50), we find that ϕ¯n,i satisfies for any g¯∈E¯n,
(54)∫ℝ3∇ϕ¯n,i∇g¯+Kϵnx+yn,iϕ¯n,ig¯dx−18π∫ℝ3∫ℝ3∑j=1kw¯ϵn,yn,j2x−ydyϕ¯n,ig¯dx−14π∫ℝ3∫ℝ3∑j=1kw¯ϵn,yn,jϕ¯n,ix−ydy∑j=1kw¯ϵn,yn,jg¯dx=on1g¯∗,where
(55)g¯∗2=∫ℝ3∇g¯2+Kϵnx+yn,ig¯2dx,w¯ϵn,yn,jx=wx+yn,i−yn,jϵn,E¯n=g¯:g¯x−yn,iϵn∈Eϵn,k,∫ℝ3∇∂w¯ϵn,yn,j∂ysn,j∇g¯dx+∫ℝ3Kϵnx+yn,i∂w¯ϵn,yn,j∂ysn,jg¯dx=0,for j=1,⋯,k and s=1,2,3..
But, forg∈C0∞ℝ3, we can decompose g as follows
(56)g=gn−∑j=1k∑s=13an,j,s∂w¯ϵn,yn,j∂ysn,j,where gn∈E¯n and an,j,s∈ℝ. Then, by the exponential decay of ∂w¯ϵn,yn,j/∂ysn,j, we have
(57)∫ℝ3∇∂w¯ϵn,yn,j∂ysn,i∇gndx+∫ℝ3Kϵnx+yn,i∂w¯ϵn,yn,j∂ysn,igndx=on1,∫ℝ3∇∂w¯ϵn,yn,j∂ysn,i∇∂w¯ϵn,yn,h∂ysn,idx+∫ℝ3Kϵnx+yn,i∂w¯ϵn,yn,j∂ysn,i∂w¯ϵn,yn,h∂ysn,idx=on1,for h≠j and h,j=1,⋯,k. On the other hand,
(58)∫ℝ3∇∂w¯ϵn,yn,j∂ysn,i2dx+∫ℝ3Kϵnx+yn,i∂w¯ϵn,yn,j∂ysn,i2dx≥C.
So, up to a subsequence, we can easily check thatan,j,s⟶0 as n⟶∞ for j≠i, while an,i,s⟶ai,s for some ai,s∈ℝ. Inserting gnx−yn,i/ϵn into (54) and letting n⟶+∞, we infer that
(59)∫ℝ3∇ϕ∇g+ϕgdx−18π∫ℝ3∫ℝ3w2yx−ydyϕgdx−14π∫ℝ3∫ℝ3wyϕyx−ydywgdx+∑s=13ai,s∫ℝ3∇ϕ∇∂w∂xs+ϕ∂w∂xsdx−18π∫ℝ3∫ℝ3w2yx−ydyϕ∂w∂xsdx−14π∫ℝ3∫ℝ3wyϕyx−ydyw∂w∂xsdx=0.
Sincew solves
(60)−Δw+w=18π∫ℝ3w2yx−ydywxinℝ3.
We find that(61)−Δ∂w∂xs+∂w∂xs=18π∫ℝ3w2yx−ydy∂w∂xs+14π∫ℝ3wy∂w/∂xsx−ydywx,which implies that
(62)∫ℝ3∇ϕ∇∂w∂xs+ϕ∂w∂xsdx=18π∫ℝ3∫ℝ3w2yx−ydyϕ∂w∂xsdx+14π∫ℝ3∫ℝ3wyϕyx−ydyw∂w∂xsdx.
Combining (59) and (62), we have
(63)∫ℝ3∇ϕ∇g+ϕgdx−18π∫ℝ3∫ℝ3w2yx−ydyϕgdx−14π∫ℝ3∫ℝ3wyϕyx−ydywgdx=0.
Considering thatg∈C0∞ℝ3 is arbitrary and w is nondegenerate, there exist as,s=1,2,3 such that
(64)ϕ=∑s=13as∂w∂xs.
Moreover, beingϕn∈Eϵn,k, we have
(65)∫ℝ3∇ϕ∇∂w∂xs+ϕ∂w∂xsdx=0,which, together with (64), yields ϕ=0. Finally, by Lemma A.1, we deduce that
(66)18πϵn2∫ℝ3∫ℝ3Wϵn,yn2yx−ydyϕn2dx=ϵn38π∫ℝ3∫ℝ3wy+∑i≠jwy+yn,j−yn,i/ϵn2x−ydyϕn2ϵnx+yn,jdx≤Cϵn3∫ℝ3w2yx−ydyϕn2ϵnx+yn,jdx+Cϵn3∫ℝ3∑i≠je−x−yn,i−yn,j/ϵn∑m=−12x−yn,i−yn,jϵnm+x−yn,i−yn,jϵn−1ϕn2ϵnx+yn,jdx≤Cϵn3∫ℝ3e−∣x∣∑m=−12xm+x−1ϕ¯n,j2dx+∫ℝ3e−x−yn,i/ϵn∑m=−12x−yn,iϵnm+x−yn,iϵn−1ϕ2xdx≤Cϵn3∫BR0+∫BRc0e−∣x∣∑m=−12xm+x−1ϕ¯n,j2dx+Cϵn3∫∪i=1kBϵnRyn,i+∫ℝ3∪i=1kBϵnRyn,ie−x−yn,i/ϵn∑m=−12x−yn,iϵnm+x−yn,iϵn−1ϕ2xdx≤Cϵn3∫∪i=1kBϵnRyn,ie−x−yn,i/ϵn∑m=−12x−yn,iϵnm+x−yn,iϵn−1ϕ2xdx+oϵn3+Ce−1−θR+1Rϵn3=oϵn3+oR1ϵn3,where oR1⟶0 as R⟶+∞.
Similarly, the Hardy-Littlewood-Sobolev inequality (45) implies that
(67)14πϵn2∫ℝ3∫ℝ3Wϵn,ynyϕny∣x−y∣dyWϵn,ynϕndx≤Cϵn2∫∪i=1kBϵnRyn,iWϵn,yn6/5ϕn6/5dx+∫ℝ3∪i=1kBϵnRyn,iWϵn,yn6/5ϕn6/5dx5/3≤oϵn3+Ce−1−θR=oϵn3+oR1ϵn3.
As a result, by (50),
(68)on1ϵn3=on1ϕnϵn2=Lϵn,ynϕn,ϕn=ϕnϵn2−18πϵn2∫ℝ3∫ℝ3Wϵn,yn2yx−ydyϕn2dx−14πϵn2∫ℝ3∫ℝ3Wϵn,ynϕnx−ydyWϵn,ynϕndx≥ϕnϵn2+oϵn3+oR1ϵn3,which is impossible. So we complete this proof.Proposition 7.
Suppose thatK1 and K2 hold. Then, for any given k=1,2⋯, there exist ϵ0,δ0,R0>0 such that for R≫R0,ϵ∈0,ϵ0,δ∈0,δ0, there is a C1 map from Dkϵ,δ to Eϵ,k, ϕϵ=ϕϵy satisfying
(69)∂Jϵy,ϕϵ∂ϕϵ,ψ=0,∀ψ∈Eϵ,k,ϕϵϵ≤Cϵ3/2ϵθ+∑j=1kKyj−1+∑i≠je−yi−yj/2ϵ.Proof.
We use the contraction mapping theorem to prove the wanted result. It follows from Lemma5 that ℓϵ,yϕ is a bounded linear map in Eϵ,k. So applying Reisz representation theorem, there exists an ℓϵ,yk∈Eϵ,k such that
(70)ℓϵ,yϕ=ℓϵ,yk,ϕ.
Thus, finding a critical point forJϵy,ϕ is equivalent to solving
(71)ℓϵ,yk+Lϵ,yϕ−Rϵ,y′ϕ=0.
SinceLϵ,y is invertible in Eϵ,k from Proposition 6, (71) can be rewritten as
(72)ϕ=Lϵ,y−1Rϵ,y′ϕ−Lϵ,y−1ℓϵ,yk≔Aϕ.
Define(73)Sϵ=ϕ∈Eϵ,k:ϕϵ≤Cϵ3/2ϵθ+∑j=1kKyj−1+∑i≠je−1/2yi−yj/ϵ.
We shall verify thatA is a contraction mapping from Sϵ to itself. First, for ∀ϕ∈Sϵ, by Lemmas 4 and 5, we have
(74)Aϕ≤Cℓϵ,ykϵ+Rϵ,y′ϕ≤Cℓϵ,ykϵ+ϵ−3/2ϕϵ2≤Cϵ3/2ϵθ+∑j=1kKyj−1+∑i≠je−yi−yj/2ϵ+Cϵ3/2ϵθ+∑j=1kKyj−1+∑i≠je−1/2yi−yj/ϵ2≤ϵ3/2ϵθ+∑j=1kKyj−1+∑i≠je−yi−yj/2ϵ,which tells that A maps Sϵ to Sϵ. On the other hand, for any ϕ1,ϕ2∈Sϵ, using Lemma 4,
(75)Aϕ1−Aϕ1=Lϵ,y−1Rϵ,y′ϕ1−Lϵ,y−1Rϵ,y′ϕ2≤CRϵ,y′ϕ1−Rϵ,y′ϕ2≤CR′ϵ,y′νϕ1+1−νϕ2ϕ1−ϕ2ϵ≤12ϕ1−ϕ2ϵ,where ν∈0,1. Therefore, A is a contraction map from Sϵ to Sϵ, and then, applying the contraction mapping theorem, we can find a unique ϕϵ satisfying (71). So the conclusion follows.


## 3. Proof of the Main Results

In this section, we come to prove our main results. LetR≫R0,ϵ∈0,ϵ0,δ∈0,δ0 and ϕϵy be as in Proposition 7. Define
(76)Fϵy=Jϵy,ϕϵy,y∈Dkϵ,δ,and let yϵ=yϵ1,⋯,yϵk∈Dkϵ,δ satisfies
(77)Fϵyϵ=maxFϵy:y∈Dkϵ,δ.Next, we can show thatyϵ is an interior point of Dkϵ,δ and thus a critical point of Fϵ for small ϵ.Lemma 8.
Suppose thatyϵ satisfies (77). Then, as ϵ⟶0, yϵj⟶y0,j=1,⋯,k, and yϵi−yϵj/ϵ⟶∞,ifi≠j.Proof.
It follows from LemmaA.2 and Proposition 7 that
(78)Jϵyϵ,ϕϵyϵ=Jϵyϵ,0+Oℓϵ,yϵkϵϕϵyϵϵ+ϕϵyϵϵ2=Jϵyϵ,0+Oϵ3ϵ2θ+∑j=1kKyϵj−12+∑i≠je−yϵi−yϵj/ϵ=ϵ3k2wH12−12∫ℝ3w2dx∑j=1kKy0−Kyϵj−k32π∫ℝ3∫ℝ3w2yx−ydyw2xdx+Oϵ3ϵθ+∑j=1kKyϵj−12+∑i≠je−yϵi−yϵj/ϵ.
Takeλϵi=ϵσξi,i=1,⋯,k for some σ∈1/2,1 and some vectors ξ1,⋯,ξk with ξi≠ξji≠j. Then as ϵ⟶0,
(79)λϵi−λϵjϵ=ξi−ξjϵ1−σ⟶∞,which means that λϵ=λϵ1,⋯,λϵk∈Dkϵ,δ if ϵ>0 is small enough. Combining (77) and (78), we have
(80)Jϵyϵ,ϕϵyϵ≥Jϵλϵ,ϕϵλϵ≥ϵ3k2wH12−k32π∫ℝ3∫ℝ3w2yx−ydyw2xdx+Oϵ3ϵθ+∑j=1kKλϵj−12+∑i≠je−ξi−ξj/ϵ1−σ=ϵ3k2wH12−k32π∫ℝ3∫ℝ3w2yx−ydyw2xdx+Oϵ3+σ¯,where σ¯=min1−σ,σθ. Hence,
(81)ϵ3k2wH12−12∫ℝ3w2dx∑j=1kKy0−Kyϵj−k32π∫ℝ3∫ℝ3w2yx−ydyw2xdx+Oϵ3ϵθ+∑j=1kKyϵj−12+∑i≠je−yϵi−yϵj/ϵ≥ϵ3k2wH12−k32π∫ℝ3∫ℝ3w2yx−ydyw2xdx+Oϵ3+σ¯,which yields that
(82)12∫ℝ3w2dx∑j=1kKy0−Kyϵj≤Oϵ3ϵσ¯+∑j=1kKyϵj−Ky02+∑i≠je−yϵi−yϵj/ϵ.
So, asϵ⟶0 and for i,j=1,⋯,k,i≠j, we find
(83)Kyϵj⟶Ky0=1,yϵj⟶y0,yϵi−yϵjϵ⟶∞,from which, the conclusion follows.Proof of Theorem 2.
By Lemma8, we can check that (77) can be obtained by some yϵ=yϵ1,⋯,yϵk∈Dkϵ,δ, which is an interior point of Dkϵ,δ for small ϵ and satisfies
(84)yϵj⟶y0,yϵi−yϵjϵ⟶∞,∂Jϵyϵ,ϕϵyϵ∂yϵj=0for i,j=1,⋯,k,i≠j. Moreover, from Proposition 7, ϕϵyϵϵ=oϵ3/2 as ϵ⟶0. Finally, it is well-known that if yϵ is a critical point of Fϵyϵ, then Wϵ,yϵ+ϕϵyϵ is a solution of (2)Thus, we finish this proof.



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