# The Existence of Normalized Solutions for a Nonlocal Problem inℝ3

Authors: Jing Yang
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this paper, we study the following fractional Schrödinger equation inℝ3−Δσu−λu=up−2u, in ℝ3 with σ∈0,1,λ∈ℝ and p∈2+σ,2+4/3σ. By using the constrained variational method, we show the existence of solutions with prescribed L2 norm for this problem.


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## Body

## 1. Introduction

This paper concerns with the following fractional Schrödinger problem:(1)−Δσu−λu=up−2u,inℝ3,where σ∈0,1, p∈2+σ,2+4/3σ, and λ∈ℝ. Here, the fractional Laplacian −Δσ in ℝn is defined by
(2)−Δσu=Cn,σPV∫ℝnux−uyx−yn+2σdy,where PV stands for the Cauchy principal value and Cn,σ is a normalization constant.In the present paper, the motivation for studying such equations comes from mathematical physics: searching for the form of standing waveψ=e−ihtu of the evolution equation
(3)i∂ψ∂t+−Δσψ+λ−hψ=ψp−2ψinℝ+×ℝ3leads to looking for solutions of (1). Here, i is the imaginary unit and h∈ℝ. This class of Schrödinger-type equations is of particular interest in fractional quantum mechanics for the study of particles on stochastic fields modelled by Lévy processes. A path integral over the Lévy flight paths and a fractional Schrödinger equation of fractional quantum mechanics are formulated by Laskin [1, 2] from the idea of Feynman and Hibbs’s path integrals.On the other hand, problem (1) has attracted considerable attention in the recent period. Part of the motivation is to consider h∈ℝ as a fixed parameter and then to search for a solution u∈Hσℝ3 solving (1). In this direction, mainly by variational methods, many researches have been devoted to the study of the existence, multiplicity, uniqueness, regularity, and asymptotic decay properties of the solutions to fractional Schrödinger equation (1). For this information, we can refer to [3–11] and the references therein. Besides, some more complicated fractional equations and systems were also studied, and indeed, some interesting results were obtained. Nearly, Mingqi et al. [12] investigated a critical Schrödinger-Kirchhoff type systems driven by nonlocal integrodifferential operators and by applying the mountain pass theorem and Ekeland’s variational principle; the authors obtained the existence and asymptotic behavior of solutions for this system under some suitable assumptions. Later, in [13], the same authors as in [12] studied a diffusion model of Kirchhoff-type. Under some appropriate conditions, by employing the Galerkin method, the local existence of nonnegative solutions was obtained, and then by virtue of a differential inequality technique, they proved that the local nonnegative solutions blow up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, in [14], Mingqi et al. concerned with a class of fractional Kirchhoff-type problems with the Trudinger-Moser nonlinearity. By applying minimax techniques combined with the fractional the Trudinger-Moser inequality, they found the existence of a ground state solution with positive energy and the existence of nonnegative solutions with negative energy by using Ekeland’s variational principle. In [15], the three authors considered a fractional Choquard-Kirchhoff-type problem involving an external magnetic potential and a critical nonlinearity and established a fractional version of the concentration-compactness principle with magnetic field, and then together with the mountain pass theorem, they verified the existence of nontrivial radial solutions in nondegenerate and degenerate cases. Furthermore, Mingqi et al. [16] concerned the Schrödinger-Kirchhoff-type problems involving the fractional p-Laplacian and critical exponent. By using the concentration-compactness principle in fractional Sobolev spaces, they showed the existence of m pairs of solutions for any m∈ℕ, and by applying Krasnoselskii’s genus theory, they also got the existence of infinitely many solutions under some suitable conditions for the parameter. For more information on this direction, one can refer to [17–24] and the references therein.In the present paper, inspired by the fact that physicists are often interested in normalized solutions, we look for solutions inHσℝ3 having a prescribed L2 norm to equation (1). Such types of problems were studied extensively in recent years for the classical Schrödinger equations with the standard Laplacian operator. We refer the interested reader to [25–31] and to the references therein. But up to our knowledge, not much is obtained for the existence of normalized solutions of equation (1) in Hσℝ3 with a fractional Laplacian operator. So, in this paper, the aim is to get the normalized solutions of equation (1). Here, we give the definition of prescribed ρ-L2 norm solutions. For fixed ρ>0, if uρ∈Hσℝ3 is a solution of problem (1) such that
(4)uρ2≔∫ℝ3uρ21/2=ρ,we call it a prescribed ρ-L2 norm solution. Naturally, a prescribed ρ-L2 norm solution uρ∈Hσℝ of (1) can be a constrained critical point of the functional
(5)Iu=12∫ℝ3−Δσ/2u2−1p∫ℝ3up,on the L2 sphere Bρ in Hσℝ3, where
(6)Bρ=u∈Hσℝ3:u2=ρ,ρ>0.Note that for anyp∈2,6/3−2σ, Iu is a well-defined and C1 functional. Set
(7)mρ=infBρIu.It is standard that ifuρ is a minimizer of (7), then uρ is a solution of (1) with prescribed ρ-L2 norm with the constraint λρ∈ℝ being the Lagrange multiplier. However, it is worth mentioning that dealing with this kind of problem, one has to face the main difficulty concerning with the lack of compactness of the minimizing sequence un⊂Bρ. In fact, we will encounter two possible bad scenarios that un⇀0 and un⇀u¯≠0 with 0<u¯2<ρ. In order to avoid the possible cases and to get that the infimum is obtained, we prove an important lemma (Lemma 6) that guarantees the compactness of minimizing sequence. As a consequence of this lemma, setting
(8)ρ1∗=infρ>0:mρ<0,ρ2∗=infρ>0:∃u∈BρsuchthatIu≤0,we can get our main result as follows:Theorem 1.
If p∈2+σ,2+4/3σ, then mρ has a minimizer if and only if ρ∈ρ1∗,+∞.
In particular, there is a prescribed ρ-L2 norm solution uρ∈Hσℝ3 of (1) with the constraint λρ∈ℝ. But, when p=2+4/3σ, mρ has no minimizer for any ρ∈ρ2∗,+∞.Remark 2.
In fact, for anyρ>0, we can infer that mρ≤0. To see this, letting u∈Bθ−1ρ be arbitrary and considering uθy=θ5/2uθy for any θ>0, we find uθ∈Bρ and
(9)Iuθ=12θ2+2σ∫ℝ3−Δσ/2u2−1pθ5/2p−3∫ℝ3up.
Hence,Iuθ⟶0 as θ⟶0 and the conclusion is as follows:
Finally, we give the following notations which can be used in this paper:(i)
Hσℝ3 is the usual Sobolev space endowed with the standard norm
(10)uHσ2=∫ℝ3−Δσ/2u2+u2
(ii)
Denoteuσ2≔∫ℝ3−Δσ/2u2
(iii)
uq is the norm of the Lebesgue space Lqℝ3 for 1<q<∞
(iv)
DenoteC>0 by various positive constants which may vary from one line to another and which are not important for the analysis of the problemThis paper is organized as follows: In Section2, we will give some preliminary results which are crucial to prove our main result. And then the proof of our main result is given in Section 3.


## 2. Preliminaries

In this part, we give some important results. First, similar to the classical Gagliardo-Nirenberg inequality to the Laplacian operator, we introduce the fractional version of Gagliardo-Nirenberg inequality as follows:Lemma 3 (see [32]).
Let 1≤q, p2<∞,0<α<q<∞, 0<σ<n, and 1<p1<n/σ. We have
(11)uLqℝn≤C0α/q−Δσ/2uLp1Rnα/quLp2ℝnq−α/q,with α1/p1−σ/n+q−α/p2=1and
(12)C0=2−σπ−σ/2Γn−σ/2Γn+σ/2ΓnΓn/2σ/n.
In particular, when n=3, one has
(13)uq≤C0ϑ−Δσ/2u2ϑu21−ϑ,with ϑ=3q−2/2qσ.In [33], the authors have established the Pohozaev identity for the fractional Laplacian operator.Applying the Pohozaev identity, we have the following:Lemma 4.
Ifu∗ is a critical point of Iu on Bρ, then ηu∗=0, where
(14)ηu=2σ∫ℝ3−Δσ/2u2−3p−2p∫ℝ3up.Proof.
Define the functional energy corresponding to (1) as
(15)Fλu≔12∫ℝ3−Δσ/2u2−λ2∫ℝ3u2−1p∫ℝ3up.
Then any critical pointu of Fλu satisfies the Pohozaev identity for (1) (see [33]), that is,
(16)Dλu≔3−2σ∫ℝ3−Δσ/2u2−3λ∫ℝ3u2−6p∫ℝ3up=0.
On the other hand, ifu is a critical point of Iu restricted to Bρ, there is a Lagrange multiplier λ∗∈ℝ such that
(17)I′u∗=λ∗u∗.
So, for anyψ∈Hσℝ3, we have
(18)Fλ∗u∗,ψ=I′u∗−λ∗u∗,ψ=0.Furthermore, if we now know that(19)Aλu≔Fλu,u=∫ℝ3−Δσ/2u2−λ∫ℝ3u2−∫ℝ3up,by (18), we find Aλ∗u∗=0. As a result,
(20)ηu∗=3Aλ∗u−Dλ∗uu=u∗=0.Using Lemma3, the estimate (13) leads to the following fact:Lemma 5.
Ifp∈2+σ,2+4/3σ, then for any ρ>0, functional I is bounded from below and coercive on Bρ.Proof.
Ifu2=ρ, from Lemma 3, we have
(21)up≤Cuσ3p−2/2pσ.
So,(22)Iu=12∫ℝ3−Δσ/2u2−1p∫ℝ3up≥12uσ2−Cuσ3p−2/2σ.
Sincep<2+4/3σ and 3p−2/2σ<2, our result follows.From the above Lemma, we can prove thatLemma 6.
Ifp∈2+σ,2+4/3σ, then
(i)
there existsρ1>0 such that for all ρ∈ρ1,∞, mρ<0
(ii)
for anyρ>0 such that mρ<0, mρ admits a minimizer
(iii)
the functionalρ↦mρ is continuous about each ρ>0Proof.
(i)
Lettinguβy=β1−3/2θuβ−θy such that uβ2=βu2 with β,θ∈ℝ, we see
(23)∫ℝ3−Δσ/2uβ2=β2−2σθ∫ℝ3−Δσ/2u2,∫ℝ3uβp=βp1−3/2θ+3θ∫ℝ3up.
If we takeθ=−2, one has
(24)Iuβ=12β21+2σ∫ℝ3−Δσ/2u2−1pβ4p−6∫ℝ3up.
Sincep>2+σ and 4p−6>2+4σ, for β is large enough, we get Iuβ<0 and then (i) has been proved.
(ii)
We will divide it into three steps to show (ii).
Step 1. We claim that for any0<ν<ρ, mρ<mν+mρ2−v2. To see this, let un be a minimizing sequence on Bρ for mρ. Since mρ<0, from (22), we can get
(25)0<C1<unσ2<C2,0<C3<unpp<C4.
On the other hand, applying (24), we obtain that for θ=−2,
(26)Iunβ=12β21+2σ∫ℝ3−Δσ/2un2−1pβ4p−6∫ℝ3unp=β2Iun+fβ,un,where
(27)fβ,un=12β4σ−1∫ℝ3−Δσ/2un2−1pβ4p−8−1∫ℝ3unp,with 4p−8>4σ. Moreover, using Iun<0, we can deduce that
(28)dfβ,undββ=1=2σ∫ℝ3−Δσ/2un2−1p4p−8<0,and for all β>1,
(29)d2fβ,undβ2=2σ4σ−1β4σ−2∫ℝ3−Δσ/2un2−1p4p−84p−9β4p−10∫ℝ3unp<0.
Thus, combining (28) and (29), for all β>1, we find fβ,un<0 and from (26),
(30)mβρ<β2mρ.
As a result, ifρ2−ν2<ν<ρ,
(31)mρ=mρ/νν<ρ2ν2mν=ρ2−ν2+ν2ν2mν=ρ2−ν2ν2mν/ρ2−ν2ρ2−ν2+mν<mρ2−ν2+mν,
and ifν<ρ2−ν2<ρ,
(32)mρ=mρ/ρ2−ν2ρ2−ν2<ρ2ρ2−ν2mρ2−ν2=ρ2−ν2+ν2ρ2−ν2mρ2−ν2=mρ2−ν2+ν2ρ2−ν2mνρ2−ν2/ν<mρ2−ν2+mν.
It follows from (31) and (32) that the claim holds.
Step 2. We will show that all the minimizing sequencesun for mρ have a weak limit, up to translations, different from zero. Let un be a minimizing sequence on Bρ for mρ. Note that for any sequence yn⊂ℝ3, un⋅+yn is still a minimizing sequence for mρ. So the proof of this step can be finished if we can prove the existence of a sequence yn⊂ℝ3 such that the weak limit of un⋅+yn is different from zero.
Applying Lion’s lemma, we know that if(33)limn→∞supx∈ℝ3∫Bx,1un2dy=0,then un⟶0 in Lqℝ3 for any q∈2,6/3−2σ, where Ba,r=x∈ℝ3:x−a≤r. Since mρ<0, we see that
(34)∫B0,1un⋅+yn2dy≥δ>0.
Therefore, it follows the compactness of the embeddingHσB0,1↪L2B0,1 that the weak limit of the sequence un⋅+yn is not the trivial function.
Step 3. Finally, we verify thatmρ has a minimizer for mρ<0. Suppose un to be a minimizing sequence on Bρ for mρ with mρ<0. Then by Lemma 5, un is bounded in Hσℝ3 and Lqℝ3 for any q∈2,6/3−2σ. So there exists u¯∈Hσℝ3 such that un⇀u¯ in Hσℝ3 and then we can get
(35)12∫ℝ3−Δσ/2un2−1p∫ℝ3unp=mρ+o1,(36)12∫ℝ3−Δσ/2un−u¯2−1p∫ℝ3un−u¯p+12∫ℝ3−Δσ/2u¯2−1p∫ℝ3u¯p=mρ+o1.
If we setν≔u¯2 and ϵn≔ρ2−v2/un−u¯2, then by the Step 2, 0<ν≤ρ. Now, we want to prove that ν=ρ. To see this, we assume that 0<ν<ρ. From (36), we find
(37)12ϵnun−u¯σ2−1pϵnun−u¯pp+Iu¯=mρ+o1,since ϵn⟶1. Noting that ϵnun−u¯2=ρ2−ν2, (37) tells that
(38)mρ2−ν2+mν≤mρ+o1,which contradicts to (31) and (32) and then ν=ρ.
Sinceu¯∈Bρ, we have un−u¯2=o1. Hence, if we would verify that un⟶u¯ in Hσℝ3, it remains to show that un−u¯σ2=o1 up to a subsequence. First, by assumption, there is λn⊂ℝ such that
(39)I′un−λnun,ψ=o1,∀ψ∈Hσℝ3.
So,(40)I′un−λnun,un=o1,which implies λn is bounded and up to a subsequence; there exists λ∈ℝ with λn⟶λ.
On the other hand, we findn,m⟶∞,
(41)I′un−I′um−λnun+λmum,un−um=o1,λn−λmum,un−um=o1.
Hence,(42)un−umσ2−∫ℝ3unp−2un−ump−2umun−um−λnun−um22=o1.
Notice that by the interpolation inequality, we get(43)un−ump≤Cun−um2γun−umσ1−γ,with γ/2+1−γ/2σ∗=1/p and 2σ∗=6/3−2σ, and then
(44)un−ump=o1.
As a result,(45)∫ℝ3unp−2unump−2umun−um≤∫ℝ3unpp−1/p+∫ℝ3umpp−1/pun−ump=o1.
Usingun−um2=o1, λn⟶λ, (42) and (45), one can get that un is a Cauchy sequence in Hσℝ3 and hence un−u¯Hσℝ3⟶0 as n⟶∞.
(iii)
Now, we come to prove that ifρn⟶ρ, then limn→∞mρn=mρ. For any n∈ℕ+, let wn∈Bρn such that Iwn<mρn+1/n. Using Lemma 5, we deduce that wn is bounded in Hσℝ3 and then wnσ and wnp are bounded. So it is easy to find that
(46)mρ≤Iρρnwn=12ρρn2wnσ2−1pρρnpwnpp=Iwn+oI<mρn+o1.
On the other hand, lettingvn⊂Bρ be a minimizing sequence for mρ, we have
(47)mρn≤Iρnρvn=Ivn+o1=mρ+o1.
Hence, from (46) and (47), limn→∞Iρn=mρ follows.


## 3. Proof of the Main Result

To prove our main theorem, we first give the following important results:Lemma 7.
Whenp∈2+σ,2+4/3σ, there exists ρ3>0 such that mρ has no minimizer for all ρ∈0,ρ3.Proof.
We prove it by contradiction and suppose that there existρn⊂ℝ+ with ρn⟶0+ as n⟶∞ and un⊂Bρn such that Iun=mρn.Since for anyρ>0, mρ≤0, we have mρn≤0 and then by Lemma 3,
(48)12unσ2≤1punpp≤Cunσ3p−2/2σρnp1−ϑ,with ϑ=3p−2/2pσ. Due to p<2+4/3σ and 3p−2/2σ<2, (48) tells us that
(49)unσ⟶0asn⟶∞.So, from (48) and Lemma 4, we infer that
(50)ηun=2σunσ2−3p−2punpp≥2σunσ2−Cunσ3p−2/2σρnp1−ϑ>0,which is impossible since ηun=0 from Lemma 4.Recall that(51)ρ1∗=infρ>0:mp<0,ρ2∗=infρ>0:∃u∈BρsuchthatIu≤0.We haveLemma 8.
Ifp=2+4/2σ, then p2∗∈0,+∞.Proof.
First, it follows from Lemma3 that for any u∈Bρ, if p=2+4/3σ
(52)upp≤Cuσ2ρ2.
So,(53)Iu≥12uσ2−Cpuσ2ρ2.Thus,Iu>0 for any u∈Bρ with ρ>0 small enough and then ρ2∗>0 follows.On the other hand, takingu1∈B1, then uθy=θ5/2u1θy∈Bθ for all θ>0 and
(54)Iuθ=12θ2+2σu1σ2−1pθ5/2p−3u1pp.This implies that ifθ>0 is large enough, Iuθ<0 and our result has been proved.Lemma 9.
Whenp∈2+σ,2+4/3σ, we have the following:
(i)
ρ1∗∈0,∞
(ii)
mp=0 if ρ∈0,ρ1∗
(iii)
mρ<0 and is strictly decreasing about ρ if ρ∈ρ1∗,+∞
(iv)
Moreover, whenp=2+4/3σ, then ρ2∗∈0,∞ and mρ=0 as ρ∈0,ρ2∗ and mρ=−∞ if ρ∈ρ2∗,+∞Proof.
(i)
We prove it by contradiction. Suppose thatρ1∗=0, and then from the definition of ρ1∗, for any ρ>0, we can get mρ<0. Hence, it follows from Lemma 6 (ii) that mρ has a minimizer for any ρ>0. But this gives a contradiction with Lemma 7. On the other hand, using Lemma 6 (i), we can infer that ρ1∗<∞ and so (i) follows.
(ii)
First, from the definition ofρ1∗ and mρ≤0 for any ρ>0, we know that mρ=0 for ρ∈0,ρ1∗. Furthermore, using the continuity of ρ⟶mρ (see Lemma 6 (iii)), we have mρ1∗=0 and (ii) is proved.
(iii)
By the definition ofρ1∗, we have mρ<0 for ρ∈ρ1∗,+∞. So, by Lemma 6 (ii), mρ admits a minimizer uρ∈Bρ. Setting with uρθ=θ5/2uρθy∈mθρ with θ>1, we can find that
(55)mθρ≤Iuρθ=12θ2+2σuρσ2−1pθ5/2p−3uρpp=θ2+2σ12uρσ2−1pθ5/2p−5−2σuρpp<θ2+2σ12uρσ2−1puρpp=θ2+2σIuρ,which implies that mθρ<θmρ<mρ since mρ<0 and θ>1. This tells that (iii) holds.
(iv)
It follows from Lemma8 that ρ2∗∈0,+∞. On the other hand, from the definition of ρ2∗, it is direct to see that mρ=0 for ρ∈0,ρ2∗. Now, it remains to show that mρ=−∞ if ρ∈ρ2∗,+∞. Here, we claim that for any ρ∈ρ2∗,+∞, there exists u∗∈Bρ such that Iu∗≤0. In fact, suppose that Iu>0 for all ρ∈ρ2∗,+∞. Then for an arbitrary ρ¯∈ρ2∗,ρ, taking v¯∈Bρ¯ and v¯θ=θ5/2v¯θy for θ=ρ/ρ¯, we find v¯θ∈Bρ and from (55),
(56)0<Iv¯θ<θ2+2σIv¯,which tells that Iv¯>0 for v¯∈Bρ¯. This contradicts the definition of ρ2∗ since ρ¯∈ρ2∗,ρ and so the claim holds.
Now we consider another scalinguθ∗=θ3/2u∗θy for all θ>0. Then uθ∗∈Bρ and
(57)Iuθ∗=12θ2σu∗σ2−1pθ3/2p−3u∗pp.
This yieldsIuθ∗⟶−∞ as θ⟶+∞, and we get our result.Proof of Theorem 1.
We will divide it into three steps to prove Theorem1.
Step 1. First, we prove that ifp∈2+σ,2+4/3σ, mρ1∗ has a minimizer. Set ℓn=ρ1∗+1/n for all n∈ℕ+. Then ℓn⟶ρ1∗ and using Lemmas 6 (iii) and 9 (ii), we find mℓn⟶mρ1∗=0.
So applying Lemmas9 (iii) and 6 (ii), mℓn<0 and mℓn have a minimizer, denoted by vn. By Lemma 5, vn is bounded in Hσℝ3.
Now, we claim thatvnp↛0 as n⟶∞. Suppose that vnp⟶0 as n⟶∞ by contradiction. Since Ivn<0, similar to (49) and (48), we find
(58)vnσ⟶0,(59)Ivn≥12vnσ2−Cpvnσ3p−2/2σℓnp1−ϑ,with ϑ=3p−2/2pσ, which, combining (58), implies that Ivn≥0 for n is large enough. This contradicts Ivn=mℓn<0, and then the claim holds.
With the same argument as the proof of (34), there exists δ>0 and a sequence yn⊂ℝ3 such that
(60)∫B0,1vn⋅+yn2dy≥δ>0.
Lettingv¯n=vn⋅+yn, then v¯n is bounded in Hσℝ3 and there exists v0∈Hσℝ3 such that v¯n⇀v0inHσℝ3, and v¯n⟶v0inLloc2ℝ3.
Thus, by (60), we can check that v0≠0. Finally, we come to show v0 is a minimizer of mρ1∗. First, we have
(61)limn→∞v¯n2=v02+limn→∞v¯n−v02=ρ1∗,and then by Lemma 6 (iii) and 9 (ii),
(62)limn→∞Iv¯n−v0≥limn→∞mv¯n−v02=mρ1∗v02=0.
On the other hand, from Lemma9 (ii), we have v02≤ρ1∗ and mv02=0 and then Iv0<0 is impossible. So by (61) and (62), we can get Iv0=0=mρ1∗ and v0 is a minimizer of mv02=0. Now if we assume that v02<ρ1∗, using (55), one has
(63)mρ1∗=mρ1∗/v02v02<ρ1∗v02mv02=0.
This yields a contradiction withmρ1∗=0, and thus v02=ρ1∗.
Step 2. We will prove thatmρ has a minimizer if and only if ρ∈ρ1∗,+∞ if p∈2+σ,2+4/3σ.
Suppose that there isρ0∈0,ρ1∗ such that mρ0 has a minimizer. Then from the definition of ρ1∗ and (55), we get that mρ0=0 and mρ<0 for any ρ>ρ0. This contradicts the definition of ρ1∗.
On the other hand, ifρ∈ρ1∗,+∞, using Lemmas 9 (iii) and 6 (ii), mρ admits a minimizer.
Step 3. Finally, it follows from the definition ofηu that for any u∈Bρ and if p=2+4/3σ,
(64)Iu−13p−2ηu=12−2σ3p−2−Δσ/2u22.
Thus, if we assume thatmρ has a minimizer uρ∈Bρ for some ρ∈ρ2∗,+∞, then from (64) and Lemma 9 (iv), we have
(65)−∞=mρ=Iuρ=0,which completes the Proof of Theorem 1.



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