Ma hema ische Zei sch i (2024) 306:50
h ps://doi.o g/10.1007/s00209-024-03432-9
Ma hema ische Zei sch i
Mul iple no malized solu ions o he plana
Sch ödinge –Poisson sys em wi h c i ical exponen ial g ow h
Si ong Chen1·Vicen¸ iu D. R˘adulescu2,3,4,5,6 ·Xianhua Tang1
Recei ed: 14 Ma ch 2023 / Accep ed: 16 Decembe 2023 / Published online: 16 Feb ua y 2024
© The Au ho (s) 2024
Abs ac
The pape deals wi h he exis ence o no malized solu ions o he ollowing Sch ödinge –
Poisson sys em wi h L2-cons ain :
−u+λu+μlog |·|∗u2u=eu2−1−u2u,x∈R2,
R2u2dx=c,
whe e μ>0, λ∈Rwill a ise as a Lag ange mul iplie and he nonlinea i y enjoys c i -
ical exponen ial g ow h o T udinge -Mose ype. By speci ying explici condi ions on he
ene gy le el c, we de ec a geome y o local minimum and a minimax s uc u e o he
co esponding ene gy unc ional, and p o e he exis ence o wo solu ions, one being a
local minimize and one o moun ain-pass ype. In pa icula , o ca ch a second solu ion
o moun ain-pass ype, some sha p es ima es o ene gy le els a e p oposed, sugges ing a
new h eshold o compac ness in he L2-cons ain . Ou s udy ex ends and complemen s he
esul s o Cingolani–Jeanjean (SIAM J Ma h Anal 51(4): 3533-3568, 2019) dealing wi h he
powe nonlinea i y a|u|p−2uin he case o a>0andp>4, which seems o be he i s
con ibu ion in he con ex o no malized solu ions. Ou model p esen s some new di icul-
ies due o he in ica e in e play be ween a loga i hmic con olu ion po en ial and a nonlinea
BVicen¸ iu D. R˘adulescu
[email p o ec ed]. o
Si ong Chen
[email p o ec ed]
Xianhua Tang
[email p o ec ed]
1School o Ma hema ics and S a is ics, HNP-LAMA, Cen al Sou h Uni e si y, Changsha 410083,
Hunan, People’s Republic o China
2Facul y o Applied Ma hema ics, AGH Uni e si y o Science and Technology, al. Mickiewicza 30,
30-059 K aków, Poland
3Facul y o Elec ical Enginee ing and Communica ion, B no Uni e si y o Technology, Technická
3058/10, B no 61600, Czech Republic
4Depa men o Ma hema ics, Uni e si y o C aio a, 200585 C aio a, Romania
5Simion S oilow Ins i u e o Ma hema ics o he Romanian Academy, 010702 Bucha es , Romania
6School o Ma hema ics, Zhejiang No mal Uni e si y, Jinhua 321004, Zhejiang, China
123
50 Page 2 o 32 S. Chen e al.
e m o c i ical exponen ial ype and equi es a no el analysis and he implemen a ion o new
ideas, especially in he compac ness a gumen . We belie e ha ou app oach will open he
doo o he s udy o o he L2-cons ained p oblems wi h c i ical exponen ial g ow h, and he
new unde lying ideas a e o u u e de elopmen and applicabili y.
Keywo ds Plana Sch ödinge –Poisson sys em ·Loga i hmic con olu ion po en ial ·
No malized solu ion ·C i ical exponen ial g ow h ·T udinge –Mose inequali y
Ma hema ics Subjec Classi ica ion 35J20 ·35J62 ·35Q55
1 In oduc ion
In his pape , we s udy he ollowing plana Sch ödinge –Poisson equa ion wi h L2-cons ain
−u+λu+μlog |·|∗u2u=eu2−1−u2u,x∈R2,
R2u2dx=c,
(1.1)
whe e μ>0, c>0 is a gi en cons an , λ∈Rappea s as a Lag ange pa ame e and is pa
o he unknowns. Pa icula ly, he nonlinea i y has c i ical exponen ial g ow h in he sense
o T udinge –Mose , which is a no el y o L2-cons ained p oblems. He e, we ecall ha
he nonlinea e m is said o ha e c i ical exponen ial g ow h i sa is ies
( 1) ∈C(R,R)and he e exis s α0>0 such ha
lim
| |→∞ | ( )|
eα 2=0, o all α>α
0,
+∞, o all α<α
0,
which is he maximal g ow h allowing o ea he p oblem a ia ionally in H1(R2),see
Adimu hi and Yada a [2] and also de Figuei edo, Miyagaki and Ru [22].
Solu ions ha ing ap io ip esc ibed L2-no m a e e e ed o as no malized solu ions in
he li e a u es. Physicis s a e o en in e es ed in no malized solu ions because he L2-no m o
such solu ions is a p ese ed quan i y o he e olu ion and hei a ia ional cha ac e iza ion
can help o analyze he o bi al s abili y o ins abili y, see, o example, [5,39,40]. Besides
ha i s solu ions ha e ap io ip esc ibed mass, ano he in e es ing ea u e o (1.1)is ha
a loga i hmic con olu ion po en ial appea s, which is unbounded and changes sign. As one
will see, o p esc ibed c>0, a solu ion o p oblem (1.1) can be ob ained as a c i ical poin
o he unc ional :X→Rde ined by
(u)=1
2R2|∇u|2dx+μ
4R2R2
log |x−y|u2(x)u2(y)dxdy
−1
2R2eu2−1−u2−u4
2dx(1.2)
on he cons ain
Sc=u∈X:u2
2=c,(1.3)
whe e
X:= u∈H1(R2):R2
log(1+|x|)u2dx<∞.(1.4)
123
Mul iple no malized solu ions... Page 3 o 32 50
This p oblem a ises when one looks o solu ions o he Sch ödinge -Poisson sys em o he
ype
−u+λu+μφu= (u), x∈RN,
φ =u2,x∈RN(1.5)
wi h N≥2, λ∈R {0}and ∈C(R,R), which has a s ong physical meaning because i
o igina es in quan um mechanics models (see e.g. [8,13,33]) and in semiconduc o heo y
[7,34,35]. The second equa ion (1.5) de e mines only up o ha monic unc ions, and i is
na u al o choose φas he nega i e New on po en ial o u2, i.e., he con olu ion o u2wi h
he undamen al solu ion No he Laplacian, which is gi en by
N(x)=1
2πlog |x|,N=2,
1
N(2−N)ωN|x|2−N,N≥3,
and ωNis he olume o he uni N-ball. Wi h his o mal in e sion, sys em (1.5)iscon e ed
in o an equi alen nonlocal equa ion
−u+λu+μ(N∗u2)u= (u), x∈RN.(1.6)
In he las decades, his equa ion has been ex ensi ely in es iga ed by using a ia ional
me hods. The majo i y o he li e a u e ocuses on he s udy o (1.6) wi h N=3, i seems
ha i is impossible o summa ize i o he case ha λ>0 is a ixed and assigned a pa ame e
since he ela ed li e a u e is oo la ge, we jus e e o [5,6,28,38] o he case ha λappea s
as a Lag ange pa ame e .
In con as wi h he highe -dimensional case N=3, much less is known o (1.6) wi h
N=2. In his case, he applicabili y o a ia ional me hods is no s aigh o wa d because
he co esponding ene gy unc ional is no well-de ined on H1(R2)unde he e ec o he
loga i hmic con olu ion po en ial. This di ec ion o esea ch was likely b ough o he a en-
ion o he communi y o nonlinea PDEs by he pape [21] published in 2016. In ha pape ,
Cingolani and We h, inspi ed by S ubbe [41], de eloped a a ia ional amewo k o deal wi h
(1.6) wi h N=2, wi hin he smalle Hilbe space Xde ined by (1.4),andp o ed heexis-
ence o g ound s a e solu ions when (u)=|u|q−2u o q>4. The key ool o p o e he
compac ness is a new sma s ong compac ness condi ion (modulo ansla ion) o Ce ami
sequences in he pe iodic se ing. This ool was subsequen ly used by Du-We h [23] o he
case ha (u)=|u|q−2u o 2 <q≤4, and by Chen-Shi-Tang [14] o he mo e gene al
case ha (u)∼|u|q−2u o q>2. In ecen pape s [17]and[18], we in oduced ano he
axially symme ic a ia ional amewo k wi hin a na u al cons ain
Eas := X∩u∈H1(R2):u(x):= u(x1,x2)=u(|x1|,|x2|), ∀x∈R2,(1.7)
and p o ed espec i ely he exis ence o axially symme ic solu ions o (1.6) wi h N=2
when (u)∼|u|q−2u o q>2andwhen has c i ical exponen ial g ow h sa is ying ( 1).
Compa ed wi h he abo e case whe e λ>0 is ixed, he sea ch o no malized solu ions
o (1.6) wi h N=2 is mo e challenging due o he ex a need o espec he L2-cons ain ,
which is ou ocus o he p esen pape . I seems ha he i s con ibu ion o his opic was
made ecen ly by Cingolani-Jeanjean [20], in which he exis ence o no malized solu ions
o he ollowing equa ion wi h he powe nonlinea i y
−u+λu+μlog |·|∗u2u=a|u|p−2u,x∈R2,
R2u2dx=c,(1.8)
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50 Page 4 o 32 S. Chen e al.
was es ablished, and a comple e analysis o he a ious cases on pa ame e s μ, a∈Rand
p>2 ha may happen o (1.8) was p o ided. In he s udy o (1.8), an impo an ole is
played by he so-called L2-c i ical exponen 4. I p>4o 2<p<4, one speaks o an L2-
supe c i ical case o an L2-subc i ical case. P ecisely, i was p o ed ha (1.8) has a g ound
s a e p o ided ha μ>0 and one o he ollowing h ee condi ions: i) a≤0andp>2; ii)
a>0andp<4; iii) a>0, p=4andc<2/(aC4), unde which he associa ed ene gy
unc ional is bounded om below on he cons ain Sc o any c>0 and a global minimum
on Sccan be achie ed, whe e he cons an C4>0 comes om he Gaglia do-Ni enbe g
inequali y (see (2.9) la e ). In all he o he cases, al hough i is no possible o ind a global
minimize , he in e play be ween a loga i hmic con olu ion po en ial and a powe unc ion
adds some ichness o he geome ic pic u e o he associa ed ene gy unc ional. In pa icula ,
when μ, a>0andp>4, i was p o ed ha he e exis s an explici alue c0=c0(μ, a,p)
such ha o c∈(0,c0),(1.8) has wo no malized solu ions, one being a local minimize
and one o moun ain-pass ype. This is eminiscen o he ecen wo k by Soa e [39], whe e
a simila s uc u e has been obse ed o he ollowing Sch ödinge equa ion wi h combined
nonlinea i ies o powe ype:
−u+λu=γ|u|q−2u+|u|p−2u,x∈RN,
RNu2dx=c,(1.9)
wi h N≥1, γ>0and2<q<2+4/N<p<2∗:= 2N/(N−2), N≥3,
+∞,N=1,2,see also
subsequen pape s [26,27,30,42] o ex ensions om p<2∗ o Sobole c i ical exponen
p=2∗. Howe e , he appea ance o he nonlocal con olu ion e m log |·|∗u2uin (1.8)
exhibi s some se ious ma hema ical di e ences o a local nonlinea e m o he o m |u|q−2u.
To add ess his ouble, Cingolani and Jeanjean used he combina ion o he ib a ion me hod
o Pohozae ( elying on he decomposi ion o L2-Pohozae mani old used in [39]) and he
s ong compac ness condi ion de eloped by Cingolani–We h [21], whe e some new es ima es
o ene gy on he dila ed unc ion su(s·) o u∈L2and s>0 belonging o Scwe e gi en.
I is wo h men ioning ha he a gumen s ongly depends on he o de po powe unc ion
and is no adequa e o he ollowing p oblem
−u+λu+μlog |·|∗u2u= (u), x∈R2,
R2u2dx=c,(1.10)
wi h he mo e gene al nonlinea e m , e en he sum o powe unc ions wi h supe -cubic
g ow h. In he ecen p ep in pape , Al es–Böe –Miyagaki [3] conside ed (1.10) wi h c i ical
exponen ial g ow h sa is ying ( 1) wi h α0=4π. In pa icula , i also sa is ies
( 2) (0)=0 and he e exis s τ>3such ha lim
→0| ( )|
| |τ=0;
( 3) he e exis s θ>6 such ha ( ) ≥θF( )>0,∀ = 0, whe e F( ):= 
0 (s)ds;
( 4) he e exis q>4andν>ν
0such ha F( )≥ν| |q,∀ ∈R,
i was p o ed ha o any c∈(0,1), he e a e implici pa ame e s μ0,ν
0>0 such ha
p oblem (1.10) has a solu ion o μ∈(0,μ
0)and ν>ν
0. No e ha his s a emen is o
pe u ba i e na u e in wo espec s: i) μ0is su icien ly small such ha (1.10) can be iewed
as a pe u ba i e o m o he plana Sch ödinge equa ion; ii) ν0is su icien ly la ge such
ha he ob ained moun ain-pass le el is small enough om which he compac ness can be
ob ained in he same way as ha o (1.8).
Clea ly, he pe u ba i e a gumen excludes many conc e e models, and ci cum en s he
added di icul ies a ising om he loga i hmic na u e o con olu ion ke nel and he c i ical
123
Mul iple no malized solu ions... Page 5 o 32 50
exponen ial g ow h o nonlinea i y compa ed o (1.8)and(1.10) wi h μ=0. To ou knowl-
edge, i s ill emains open exac ly how he in e play be ween he loga i hmic con olu ion
e m log |·|∗u2uand he nonlinea e m sa is ying ( 1) e ec s he geome y s uc-
u e o he co esponding unc ional, which is unbounded om below on Sc o all c>0
since lim|u|→∞ | (u)|
|u|3=+∞i ( 1) holds.
Mo i a ed by he s udy o (1.8)in heL2-c i ical case ha a|u|p−2uwi h a>0and
p>4, conside ed in [20], a na u al ques ion a ises:
(Q) Is i possible o ob ain an analogous s uc u e o local minima o plana Sch ödinge –
Poisson p oblems wi h c i ical exponen ial g ow h?
In he p esen pape , we will gi e an a i ma i e answe o abo e ques ion. Mo e p ecisely,
a e he sea ch o a s uc u e o local minima, di e en ly om he pe u ba i e a gumen
o [3], by speci ying explici condi ions on c, we s udy he exis ence o mul iple no malized
solu ions o (1.10) wi h c i ical exponen ial g ow h, and achie e a signi ican ex ension o
nonlinea i y om he powe ype o he c i ical exponen ial ype. To be e illus a e ou
app oach, we p o ide a conc e e nonlinea model (u)=eu2−1−u2u, which clea ly
sa is ies condi ion ( 1). This model is somehow inspi ed by Cassani–Ta a es–Zhang [12] o
he s udy o posi i e solu ions o he Bose–Eins ein ype sys ems in R2.
Compa ed o (1.8) wi h a>0andp>4 conside ed in [20], addi ional di icul ies a ise
in he s udy o (1.1) since he combina ion o he loga i hmic na u e o con olu ion ke nel
and he c i ical exponen ial g ow h o nonlinea i y mixes hings up.
Indeed, i s , a nonlinea e m o exponen ial ype beha es like in ini e se ies o powe s
nonlinea in e ac ions, he in e play be ween i and he nonlocal e m log |·|∗u2uis
mo e in ica e, which s ongly e ec s he geome y s uc u e o on Sc.E eni sucha
geome y may somehow be expec ed o su icien ly small alues o c>0 along he esea ch
lines o [20] conside ing (1.8) wi h a|u|p−2u(a>0andp>4), he a gumen s o [20]a e
insu icien o ind an explici exis ence ange o c o (1.1). I equi es us o de elop mo e
obus a gumen s in he sea ch o a geome y o local minima o on Sc. No e ha such
a s uc u e sugges s he possibili y o sea ch o ano he solu ion lying a a moun ain pass
le el, as well as a solu ion cha ac e izing as a local minima. I such a s uc u e exis s, hen
he nex mos complica ed pa lies in he compac ness analysis o minimizing sequences
and (PS) sequences, since i is no clea whe he
lim
n→∞R2|un|seαu2
n−1dx=R2|¯u|seα¯u2−1dx(1.11)
o s≥2i unuin X, despi e he compac ness o embedding X→Lq(R2) o all q≥2.
This ac p e en s us om using he compac ness a gumen o [20]. These di icul ies
en o ce he implemen a ion o new ideas since he app oach due o Cingolani–Jeanjean [20],
ea ing he powe case (u)=a|u|p−2u, is no a ailable o (1.1).
In pa icula , ins ead o wo king di ec ly in space X, we shall ake ad an age o he axially
symme ic a ia ional amewo k wi hin Eas de ined by (1.7), endowed wi h he no m gi en
by
uEas := (∇u2
2+u2
∗)1/2,whe e u2
∗=R2
log(2+|x|)u2(x)dx,(1.12)
and wo k wi h he cons ain
ˆ
Sc:= Eas ∩Sc=u∈Eas :u2
2=c.(1.13)
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50 Page 6 o 32 S. Chen e al.
As in ou pape [18], i uis a c i ical poin o  es ic ed o ˆ
Sc, henuis a c i ical poin o
on Sc. As one will obse e, besides helping o o e come he lack o compac ness caused
by he c i ical exponen ial g ow h, his ype o axially symme ic se ing is o ex emely
bene i o he p oo o he L2-con e gence o minimizing sequences and (PS) sequences,
which is a well-iden i ied obs acle dealing wi h he L2-cons ained p oblems due o he lack
o compac ness o he embedding H1
ad(R2)→L2(R2).
Ou main esul s ead as ollows.
Theo em 1.1 Fo any μ>0, he e exis s c1=c1(μ) > 0such ha , o any c ∈(0,c1),
(1.1)has a couple solu ion (uc,λ
c)∈Sc×Rsuch ha
uc∈ˆ
Sc,uc≥0,(uc)=m(c):= in (u):u∈ˆ
Sc,∇u2
2<π/3.(1.14)
Theo em 1.2 Fo any μ>0, he e exis s c0=c0(μ) > 0such ha , o any c ∈(0,c0),
(1.1)has a second couple solu ion (ˆuc,ˆ
λc)∈Sc×Rsuch ha
0<(ˆuc)<m(c)+2π. (1.15)
Rema k 1.3 The condi ion c∈(0,c1)in Theo em 1.1 en e s in he s udy o a geome y o
local minima o , while he condi ion c∈(0,c0)in Theo em 1.2, which appea s o be mo e
delica e, is used in o de o u he ensu e ha a minimax s uc u e o he moun ain-pass ype
exis s and he ob ained ene gy le el is less han m(c)+2π ha is a h eshold o compac ness,
which is an essen ial and s iking ing edien in ou compac ness a gumen .
De ine he L2-Pohozae unc ional P:X→Rby
P(u)=R2|∇u|2dx−μc2
4−R2u2−1eu2+1−u4
2dx.(1.16)
As one will see in Lemma 3.4, any solu ion o (1.1) sa is ies he L2-Pohozae iden i y
P(u)=0.
Le us now ske ch ou esea ch s a egies and poin ou key elemen s o he p oo s o
Theo ems 1.1 and 1.2.
Fi s , we sea ch o a geome y o local minima o on ˆ
Sc=Sc∩Eas unde explici
condi ions on c. Fo his, we in oduce a c ucial se Aπ/3={u∈Eas :∇u2
2<π/3}such
ha o any u∈ˆ
Sc∩∂Aπ/3,P(u)>0 and he e exis s u∈(0,1)such ha P( uu u)=0,
wi h his impo an p ope y and sub le es ima es o ene gy, o any μ>0, we succeed in
inding an explici alue c1=c1(μ) > 0 such ha o any c∈(0,c1),has a geome y o
local minima
m(c):= in
ˆ
Sc∩Aπ/3
< in
ˆ
Sc∩∂Aπ/3
. (1.17)
In ou a gumen , he uppe bound π/3 is no essen ial bu b ings a con enience in ob ain-
ing he explici exis ence ange c∈(0,c1)and p o ing he compac ness in he ollowing
discussion. In his ega d, ou s a egy is o ally di e en om ha o [20], since he bounda y
o he co esponding auxilia y se used in [20] depends on he mass u2
2=cas well as he
o de po powe unc ion in (1.8), ins ead o being gi en in ad ance like us.
Second, we p o e ha he local minima m(c)de ined by (1.17) is a ained, ha is, le ing
{un}⊂ ˆ
Sc∩Aπ/3be such ha (un)→m(c),we e i y ha un→uin Eas, p o ing
Theo em 1.1. The c ucial ing edien o he p oo is o ob ain he boundedness o {unX},
o mo e p ecisely, p o e ha un2
∗=R2log(2+|x|)u2
n(x)dx≤C o some C>0 due
o he ac ha {un}⊂ ˆ
Sc∩Aπ/3and he de ini ion o ·Xgi en by (1.12). No e ha a
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Mul iple no malized solu ions... Page 7 o 32 50
his s age, he sign o m(c)can no be judged unde he unpleasan e ec o a nonlinea e m
o exponen ial ype. This ac esul s in he ailu e o he me hod used in [20] elying on he
s ong compac ness condi ion. Indeed, ollowing he lines o [20], i is essen ial o e i y
ha un→u∈L2(R2) {0}poin wise a.e. on R2such ha he s ong compac ness condi ion
wo ks which leads o he boundedness o {un∗}up o ansla ions. Bu i seems impossible
o make i in ou case since he anishing o {un}can no be uled ou when he si ua ion o
m(c)=0 may occu . Somewha su p isingly, ou axially symme ic a ia ional amewo k
allows us o a oid he obs acle since he boundedness o {un2
∗} ollows di ec ly om he
speci ic inequali y ela ed wi h he coupling e m o equa ion es ic ed on Eas
R2R2
log (2+|x−y|)u2(x) 2(y)dxdy
≥1
4R2
u2(x)dxR2
log(2+|x|) 2(x)dx.(1.18)
This also explains why we wo k wi h Eas ∩Sca he beginning. The emaining p oo o
con e gence is s anda d, since (1.11) ollows di ec ly om T udinge -Mose inequali y (see
Lemma 2.3) due o he ac ha ∇un2
2≤π/3<2π o all n∈N.
Las bu no leas , we u he speci y an explici ange on c o gua an ee he exis ence
o ano he solu ion o moun ain-pass ype, p o ing Theo em 1.2, which is he hea o he
pape . Se e al c ucial s eps a e summa ized as ollows.
S ep 1. Cons uc a (PS) sequence {un}⊂ ˆ
Sco ˆ
Scpossessing addi ional p ope y
P(un)→0a a moun ain pass le el M(c).The condi ion ha P(un)→0helps o
deduce he boundedness o {∇un2}.
This s ep is eminiscen o he one de eloped by Jeanjean [25] bu he e he ac ha has
a s uc u e o local minimum ins ead o a di ec moun ain-pass geome y and he appea ance
o a loga i hmic con olu ion po en ial make he p oo mo e delica e.
To de ec a minimax s uc u e o ˆ
Sc, we use se e al c i ical poin heo ems on a mani old,
de eloped ecen ly by us in [16] conside ing p oblem (1.9). Ou app oach is applicable o
mo e gene al nonlinea i ies and o ally di e en om ha o [20] dealing wi h (u)=
a|u|p−2uin he case o a>0andp>4. No ing ha he si ua ion m(c)=0 can no be uled
ou in ad ance, i is om he speci ic inequali y (1.18) ha we deduce he boundedness o
{un2
∗}, and hus {unX}is bounded and he e exis s ¯u∈ˆ
Scsuch ha , up o a subsequence,
un¯uin Eas and un→¯uin Ls(R2) o s≥2. Howe e , i is insu icien o show ha ¯uis
asolu ion o(1.1) since i is unclea whe he
R2R2
log |x−y|u2
n(x)[un(y)−¯u(y)] (y)dxdy=0,∀ ∈C∞
0(R2). (1.19)
This equi es o u he p o e he s ong con e gence. Inspi ed by he B ezis-Ni enbe g
p oblem, he c ucial poin in p o ing he compac ness is o ob ain a good ene gy es ima e o
he ob ained (PS) sequence, which is wha o do nex .
S ep 2. Es ablish a p ecise uppe es ima e o he ene gy le el M(c),gi en by
M(c)<m(c)+2π, (1.20)
such ha he compac ness o he ob ained (PS) sequences s ill holds.
In he uncons ained case (1.5), his kind o sha p uppe es ima e is known, see ou
pape s [15,17], and he usual way o de i e such s ic inequali y is h ough he use o
es ing unc ions, ha is a sequence o Mose - ype unc ions in oduced by de Figuei edo,
Miyagaki and Ru [22], ela ed wi h he T udinge –Mose inequali y. Bu , he e seems no an
analogue in ou case due o he need o espec L2-cons ain and he loga i hmic na u e o
123
50 Page 8 o 32 S. Chen e al.
con olu ion ke nel. This s ep gi es i s ly a coun e pa in ha di ec ion, whose p oo
is a he complica ed, and equi es a lo o sub le ene gy es ima es as well as a be e
unde s anding o s uc u e o on ˆ
Sc, see Rema k 1.4 o u he desc ip ion.
S ep 3. P o e he limi (1.11)and hen un→¯uin Eas,up o a subsequence.
To ensu e ha he T udinge –Mose inequali y ii) o Lemma 2.3 wo ks in he p oo o
(1.11), one needs o con ol app op ia ely he alue o ∇un2
2 om abo e, which is why
one equi es a sha p uppe es ima e o M(c)be o e. Un o una ely, i seems impossible o
ob ain ∇un2
2<4π o la ge n.Ins ead,wep o e∇(un−¯u)2
2<4π o la ge nin a
ac ully ound-abou way, o hese a gumen s, wo main di icul ies a e o p o e P(¯u)≥0
and (¯u)≥m(c), see he p oo o (4.98), and hen show indi ec ly (1.11) wi h he Young’s
inequali y.
Rema k 1.4
i) To ob ain a cons ained (PS) sequence wi h addi ional p ope y, he app oach in [20],
ea ing (1.8) wi h (u)=a|u|p−2uin he case o a>0andp>4, no only
elied on he decomposi ion o he L2-Pohozae mani old in o h ee disjoin subse s,
bu used he Ghoussoub minimax p inciple [24], whe e he o me jus wo ks o an
easy calcula ing o m o nonlinea i y, and he la e equi es echnical opological, e y
complica ed, a gumen s based on σ-homo opy s able amily o compac subse s. This
app oach was also applied o p oblem (1.9) wi h mixed nonlinea i ies, see [27,29,30,39,
40,42], ne e heless, i is no a ailable in ou case due o he complex beha io s o he
e ms log |·|∗u2uand eu2−1−u2u. In con as , ou me hod does no equi e he
decomposi ion o he L2-Pohozae mani old, and ou ool o de ec he minimax s uc-
u es jus depends on he gene al de o ma ion lemma on a mani old, and is echnically
simple han opological a gumen s in ol ed in he Ghoussoub minimax p inciple [24].
ii) No e ha (1.20) gi es a new h eshold o compac ness o plana Sch ödinge –Poisson
sys ems wi h c i ical exponen ial g ow h in he L2-cons ain . To ob ain he s ic inequal-
i y (1.20), oughly speaking, we use a nice supe posi ion o a minima ob ained in
Theo em 1.1 and a modi ied sequence o Mose - ype unc ions wi h ine suppo s whe e
he suppo s would be disjoin , see Lemma 4.4 o mo e de ails. The idea behind he p oo
is ha he in e ac ion dec eases he in ol ed ene gy alue. E en i his idea is somehow
inspi ed by [42] conce ning he Sobole c i ical si ua ion in he highe dimensions, he
ma hema ical s a egies and p oo echniques a e di e en , o example, he a ia ional
cha ac e iza ions o a minima a e a ious in he use o es ing unc ions; ou ool o
ene gy es ima e is he nea ly combina ion o he Gaglia do-Ni enbe g inequali y and he
T udinge -Mose inequali y ins ead o he Sobole inequali y; ex a e o s a e always
equi ed o o e come he unpleasan e ec due o he loga i hmic na u e o con olu ion
ke nel.
iii) We belie e ha ou app oach may be adap ed o a ack mo e L2-cons ained p oblems
wi h c i ical exponen ial g ow h, and he new unde lying ideas and he s a egy o ene gy
es ima es a e o u u e de elopmen and applicabili y.
The pape is o ganized as ollows. Sec ion 2is de o ed o some p elimina ies. In pa icula ,
we p esen se e al c i ical poin heo ems on a mani old, we ha e de eloped ecen ly in [16],
which play a c ucial ole in he p oo s o heo ems. In Sec .3, we conside he exis ence o a
local minima o on Eas ∩Sc, and gi e he p oo o Theo em 1.1. In Sec .4, we s udy he
exis ence o a c i ical poin o moun ain-pass ype o on Eas ∩Sc, and inish he p oo o
Theo em 1.2.
Th oughou he pape , we make use o he ollowing no a ions:
123
Mul iple no malized solu ions... Page 9 o 32 50
•H1(R2)deno es he usual Sobole space equipped wi h he inne p oduc and no m
(u, )=R2
(∇u·∇ +u )dx,u=(u,u)1/2,∀u, ∈H1(R2);
•H1
ad(R2)deno es he space o sphe ically symme ic unc ions belonging o H1(R2):
H1
ad(R2):= {u∈H1(R2)u(x)=u(|x|)a.e. in R2};
•Ls(R2)(1≤s<∞)deno es he Lebesgue space wi h he no m us=R2|u|sdx1/s;
•Fo any u∈H1(R2) {0},u (x):= u( x) o >0;
•Fo any x∈R2and >0, B (x):= {y∈R2:|y−x|< }and B =B (0);
•C1,C2,··· deno e posi i e cons an s possibly di e en in di e en places, which a e
dependen on c>0.
2 P elimina y esul s
As in [17], we de ine he ollowing symme ic bilinea o ms
(u, )→ A1(u, ):= R2R2
log (2+|x−y|)u(x) (y)dxdy,(2.1)
(u, )→ A2(u, ):= R2R2
log 1+2
|x−y|u(x) (y)dxdy,(2.2)
(u, )→ A0(u, ):= A1(u, )−A2(u, )=R2R2
log |x−y|u(x) (y)dxdy,(2.3)
whe e he de ini ion is es ic ed, in each case, o measu able unc ions u, :R2→R
such ha he co esponding double in eg al is well de ined in Lebesgue sense. No ing ha
0≤log(1+ )≤ o ≥0, i ollows om he Ha dy–Li lewood–Sobole inequali y(see
[31]o [32, p. 98]) ha
|A2(u, )|≤2R2R2
1
|x−y||u(x) (y)|dxdy≤C0u4/3 4/3(2.4)
wi h a cons an C0>0. Using (2.1), (2.2)and(2.3), we de ine he ollowing ene gy unc-
ionals:
I1:H1(R2)→[0,∞],
I1(u):= A1(u2,u2)=R2R2
log (2+|x−y|)u2(x)u2(y)dxdy,
I2:L8/3(R2)→[0,∞),
I2(u):= A2(u2,u2)=R2R2
log 1+2
|x−y|u2(x)u2(y)dxdy,
I0:H1(R2)→R∪{∞},
I0(u):= A0(u2,u2)=R2R2
log |x−y|u2(x)u2(y)dxdy.
He e I2only akes ini e alues on L8/3(R2). Indeed, (2.4) implies
|I2(u)|≤C0u4
8/3,∀u∈L8/3(R2). (2.5)
123
50 Page 16 o 32 S. Chen e al.
I ollows om (3.20) ha ∇ n2
2=o(1)and I1( n)=o(1),andso( n)=o(1).I
ollows om (3.19) ha A1(¯u2, 2
n)=o(1), and so by Lemma 2.4, n→0inEas, i.e.
un→¯uin Eas.Sinceun≥0, i ollows ha ¯u≥0.
S ep 4. Ob iously u∈¯
Aπ/3and (¯u)=m(c). Nex , we show ha ∇¯u2
2<π
3.Le
us assume by con adic ion ha ∇¯u2
2=π
3. Then we see di ec ly om Co olla y 3.2 ha
necessa ily P(¯u)>0. Bu hen we conside 0wi h 0<1 close o 1. Reco ding (3.9), i
ollows ha 0¯u 0∈Aπ/3and ( 0¯u 0)<(¯u)=m(c), p o iding a con adic ion. Hence,
Co olla y 2.11 implies ha |ˆ
Sc
(¯u)=0, and so he e exis s a Lag ange mul iplie λc∈R
such ha (¯u)+λc¯u,φ=0 o anyφ∈Eas. By Lemma 2.6,weha e(¯u)+λc¯u,φ=0
o any φ∈X, ha is
−¯u+μlog |x|∗¯u2¯u−e¯u2−1−¯u2¯u=−λc¯u,x∈R2.(3.21)
This comple es he p oo . 
4 P oo o Theo em 1.2
In his sec ion, we conside he exis ence o a c i ical poin o moun ain-pass ype o on
ˆ
Sc=Eas ∩Sc, and gi e he p oo o Theo em 1.2.
Lemma 4.1 Le μ>0and c ∈(0,c2). Fo any u ∈ˆ
Sc, he ollowing exis :
(i) A unique s+
u>0such ha s+
uis a s ic local minimum poin o gu.
(ii) A unique s−
u>0such ha s−
uis a s ic local maximum poin o gu.
P oo Fo any u∈ˆ
Sc,le τ:= 1/∇u2and ˆu:= τuτ.Then∇ˆu2
2=1and ˆu =( τ)u τ
o >0. The e o e, we only p o e his lemma o u∈ˆ
Scwi h ∇u2
2=1.
Fix u∈ˆ
Scwi h ∇u2
2=1, we ha e
g
u( )=1
P( u ), ∀ >0.(4.1)
Le ∗>0 such ha
−4
∗R21− 2
∗u2+ 4
∗u4e 2
∗u2−1− 4
∗u4
2dx=1.(4.2)
I ollows ha
2> −2R21− 2u2+ 4u4e 2u2−1− 4u4
2dx,0< < ∗(4.3)
and
2< −2R21− 2u2+ 4u4e 2u2−1− 4u4
2dx, ∗< <+∞.(4.4)
By (3.9)and(4.3), one has
g
u( )=1
 2−μc2
4− −2R2 2u2−1e 2u2+1− 4u4
2dx
≥1
 2−μc2
4− −2
2R21− 2u2+ 4u4e 2u2−1− 4u4
2dx
123

Mul iple no malized solu ions... Page 17 o 32 50
>1
2  2−μc2
2,0< < ∗.(4.5)
I ∗<π, hen om (2.14)and(4.2), we ha e
1= −4
∗R21− 2
∗u2+ 4
∗u4e 2
∗u2−1− 4
∗u4
2dx
=∞

k=3
(k−1)2
k!u2k
2k 2(k−2)
∗
≤2c
π
∞

k=3
(k−1)24k−1(k−2)+1
(k−2)k! 2
∗√c
πk−2
+1
2π
∞

k=3
(k−1)2 2
∗
πk−2
=2c
π
∞

k=3
(k−1)24k−1(k−2)+1
(k−2)k! 2
∗√c
πk−2
+ 2
∗4π2−3π 2
∗+ 4
∗
2ππ− 2
∗3.(4.6)
Combining (3.3) wi h (4.6), we deduce ∗≥η(c). I ollows om (3.3) ha η(c)is dec easing
on c>0. Hence, by (3.2), we ha e
μc2
2<μc2
2
2=η2(c2)<η
2(c)≤ 2
∗,∀c∈(0,c2). (4.7)
Hence, (4.7) shows ha he e exis s δ>0 such ha 2−μc2
2>0 o any ∈( ∗−δ, ∗).
Hence, by (4.5), we in e ha g
u( )>0 o any ∈( ∗−δ, ∗), and hus gu( )is inc easing
in ( ∗−δ, ∗).
Taking in o accoun ha he unc ion gu( )→+∞as →0+and gu( )→−∞as
→+∞, we conclude ha he e exis s a leas a c i ical poin s+
u< ∗which is a local
minimum poin o guand a c i ical poin s−
u> ∗which is a local maximum poin o gu.
Since s−
u> ∗, om (4.4)wede i e ha
(s−
u)2<(s−
u)−2R21−(s−
u)2u2+(s−
u)4u4e(s−
u)2u2−1−(s−
u)4u4
2dx
=∞

k=3
(k−1)2
k!u2k
2k(s−
u)2(k−1).(4.8)
Mo eo e , om (3.9), (4.8) and he ac ha g
u(s−
u)=0, we de i e ha
g
u(s−
u)=1
(s−
u)2(s−
u)2−∞

k=3
(k−1)(2k−3)
k!u2k
2k(s−
u)2(k−1)+μc2
4
=2
(s−
u)2(s−
u)2−∞

k=3
(k−1)2
k!u2k
2k(s−
u)2(k−1)<0.(4.9)
The e o e s−
uis a s ic maximum poin o gu.
We ha e o show ha s−
uis unique. By con adic ion we assume ha he e exis s ˆs−
u>0,
ano he c i ical poin o guwhich is a local maximum poin .
Fi s , we obse e ha i 0 <ˆs−
u< ∗, hen om g
u(ˆs−
u)=0and(4.3) we ob ain
g
u(ˆs−
u)=1
(ˆs−
u)2(ˆs−
u)2−∞

k=3
(k−1)(2k−3)
k!u2k
2k(ˆs−
u)2(k−1)+μc2
4
123
50 Page 18 o 32 S. Chen e al.
=2
(ˆs−
u)2(ˆs−
u)2−∞

k=3
(k−1)2
k!u2k
2k(ˆs−
u)2(k−1)>0,(4.10)
which is a con adic ion. This implies ha ˆs−
u> ∗, and hus a guing as be o e we ha e
g
u(ˆs−
u)<0. We de i e he exis ence o ano he c i ical poin : θu∈(ˆs−
u,s−
u)o θu∈(s−
u,ˆs−
u),
which is a local minimum o gu. Taking in o accoun (4.4), we again deduce g
u(θu)<0,
which is a con adic ion. The e o e he poin s−
uis unique.
Now a di ec adap a ion o he a gumen used o s−
uleads us o conclude ha s+
uis he
unique local minimum poin o gu.
Lemma 4.2 Le μ>0. Fo any c ∈(0,c1), he e exis s κc>0such ha
M(c):= in
γ∈c
max
∈[0,1](γ ( )) ≥κc>sup
γ∈c
max {(γ (0)), (γ (1))},(4.11)
whe e
c=γ∈C([0,1],ˆ
Sc):γ(0)=uc,(γ(1)) < m(c)−1,(4.12)
and ucis de e mined by Theo em 1.1.
P oo Se κc:= in u∈∂( ˆ
Sc∩Aπ/3)(u). By Theo em 1.1 and Co olla y 3.2,κc>m(c)=
(uc).Le γ∈cbe a bi a y. Since γ(0)=uc,and(γ (1)) < m(c)−1, necessa ily
in iew o Theo em 1.1,γ(1)/∈ˆ
Sc∩Aπ/3. By con inui y o γ( )on [0,1], he e exis s a
0∈(0,1)such ha γ( 0)∈∂( ˆ
Sc∩Aπ/3),andsomax
∈[0,1](γ ( )) ≥κc.Thus,(4.11)
holds. 
To apply Lemma 2.12,wele E=Eas and H=L2(R2). De ine he no ms o Eand H
by
uE:= ∇u2+u2
∗1/2,u2
H:= 1
√cR2
u2dx1/2
,∀u∈E.(4.13)
By Lemma 2.6, a e iden i ying Hwi h i s dual, we ha e E→H→E∗wi h con inuous
injec ions. Se
M:= u∈E:u2
2=R2
u2dx=c.(4.14)
Ob iously, Lemma 2.3 shows ha ∈C1(E,R),and
(u), u=R2|∇u|2dx+μI0(u)−R2eu2−1−u2u2dx.(4.15)
Se F(u):= 1
2eu2−1−u2−u4
2and (u):= eu2−1−u2u. Inspi ed by [25], le us
de ine a con inuous map β:Eas ×R→Eas by
β( , )(x):= e (e x) o ∈Eas, ∈R,x∈R2,(4.16)
and conside he ollowing auxilia y unc ional:
˜
( , ):= (β( , )) =e2
2∇ 2
2+μ
4I0( ) −μc2
4−1
e2 R2
F(e )dx.(4.17)
We see ha ˜
is o class C1,and o any(w, s)∈Eas ×R,
˜
( , ), (w, s)=˜
( , ), (w, 0)+˜
( , ), (0,s)
123
Mul iple no malized solu ions... Page 19 o 32 50
=e2 R2∇ ·∇wdx+μR2R2
log |x−y| 2(x) (y)w(y)dxdy
−1
e2 R2
(e )e wdx+e2 s∇ 2
2−μc2s
4
+s
e2 R22F(e ) − (e )e dx
=(β( , )), β(w, ) +sP(β( , )). (4.18)
Le
u(x):= β( , )(x)=e (e x), φ(x):= β(w, )(x)=e w(e x). (4.19)
Then
(u,φ)H=1
cR2
u(x)φ(x)dx=1
cR2
(x)w(x)dx=( , w)H.(4.20)
This shows ha
φ∈Tu(ˆ
Sc)⇔(w, s)∈˜
T( , )(ˆ
Sc×R), ∀ ,s∈R.(4.21)
I is easy o e i y ha
log 2+e−| | ≥e−| |log(2+ ), ∀ >0, ∈R.(4.22)
I ollows om (1.12), (4.18), (4.19), (4.21)and(4.22) ha
|P(u)|=˜
( , ), (0,1)≤

˜
|ˆ
Sc×R( , )

(4.23)
and


|ˆ
Sc(u)

=sup
φ∈Tu(ˆ
Sc)
1
φE(u), φ 
=sup
φ∈Tu(ˆ
Sc)
1
∇φ2
2+φ2
∗(β( , )), β(w, ) 
=sup
φ∈Tu(ˆ
Sc)
1
∇φ2
2+φ2
∗˜
( , ), (w, 0)
≤sup
(w,0)∈˜
T( , )(ˆ
Sc×R)
e| |
(w, 0)E×R˜
( , ), (w, 0)
≤e| |

˜
|ˆ
Sc×R( , )

.(4.24)
Lemma 4.3 Le μ>0. Then o any c ∈(0,c1), he e exis s a sequence {un}⊂ ˆ
Scsuch ha
(un)→M(c)>m(c), |ˆ
Sc(un)→0and P(un)→0.(4.25)
P oo Se
˜
c:= ˜γ∈C([0,1],ˆ
Sc×R):˜γ(0)=(uc,0), ˜
( ˜γ(1)) < m(c)−1(4.26)
and
˜
M(c):= in
˜γ∈˜
c
max
∈[0,1]˜
( ˜γ( )). (4.27)
123
50 Page 20 o 32 S. Chen e al.
Fo any ˜γ∈˜
c, i is easy o see ha γ=β◦˜γ∈cde ined by (4.12). Le κ
c:=
supγ∈cmax {(γ (0)), (γ (1))}. Then i ollows om (4.11) ha
max
∈[0,1]˜
( ˜γ( )) =max
∈[0,1](γ ( )) ≥κc>κ

c≥max {(γ (0)), (γ (1))}
=max ˜
( ˜γ(0)), ˜
( ˜γ(1)).
I ollows ha ˜
M(c)≥M(c),and
˜
M(c)=in
˜γ∈˜
c
max
∈[0,1]˜
( ˜γ( )) ≥κc>κ

c≥sup
˜γ∈˜
c
max ˜
( ˜γ(0)), ( ˜γ(1)).(4.28)
This shows ha (2.29) holds.
On he o he hand, o any γ∈c,le ˜γ( ):= (γ ( ), 0). I is easy o e i y ha ˜γ∈˜
c
and (γ ( )) =˜
( ˜γ( )), and so, we i ially ha e ˜
M(c)≤M(c). Thus ˜
M(c)=M(c).
Fo any n∈N,(4.11) implies ha he e exis s γn∈csuch ha
max
∈[0,1](γn( )) ≤M(c)+1
n.(4.29)
Se ˜γn( ):= (γn( ), 0). Then apply Lemma 2.12 o ˜
, he e exis s a sequence {( n, n)}⊂
ˆ
Sc×Rsa is ying
(i) M(c)−2
n≤˜
( n, n)≤M(c)+2
n;
(ii) min ∈[0,1]( n, n)−(γn( ), 0)E×R≤2
√n;
(iii) 

˜
|ˆ
Sc×R( n, n)

≤8
√n.
Le un=β( n, n). I ollows om (4.23), (4.24) and (i)-(iii) ha (4.25) holds. 
Now we de ine he ollowing Mose ype unc ions wn(x)suppo ed in B1(0)
wn(x)=1
√2π⎧
⎪
⎨
⎪
⎩
√log n,0≤|x|≤1/n;
log(1/|x|)
√log n,1/n≤|x|≤1;
0,|x|≥1.
(4.30)
Compu ing di ec ly, we ge ha
∇wn2
2=R2|∇wn|2dx=1,(4.31)
wn2
2=R2|wn|2dx=log n1/n
0
d +1
1/n
log2(1/ )
log n d
=1
4logn−1
4n2log n−1
2n2,(4.32)
wn8/3
8/3=R2|wn|8/3dx=O1
log4/3n,n→∞,(4.33)
wn2
∗=R2
log(2+|x|)|wn|2dx=O1
log n,n→∞ (4.34)
and
|I0(wn)|=R2R2
log |x−y|w2
n(x)w2
n(y)dxdy≤O1
log2n,n→∞.(4.35)
123
Mul iple no malized solu ions... Page 21 o 32 50
Lemma 4.4 Le μ>0. Then o any c ∈(0,c1), he e holds
M(c)<m(c)+2π. (4.36)
P oo Le ucbe de e mined by Theo em 1.1. By Theo em 1.1 and Lemma 3.4,weha e
uc2
2=c,(uc)=m(c), uc(x)≥0,∀x∈R2(4.37)
and
−λcc=μc2
4+μI0(uc)−R2eu2
c−1−u2
c−u4
c
2dx.(4.38)
Since uc∈Eas, i ollows om (2.4), (2.7), (4.30), (4.32), (4.33)and(4.34) ha
R2R2
log |x−y|uc(x)wn(x)uc(y)wn(y)dxdy=O1
log n,n→∞,(4.39)
R2R2
log |x−y|u2
c(x)w2
n(y)dxdy=O1
log n,n→∞,(4.40)
R2R2
log |x−y|uc(x)wn(x)w2
n(y)dxdy=O1
log3/2n,n→∞,(4.41)
R2
ucwndx=O1
√log n,n→∞ (4.42)
and
R2eu2
c−1−u2
cucwndx=O1
√log n,n→∞.(4.43)
By (1.1), (4.32)and(4.37), one has
R2∇uc·∇wndx=R2−μR2
log |x−y|u2
c(y)dy+eu2
c−1−u2
c−λcucwndx
(4.44)
and
uc+ wn2
2=c+ 2wn2
2+2 R2
ucwndx
=c+2 R2
ucwndx+ 2O1
log n,n→∞.(4.45)
Le τ:= uc+ wn2/√c.Then
τ2=1+2
cR2
ucwndx+ 2O1
log n,n→∞ (4.46)
and o any p≥1,
τ−2p=1−2p
cR2
ucwndx+ 2O1
log n,n→∞.(4.47)
Now, we de ine
Wn, (x):= uc(τ x)+ wn(τ x). (4.48)
Then one has
∇Wn, 2
2=∇(uc+ wn)2
2,Wn, 2
2=τ−2uc+ wn2
2=c,(4.49)
123

50 Page 22 o 32 S. Chen e al.
I0(Wn, )=R2R2
log |x−y|[uc(τ x)+ wn(τ x)]2[uc(τ y)+ wn(τ y)]2dxdy
=1
τ4R2R2
log |x−y|[uc(x)+ wn(x)]2[uc(y)+ wn(y)]2dxdy−c2log τ
(4.50)
and
R2%eW2
n, −1−W2
n, −W4
n,
2&dx
=1
τ2R2e(uc+ wn)2−1−(uc+ wn)2−(uc+ wn)4
2dx.(4.51)
F om (4.42)and(4.46), one has
τ2=1+2
cR2
ucwndx+ 2O1
log n≤1+ + 2, o la ge n∈N.(4.52)
Now, we de ine n( )by
n( )= 2
2−1
2τ2R2e 2w2
n−1− 2w2
n−1
2 4w4
ndx,∀ >0.(4.53)
We claim ha
sup
>0n( )+ 2O1
log n+ 4O1
log2n
≤2π−π
2lognlog log n
32π, o la ge n∈N.(4.54)
The e a e h ee cases o dis inguish. In he sequel, we ag ee ha all inequali ies hold o la ge
n∈Nwi hou men ioning.
Case i) ∈'0,√2π(.Thenby(4.35)and(4.53), we ha e
n( )= 2
2−1
2τ2R2e 2w2
n−1− 2w2
n−1
2 4w4
ndx≤ 2
2≤3π
2.(4.55)
I ollows ha
sup
0< ≤√2πn( )+ 2O1
log n+ 4O1
log2n
≤2π−π
2lognlog log n
32π, o la ge n∈N.(4.56)
Case ii) ∈'√2π,√6π. Then i ollows om (4.30), (4.35)and(4.52) ha
1
τ2R2e 2w2
n−1− 2w2
n−1
2 4w4
ndx
≥1
2τ2B1/n
e 2w2
ndx≥1
16n2e(2π)−1 2log n.(4.57)
123
Mul iple no malized solu ions... Page 23 o 32 50
Using (4.53)and(4.57), we a e led o
n( )= 2
2−1
2τ2R2e 2w2
n−1− 2w2
n−1
2 4w4
ndx
≤ 2
2−1
32n2e(2π)−1 2log n:= ϕn( ). (4.58)
Choosing n>0 be such ha ϕ
n( n)=0, hen we ha e
1=log n
32πn2e(2π)−1 2
nlog n.(4.59)
I ollows ha
2
n=4π1+log(32π)−log(log n)
2logn(4.60)
and
ϕn( )≤ϕn( n)= 2
n
2−π
log n,∀ ≥0.(4.61)
Using (4.60)and(4.61), we a e led o
ϕn( )≤ 2
n
2−π
log n=2π−π
log nlog elog n
32π,
which, oge he wi h (4.58), yields
n( )≤2π−π
log nlog elog n
32π.
I ollows ha
sup
√2π< ≤√6πn( )+ 2O1
log n+ 4O1
log2n
≤2π−π
2lognlog log n
32π, o la ge n∈N.(4.62)
Case iii) ∈√6π,+∞. Then i ollows om (4.30),(4.35)and(4.52) ha
n( )+ 2O1
log n+ 4O1
log2n
≤ 2
2−1
2τ2R2e 2ω2
n−1− 2ω2
n−1
2 4ω4
ndx
+ 2O1
log n+ 4O1
log2n
≤ 2
2−π
4n2τ2e(2π)−1 2log n+ 2O1
log n+ 4O1
log2n
≤ 2
2−π
4n2(1+ + 2)e(2π)−1 2log n+ 2O1
log n+ 4O1
log2n
:= 2
2−π
4n2(1+ + 2)e(2π)−1 2log n+an 2+bn 4(4.63)
123
50 Page 24 o 32 S. Chen e al.
≤3π−π
2n2(1+√6π+6π)e3logn+6πan+36π2bn≤3
2π, (4.64)
whe e we ha e used he ac ha he unc ion
φn( ):= 2
2−π
4n2(1+ + 2)e(2π)−1 2log n+an 2+bn 4
is dec easing on ∈√6π,+∞ o la ge n. In ac ,
φ
n( )=(1+2an) +4bn 3−1+ + 2 log n−(1+2 )π
4n21+ + 22e(2π)−1 2log n.
Assume ha sn>0 such ha φ
n(sn)=0 o la gen.Then
4(1+2an)sn+4bns3
n1+sn+s2
n2=1+sn+s2
nsnlog n−(1+2sn)π
n2e(2π)−1s2
nlog n,
which yields
s2
n=4π⎧
⎨
⎩
1+
log '4(1+2an)sn+4bns3
n1+sn+s2
n2(
2logn
−log 1+sn+s2
nsnlog n−(1+2sn)π
2logn).(4.65)
This implies ha limn→∞ s2
n=4π.Soφn( )is dec easing on ∈√6π,+∞ o la ge n.
F om (4.64), one has
sup
√6π≤ <+∞n( )+ 2O1
log n+ 4O1
log2n
≤2π−π
2lognlog log n
32π, o la ge n∈N.(4.66)
Cases i)–iii) show ha (4.54) holds. I is easy o e i y he ollowing inequali y:
(1+ )q≥1+q q−1+ q,∀ ≥0,q≥2.(4.67)
By (4.35), (4.39)-(4.41), we ha e
I0(uc+ wn)=R2R2
log |x−y|[uc(x)+ wn(x)]2[uc(y)+ wn(y)]2dxdy
=I0(uc)+ 4I0(wn)+4 R2R2
log |x−y|u2
c(x)uc(y)wn(y)dxdy
+4 2R2R2
log |x−y|uc(x)wn(x)uc(y)wn(y)dxdy
+2 2R2R2
log |x−y|u2
c(x)w2
n(y)dxdy
+4 3R2R2
log |x−y|uc(x)wn(x)w2
n(y)dxdy
=I0(uc)+4 R2R2
log |x−y|u2
c(x)uc(y)wn(y)dxdy
123
Mul iple no malized solu ions... Page 25 o 32 50
+ 2O1
log n+ 3O1
log3/2n+ 4O1
log2n.(4.68)
F om (1.2), (4.31), (4.37)–(4.44), (4.46)–(4.53), (4.54),(4.67)and(4.68), we ha e
(Wn, )
=1
2∇Wn, 2
2+μ
4I0(Wn, )−1
2R2%eW2
n, −1−W2
n, −W4
n,
2&dx
=1
2∇(uc+ wn)2
2+μ
4τ4I0(uc+ wn)−μc2
4log τ
−1
2τ2R2e(uc+ wn)2−1−(uc+ wn)2−(uc+ wn)4
2dx
≤1
2∇uc2
2+μτ−4
4I0(uc)−μc2
4log τ−1
2τ2R2eu2
c−1−u2
c−u4
c
2dx
+ 2
2∇wn2
2−1
2τ2R2e 2w2
n−1− 2w2
n− 4w4
n
2dx+ R2∇uc·∇wndx
+μτ−4 R2R2
log |x−y|u2
c(x)uc(y)wn(y)dxdy−τ−2 R2eu2
c−1−u2
cucwndx
+ 2O1
log n+ 3O1
log3/2n+ 4O1
log2n
=(uc)+n( )−μ1−τ−4
4I0(uc)−μc2
4log τ
+1−τ−2
2R2eu2
c−1−u2
c−u4
c
2dx
−μ1−τ−4 R2R2
log |x−y|u2
c(x)uc(y)wn(y)dxdy
+1−τ−2 R2eu2
c−1−u2
cucwndx−λc R2
ucwndx
+ 2O1
log n+ 3O1
log3/2n+ 4O1
log2n
≤m(c)+n( )−λc R2
ucwndx−μc2
4
cR2
ucwndx+ 2O1
log n
−μI0(uc)
cR2
ucwndx+ 2O1
log n
+
cR2
ucwndx+ 2O1
log nR2eu2
c−1−u2
c−u4
c
2dx
−μ4
cR2
ucwndx+ 2O1
log n R2R2
log |x−y|u2
c(x)uc(y)wn(y)dxdy
+2
cR2
ucwndx+ 2O1
log n R2eu2
c−1−u2
cucwndx
+ 2O1
log n+ 3O1
log3/2n+ 4O1
log2n
123
50 Page 32 o 32 S. Chen e al.
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Publishe ’s No e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps and
ins i u ional a ilia ions.
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