Recei ed: 8 June 2023 Re ised: 3 Oc obe 2023 Accep ed: 27 Oc obe 2023
DOI: 10.1112/blms.12961
Bulle in o he London
Ma hema ical Socie y
RESEARCH ARTICLE
Non-au onomous double phase eigen alue
p oblems wi h inde ini e weigh and lack o
compac ness
Tianxiang Gou1Vicenţiu D. Rădulescu2,3,4,5,6
1School o Ma hema ics and S a is ics,
Xi’an Jiao ong Uni e si y, Xi’an, Shaanxi,
China
2Facul y o Applied Ma hema ics, AGH
Uni e si y o Science and Technology,
K akow, Poland
3Facul y o Elec ical Enginee ing and
Communica ion, B no Uni e si y o
Technology, B no, Czech Republic
4Depa men o Ma hema ics, Uni e si y
o C aio a, C aio a, Romania
5Simion S oilow Ins i u e o Ma hema ics
o he Romanian Academy, Bucha es ,
Romania
6School o Ma hema ics, Zhejiang No mal
Uni e si y, Jinhua, China
Co espondence
Vicenţiu D. Rădulescu, Facul y o Applied
Ma hema ics, AGH Uni e si y o Science
and Technology, al. Mickiewicza 30,
30-059 K akow, Poland.
Email: icen iu. adulescu@ima . o
Funding in o ma ion
Na ional Na u al Science Founda ion o
China, G an /Awa d Numbe : 12101483;
China Pos doc o al Science Founda ion,
G an /Awa d Numbe : 2021M702620;
Romanian Minis y o Resea ch,
Inno a ion and Digi iza ion, G an /Awa d
Numbe : PNRR-III-C9-2022-I8/22
Abs ac
In his pape , we conside eigen alues o he ollow-
ing double phase p oblem wi h unbalanced g ow h and
inde ini e weigh ,
−Δ𝑎
𝑝𝑢−Δ
𝑞𝑢=𝜆𝑚(𝑥)|𝑢|𝑞−2𝑢in ℝ𝑁,
whe e 𝑁⩾2,1<𝑝,𝑞<𝑁,𝑝≠𝑞,𝑎∈𝐶
0,1(ℝ𝑁,
[0, +∞)),𝑎≢0and 𝑚∶ℝ𝑁→ℝis an inde ini e
sign weigh which may admi non- i ial posi i e and
nega i e pa s. He e, Δ𝑞is he 𝑞-Laplacian ope a o
and Δ𝑎
𝑝is he weigh ed 𝑝-Laplace ope a o de ined
by Δ𝑎
𝑝𝑢∶=di (𝑎(𝑥)|∇𝑢|𝑝−2∇𝑢).Thep oblemcanbe
degene a e, in he sense ha he in imum o 𝑎in ℝ𝑁
may be ze o. Ou main esul s dis inguish be ween he
cases 𝑝<𝑞and 𝑞<𝑝. In he i s case, we es ablish he
exis ence o a con inuous amily o eigen alues, s a ing
om he p incipal equency o a sui able single phase
eigen alue p oblem. In he la e case, we p o e he
exis ence o a disc e e amily o posi i e eigen alues,
which di e ges o in ini y.
MSC 2020
35P30 (p ima y), 35J70, 46E30, 47J10, 58C40, 58E05 (seconda y)
© 2023 The Au ho s. Bulle in o he London Ma hema ical Socie y is copy igh © London Ma hema ical Socie y. This is an open access a icle
unde he e ms o he C ea i e Commons A ibu ion License, which pe mi s use, dis ibu ion and ep oduc ion in any medium, p o ided
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734 wileyonlinelib a y.com/jou nal/blms Bull. London Ma h. Soc. 2024;56:734–755.
NON-AUTONOMOUS DOUBLE PHASE EIGENVALUE PROBLEMS 735
1 INTRODUCTION
In his pape , we in es iga e eigen alues o he ollowing double phase p oblem wi h unbalanced
g ow h and inde ini e weigh ,
−Δ𝑎
𝑝𝑢−Δ
𝑞𝑢=𝜆𝑚(𝑥)|𝑢|𝑞−2𝑢in ℝ𝑁,(1.1)
whe e 𝑁⩾2,1<𝑝,𝑞<𝑁,𝑝≠𝑞,𝑎∈𝐶
0,1(ℝ𝑁,[0,+∞)),𝑎≢0and 𝑚∶ℝ𝑁→ℝis an inde ini e
sign weigh which may admi non- i ial posi i e and nega i e pa s. He e, Δ𝑞is he 𝑞-Laplacian
ope a o and Δ𝑎
𝑝is he weigh ed 𝑝-Laplace ope a o de ined by Δ𝑎
𝑝𝑢∶=di (𝑎(𝑥)|∇𝑢|𝑝−2∇𝑢).
Th oughou o his pape , we shall always assume ha he weigh unc ion 𝑚∶ℝ𝑁→ℝsa is ies
he ollowing assump ion,
(𝐻) 𝑚 = 𝑚1−𝑚
2, whe e 𝑚1,𝑚
2⩾0,𝑚1≢0,𝑚1∈𝐿
𝑁
𝑞(ℝ𝑁)∩𝐿
∞(ℝ𝑁)and 𝑚2∈𝐿
∞(ℝ𝑁).
Rema k 1.1. In ou case, 𝑚2=0is allowable.
P oblems like (1.1) a ise when one looks o he s a iona y solu ions o eac ion–di usion
sys ems o he o m
𝑢𝑡=di [𝐷(𝑥, ∇𝑢)∇𝑢] + g(𝑥, 𝑢) (𝑥, 𝑡) ∈ ℝ𝑁× (0, ∞),
whe e𝐷(𝑥,∇𝑢) = 𝑎(𝑥)|∇𝑢|𝑝−2 +|∇𝑢|𝑞−2. Thissys em hasa wide angeo applica ions inphysics
and ela ed ields, such as biophysics, plasma physics and chemical eac ion design (see [7, 26]).
In such applica ions, he unc ion 𝑢is a s a e a iable and desc ibes densi y o concen a ion
o mul i-componen subs ances, di [𝐷(𝑥, ∇𝑢)∇𝑢] co esponds o he di usion wi h a di usion
coe icien 𝐷(𝑥, ∇𝑢) and g(𝑥, 𝑢) is he eac ion and ela es o sou ce and loss p ocesses. Typically,
in chemical and biological applica ions, he eac ion e m g(𝑥, 𝑢) has a polynomial o m wi h
espec o he unknown concen a ion deno ed by 𝑢.
The analysis o he double phase eigen alue p oblem (1.1) is closely associa ed wi h he
ollowing single phase quasilinea eigen alue p oblem,
−Δ𝑎
𝑟𝑢=𝜇𝑚(𝑥)|𝑢|𝑟−2𝑢in ℝ𝑁.(1.2)
The i s pa o he pape is de o ed o he s udy o (1.2). The main esul s we es ablish
ega ding (1.2) a e upcoming Theo em 3.1 and P oposi ion 3.1, which e eal ha he e exis
a sequence o eigen alues o (1.2) and he i s eigen alue is simple. In he case o bounded
domains and 𝑟=2, his p oblem is ela ed o he Riesz–F edholm heo y o sel -adjoin and
compac ope a o s. The aniso opic linea case (i 𝑟=2and 𝑚(⋅)is non-cons an ) was i s
conside ed in he pionee ing pape s o Boche [6], Hess and Ka o [17]andPleijel[25]. An impo -
an con ibu ion in he case o unbounded domains is due o Alleg e o and Huang [1]and
Szulkin and Willem [27]. In [27], he au ho s assumed ha weigh unc ion may ha e singula
poin s.
Equa ion (1.1) con ains he con ibu ion o wo di e en ial ope a o s in he le -hand side, so
his p oblem is no homogeneous. In ac , he di e en ial ope a o 𝑢↦−Δ
𝑎
𝑝𝑢−Δ
𝑞𝑢is ela ed o
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736 GOU and RĂDULESCU
he ‘double-phase a ia ional unc ional de ined by
𝑢↦∫ℝ𝑁
𝑎(𝑥)|∇𝑢|𝑝+|∇𝑢|𝑞𝑑𝑥.
The in eg and o his unc ional is he unc ion
𝜉(𝑥, 𝑡) = 𝑎(𝑥)𝑡𝑝+𝑡
𝑞 o all 𝑥∈ℝ𝑁and 𝑡⩾0.
When 𝑎≡1, hen (1.1) becomes he so-called 𝑝&𝑞Laplacian p oblem, which was in es iga ed by
Benouhiba and Belyacine [4, 5]. A ea u e o his pape is ha we do no assume ha he unc ion
𝑎(⋅)is bounded away om ze o, ha is, we do no equi e ha essin 𝑥∈ℝ𝑁𝑎(𝑥) > 0.Thisimplies
ha he in eg and 𝜉(𝑥,𝑡) exhibi s unbalanced g ow h, namely he e holds ha
𝑡𝑞⩽𝜉(𝑥,𝑡) ⩽𝐶0(𝑡𝑝+𝑡
𝑞) o all 𝑥∈ℝ𝑁and 𝑡⩾0, (1.3)
whe e 𝐶0>0 is a cons an . In his scena io, he s udy is ca ied ou in he amewo k o
Musielak–O licz–Sobole spaces. Such unc ionals we e i s in es iga ed by Ma cellini [18–20]
in he con ex o p oblems o he calculus o a ia ions and o non-linea elas ici y o s ongly
aniso opic ma e ials. Fo such p oblems, he e is no global ( ha is, up o he bounda y) egula -
i y heo y. The ea e only in e io egula i y esul s, which a e p ima ily due o Ba oni e al. [3]and
Ma cellini [10, 20, 21]. In ac , mos o wo ks deal wi h double phase p oblems ha ing unbalanced
g ow h in bounded domains o ℝ𝑁, we e e he eade s o [12–15, 22–24] and e e ences he ein.
Howe e , he e exis ela i ely ew ones ea ing he p oblems in ℝ𝑁. The s udy o eigen alue
p oblems like (1.1) is open un il now. Since (1.1) is se in he whole space ℝ𝑁, lack o compac ness
is one o majo di icul ies we encoun e o discuss he eigen alue p oblem (1.1) in Musielak–
O licz–Sobole spaces and mo e ca e ul analysis is needed in sui able weigh ed unc ions spaces.
Indeed, his is mainly because he embedding 𝑊1,𝜉(ℝ𝑁)↪𝐿
𝑟(ℝ𝑁)is only con inuous o any
𝑞⩽𝑟⩽𝑞∗(see Lemma 2.3) and he weigh unc ion 𝑚∶ℝ𝑁→ℝis inde ini e, which cause ha
he e i ica ion o he compac ness o he unde lying (minimizing and Palasi–Smale) sequences
becomes di icul . Consequen ly, we manage o s udy he p oblem (1.1)inanewweigh edSobole
space 𝐸de ined by he comple ion o 𝐶∞
0(ℝ𝑁)unde he no m
‖𝑢‖𝐸∶= ‖∇𝑢‖𝜉+(∫ℝ𝑁|𝑢|𝑞max{𝑚2,𝜔}𝑑𝑥)1
𝑞,𝜔(𝑥)∶= 1
(1 + |𝑥|)𝑞,𝑥∈ℝ𝑁,
whe e ‖⋅‖𝜉deno es he s anda d no m in 𝐷1,𝜉(ℝ𝑁). He e, 𝑊1,𝜉(ℝ𝑁)and 𝐷1,𝜉(ℝ𝑁)a e Musielak–
O licz–Sobole spaces de ined in Sec ion 2. In his pape , when 𝑝<𝑞, we es ablish he exis ence
o a con inuous amily o eigen alues o (1.1), s a ing om he p incipal equency o (1.2), see
Theo ems 3.2 and 3.3.While𝑞<𝑝, we p o e he exis ence o a disc e e amily o posi i e eigen al-
ues o (1.1), which di e ges o in ini y, see Theo em 3.4 and P oposi ion 3.2. The esul s we de i e
e eal new ac s o eigen alues o double phase p oblems in ℝ𝑁. In bo h cases, we ac ually need
o assume 𝑞<𝑞
∗∶= 𝑁𝑞
𝑁−𝑞 , because o he unbalanced g ow h p ope y (1.3) wi h espec o he
double phase ope a o and he dominance is he 𝑞-Laplacian e m. Thus, he p oblem unde con-
side a ion is Sobole subc i ical and he ene gy unc ional 𝐽co esponding o (1.1) is well-de ined
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NON-AUTONOMOUS DOUBLE PHASE EIGENVALUE PROBLEMS 737
in he Sobole space 𝐸by Theo em 2.3, whe e
𝐽(𝑢) ∶= 1
𝑝∫ℝ𝑁
𝑎(𝑥)|∇𝑢|𝑝𝑑𝑥 + 1
𝑞∫ℝ𝑁|∇𝑢|𝑞𝑑𝑥 − 𝜆
𝑞∫ℝ𝑁
𝑚(𝑥)|𝑢|𝑞𝑑𝑥.
Obse e ha 𝑝
𝑞<1+ 1
𝑁implies𝑝<𝑞
∗.When double phasep oblemsa e se in boundeddomains
in ℝ𝑁, hen he condi ion 𝑝
𝑞<1+ 1
𝑁can be applied o p o e he desi ed compac embedding
esul s, o example [22, P oposi ion 4]. While double phase p oblems a e se in ℝ𝑁, he condi ion
𝑝
𝑞<1+ 1
𝑁can no longe be applicable o de i e he compac embedding esul s, which leads o
lack o compac ness o he s udy. In his pape , such a condi ion is ac ually used o gua an ee he
egula i y o solu ions o (1.1)(see[8, 9]), which along wi h he maximum p inciple de eloped in
[23, 24] can lead o he simplici y o eigen alues, see P oposi ion 3.2.
2 PRELIMINARIES
In he sec ion, we a e going o p esen some p elimina y esul s used o es ablish ou main
heo ems. To deal wi h he eigen alue p oblem (1.1), we shall wo k in he co esponding
Musielak–O licz–Sobole space. Fo he con enience o he eade s, le us i s p esen a ew
de ini ions om [11, Sec ion 2] conce ning he main no ions and unc ion spaces used in his
pape .
De ini ion 2.1. A unc ion 𝜑 ∶ [0, +∞] → [0, +∞) is called a Φ- unc ion i 𝜑is con ex and le -
con inuous on [0, +∞). In addi ion, 𝜑sa is ies ha
𝜑(0) = 0, lim
𝑡→0+𝜑(𝑡) = 0, lim
𝑡→+∞ 𝜑(𝑡) = +∞.
De ini ion 2.2. A unc ion 𝜉∶ℝ𝑁× [0, +∞] → [0, +∞) is called a gene alized Φ- unc ion i i
sa is ies he ollowing condi ions:
(i) o almos e e y 𝑥∈ℝ𝑁,𝜉(𝑥,⋅)is a Φ- unc ion;
(ii) o almos e e y 𝑡∈[0,+∞),𝜉(⋅,𝑡)is measu able.
De ini ion 2.3. A gene alized Φ- unc ion 𝜉∶ℝ𝑁× [0, +∞] → [0, +∞) sa is ies Δ2-condi ion i
he e exis s 𝐾⩾2such ha , o almos e e y 𝑥∈ℝ𝑁and 𝑡⩾0,
𝜉(𝑥,2𝑡) ⩾𝐾𝜉(𝑥, 𝑡).
De ini ion 2.4. AΦ- unc ion𝜑 ∶ [0, +∞] → [0, +∞) issaid o be an𝑁- unc ion i i is con inuous
and posi i e on [0, +∞). In addi ion, i sa is ies ha
lim
𝑡→0+
𝜑(𝑡)
𝑡=0, lim
𝑡→+∞
𝜑(𝑡)
𝑡=+∞.
A gene alized Φ- unc ion 𝜉∶ℝ𝑁× [0, +∞] → [0, +∞) is said o be a gene alized 𝑁- unc ion i ,
o almos e e y 𝑥∈ℝ𝑁,𝜉(𝑥,⋅)is an 𝑁- unc ion.
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738 GOU and RĂDULESCU
De ini ion 2.5. A gene alized 𝑁- unc ion 𝜉∶ℝ𝑁× [0, +∞] → [0, +∞) is called uni o mly
con ex i , o any 𝜖>0, he e exis s 𝛿>0such ha , o almos e e y 𝑥∈ℝ𝑁,
𝜉(𝑥, 𝑠+𝑡
2)⩽(1 − 𝛿)𝜉(𝑥,𝑠)+𝜉(𝑥,𝑡)
2,
whene e 𝑠,𝑡 ⩾0and |𝑥−𝑡|⩾𝜖max{|𝑠|,|𝑡|}.
Wi h hese de ini ions in hand, we a e now eady o in oduce he double phase unc ion 𝜉∶
ℝ𝑁× [0, +∞) → [0, +∞) co esponding o (1.1)as
𝜉(𝑥,𝑡) ∶= 𝑎(𝑥)𝑡𝑝+𝑡
𝑞,𝑥∈ℝ𝑁,𝑡⩾0. (2.1)
I is simple o check ha 𝜉is a gene alized 𝑁- unc ion. Mo eo e , 𝜉is uni o mly con ex and i
sa is ies heΔ2-condi ion. Le us deno eby 𝑀(ℝ𝑁) he spaceconsis ing o all Lebesguemeasu able
unc ion 𝑢∶ℝ𝑁→ℝ. The Musielak–O licz space 𝐿𝜉(ℝ𝑁)is de ined by
𝐿𝜉(ℝ𝑁)∶={𝑢∈𝑀(ℝ𝑁)∶𝜌
𝜉(𝑢) < +∞},
whe e 𝜌𝜉is he modula unc ion gi en by
𝜌𝜉(𝑢) ∶= ∫ℝ𝑁
𝜉(𝑥,|𝑢|)𝑑𝑥=∫ℝ𝑁
𝑎(𝑥)|𝑢|𝑝+|𝑢|𝑞𝑑𝑥. (2.2)
He e, he space 𝐿𝜉(ℝ𝑁)is equipped wi h he Luxembu g no m gi en by
‖𝑢‖𝜉∶= in {𝜆>0∶𝜌
𝜉(𝑢
𝜆)⩽1}.(2.3)
Using he abo e p ope ies sa is ied by 𝜉, we can easily check ha 𝐿𝜉(ℝ𝑁)is a Banach space,
which is also sepa able and e lexi e. The Musielak–O licz–Sobole space 𝑊1,𝜉(ℝ𝑁)is de ined
by
𝑊1,𝜉(ℝ𝑁)∶={𝑢∈𝐿
𝜉(ℝ𝑁)∶|∇𝑢|∈𝐿
𝜉(ℝ𝑁)}.
He e, he space 𝑊1,𝜉(ℝ𝑁)is equipped wi h he no m
‖𝑢‖1,𝜉 ∶= ‖𝑢‖𝜉+‖∇𝑢‖𝜉,
whe e ‖∇𝑢‖𝜉∶= ‖|∇𝑢|‖𝜉. Clea ly, 𝑊1,𝜉(ℝ𝑁)is a sepa able, e lexi e Banach space. Le us
in oduce he associa ed homogeneous Musielak–O licz–Sobole 𝐷1,𝜉(ℝ𝑁)as he comple ion o
𝐶∞
0(ℝ𝑁)unde he no m ‖∇𝑢‖𝜉.
Nex , we a e going o show some ela ions be ween he no m in 𝐿𝜉(ℝ𝑁)and he modula
unc ion 𝜌𝜉gi enby(2.2)and(2.3), espec i ely, p oo s o which can be comple ed by using he
ing edien s p esen ed in [16, Sec ion 3.2].
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NON-AUTONOMOUS DOUBLE PHASE EIGENVALUE PROBLEMS 739
Lemma 2.1. Le 𝜉∶ℝ𝑁× [0, +∞) → [0, +∞) be de ined by (2.1). Then, he ollowing asse ions
hold.
(i) ‖𝑢‖𝜉=𝜆i and only i 𝜌𝜉(𝑢
𝜆)=1.
(ii) ‖𝑢‖𝜉< 1(= 1, > 1,𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦) i and only i 𝜌𝜉(𝑢) < 1(= 1,> 1,𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦).
(iii) I ‖𝑢‖𝜉<1, hen‖𝑢‖max{𝑝,𝑞}
𝜉⩽𝜌𝜉(𝑢) ⩽‖𝑢‖min{𝑝,𝑞}
𝜉.
(i ) I ‖𝑢‖𝜉>1, hen‖𝑢‖min{𝑝,𝑞}
𝜉⩽𝜌𝜉(𝑢) ⩽‖𝑢‖max{𝑝,𝑞}
𝜉.
( ) lim𝑛→+∞ ‖𝑢𝑛‖𝜉= 0(+∞,𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦) i and only i lim𝑛→+∞ 𝜌𝜉(𝑢𝑛)=
0(+∞,𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦).
No e ha 𝑡𝑞⩽𝜉(𝑥,𝑡) o any 𝑥∈ℝ𝑁and 𝑡∈ℝ, by asse ion (ii)o Lemma 2.1, hen he e holds
he ollowing embedding esul .
Lemma 2.2. Le 𝜉∶ℝ𝑁× [0, +∞) → [0, +∞) be de ined by (2.1). Then, he embedding 𝐿𝜉(ℝ𝑁)↪
𝐿𝑞(ℝ𝑁)is con inuous.
As a consequence o Lemma 2.2 and Sobole ’s embeddings in 𝑊1,𝑞(ℝ𝑁)and 𝐷1,𝑞(ℝ𝑁) o 1<
𝑞<𝑁, we ha e he ollowing embedding esul .
Lemma 2.3. Le 𝜉∶ℝ𝑁× [0, +∞) → [0, +∞) be de ined by (2.1). Then, he embedding
𝑊1,𝜉(ℝ𝑁)↪𝑊
1,𝑞(ℝ𝑁)↪𝐿
𝑟(ℝ𝑁)is con inuous o any 𝑞⩽𝑟⩽𝑞∗. Mo eo e , he embedding
𝐷1,𝜉(ℝ𝑁)↪𝐷
1,𝑞(ℝ𝑁)↪𝐿
𝑞∗(ℝ𝑁)is con inuous.
3 MAIN RESULTS
In his sec ion, we shall conside he eigen alue p oblem (1.1) unde he assump ion (𝐻).The
hypo hesis (𝐻) is always assumed o hold in wha ollows. Fi s , we shall p esen some esul s
ela ed o he ollowing eigen alue p oblem,
−Δ𝑎
𝑟𝑢=𝜇𝑚(𝑥)|𝑢|𝑟−2𝑢in ℝ𝑁.(3.1)
Theo em 3.1. Assume (𝐻) holds, 𝑁⩾2,1<𝑟<𝑁,𝑎∈𝐶
0,1(ℝ𝑁,[0,+∞))and 𝑎≢0.Then, he e
exis s a sequence o solu ions (𝜇𝑎,𝑟,𝑘,𝑢
𝑎,𝑟,𝑘)∈ℝ×𝐷
1,𝜂(ℝ𝑁) o (3.1) wi h 𝑢𝑎,𝑟,𝑘 ∈and
0<𝜇
𝑎,𝑟,1 <𝜇
𝑎,𝑟,2 ⩽⋯⩽𝜇𝑎,𝑟,𝑘 ⩽⋯,lim
𝑘→∞ 𝜇𝑎,𝑟,𝑘 →+∞ as 𝑘 → +∞,
whe e 𝜂(𝑥, 𝑡) = 𝑎(𝑥)𝑡𝑟 o 𝑥∈ℝ𝑁and 𝑡⩾0,
𝑟∶= {𝑢∈𝐷
1,𝜂(ℝ𝑁)∶∫ℝ𝑁
𝑚(𝑥)|𝑢|𝑟𝑑𝑥 = 1}.
P oo . De ine
Ψ(𝑢) ∶= ∫ℝ𝑁
𝑎(𝑥)|∇𝑢|𝑟𝑑𝑥, 𝑀𝑟∶= 𝑟∩,
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740 GOU and RĂDULESCU
whe e he Sobole space is he comple ion o 𝐶∞
0(ℝ𝑁)unde he no m
‖∇𝑢‖𝜂+(∫ℝ𝑁|𝑢|𝑟max{𝑚2,𝜔}𝑑𝑥)1
𝑟,𝜔(𝑥)= 1
(1 + |𝑥|)𝑟,𝑥∈ℝ𝑁.
Reasoning as he p oo o [1, Lemma 1], we a e able o show Ψ(𝑢) es ic ed on 𝑀𝑟sa is ies
he Palais–Smale condi ion. Then, by adap ing Ljus e nik–Schni elman heo y as he p oo o
o hcomingTheo em 3.4,we can de i e he desi edconclusion. Thus, he p oo is comple ed. □
P oposi ion 3.1. Assume (𝐻) holds, 𝑁⩾2,1<𝑟<𝑁,𝑎∈𝐶
0,1(ℝ𝑁,[0,+∞))and 𝑎≢0.Then,
he i s eigen alue 𝜇𝑎,𝑟,1 ob ained in Theo em 3.1 is simple and he eigen unc ion 𝑢𝑎,𝑟,1 has cons an
sign. Mo eo e , i 𝑢∈𝐷
1,𝜂(ℝ𝑁)is a non- i ial solu ion o (3.1) co esponding o 𝜇>𝜇
𝑎,𝑟,1, hen𝑢
is sign-changing.
Since he unc ion 𝑚is an inde ini e sign weigh , hen p oo o P oposi ion 3.1 is no
s aigh o wa d. To p o e his, we need he ollowing auxilia y esul .
Lemma 3.1. De ine
𝐼(𝑢,𝑣) ∶= −∫ℝ𝑁(Δ𝑎
𝑟𝑢)𝑢𝑟−𝑣
𝑟
𝑢𝑟−1 𝑑𝑥 − ∫ℝ𝑁(Δ𝑎
𝑟𝑣)𝑣𝑟−𝑢
𝑟
𝑣𝑟−1 𝑑𝑥, 𝑢,𝑣 ∈ 𝐷1,𝜂(ℝ𝑁), 𝑢, 𝑣 > 0.
Then, 𝐼(𝑢,𝑣) ⩾0. Mo eo e , 𝐼(𝑢,𝑣) = 0 i and only i 𝑢=𝑘𝑣 o some 𝑘∈ℝ.
P oo . Obse e ha
∇(𝑢𝑟−𝑣
𝑟
𝑢𝑟−1 )=(1+(𝑟−1)
(𝑣
𝑢)𝑟)∇𝑢 − 𝑟(𝑣
𝑢)𝑟−1∇𝑣,
∇(𝑣𝑟−𝑢
𝑟
𝑣𝑟−1 )=(1+(𝑟−1)
(𝑢
𝑣)𝑟)∇𝑣 − 𝑟(𝑢
𝑣)𝑟−1∇𝑢.
Then, by he di e gence heo em, we see
𝐼(𝑢,𝑣) = ∫ℝ𝑁
𝑎(𝑥)((1+(𝑟−1)
(𝑣
𝑢)𝑟)|∇𝑢|𝑟−𝑟
(𝑣
𝑢)𝑟−1|∇𝑢|𝑟−2(∇𝑣 ⋅∇𝑢))𝑑𝑥
+∫ℝ𝑁
𝑎(𝑥)((1+(𝑟−1)
(𝑢
𝑣)𝑟)|∇𝑣|𝑟−𝑟
(𝑢
𝑣)𝑟−1|∇𝑣|𝑟−2(∇𝑢 ⋅∇𝑣))𝑑𝑥.
(3.2)
Using Young’s inequali y, we ha e
𝑟(𝑣
𝑢)𝑟−1|∇𝑢|𝑟−2(∇𝑣 ⋅∇𝑢)⩽𝑟(𝑣
𝑢)𝑟−1|∇𝑢|𝑟−1|∇𝑣|⩽(𝑟 − 1)(𝑣
𝑢)𝑟|∇𝑢|𝑟+|∇𝑣|𝑟,
𝑟(𝑢
𝑣)𝑟−1|∇𝑣|𝑟−2(∇𝑢 ⋅∇𝑣)⩽𝑟(𝑢
𝑣)𝑟−1|∇𝑣|𝑟−1|∇𝑢|⩽(𝑟 − 1)(𝑢
𝑣)𝑟|∇𝑣|𝑟+|∇𝑢|𝑟.
As a consequence, coming back o (3.2), we can conclude 𝐼(𝑢,𝑣) ⩾0.I 𝐼(𝑢,𝑣) = 0, hen
∇𝑢 ⋅∇𝑣 = |∇𝑢||∇𝑣|,(𝑣
𝑢)𝑟|∇𝑢|𝑟=|∇𝑣|𝑟,(𝑢
𝑣)𝑟|∇𝑣|𝑟=|∇𝑢|𝑟.
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NON-AUTONOMOUS DOUBLE PHASE EIGENVALUE PROBLEMS 741
I hen ollows ha
|𝑢∇𝑣 − 𝑣∇𝑢|=0.
This implies ha he e exis s 𝑘∈ℝsuch ha 𝑢=𝑘𝑣and he p oo is comple ed. □
P oo o P oposi ion 3.1. No e i s ha
𝜇𝑎,𝑟,1 =in
𝑢∈𝑟
Ψ(𝑢).
I 𝑢∈𝑟sa is ies Ψ(𝑢) = 𝑢𝑎,𝑟,1, hen |𝑢|∈𝑟and Ψ(|𝑢|)=𝑢
𝑎,𝑟,1. The e o e, wi hou es ic-
ion, we may assume 𝑢𝑎,𝑟,1 is non-nega i e. Obse e ha 𝑢𝑎,𝑟,1 ∈𝐷
1,𝜂(ℝ𝑁)sa is ies he equa ion
−Δ𝑎
𝑟𝑢𝑎,𝑟,1 +𝜇
𝑎,𝑟,1𝑚2(𝑥)|𝑢𝑎,𝑟,1|𝑟−2𝑢𝑎,𝑟,1 =𝜇
𝑎,𝑟,1𝑚1(𝑥)|𝑢𝑎,𝑟,1|𝑟−2𝑢𝑎,𝑟,1 ⩾0in ℝ𝑁.
By maximum p inciple, 𝑢𝑎,𝑟,1 >0.Le 𝑢𝑎,𝑟,1 ∈𝑟and 𝑣𝑎,𝑟,1 ∈𝑟be wo posi i e eigen unc ions
co esponding o 𝜇𝑎,𝑟,1, hen
−Δ𝑎
𝑟𝑢𝑎,𝑟,1 =𝜇
𝑎,𝑟,1𝑚(𝑥)𝑢𝑟−1
𝑎,𝑟,1,−Δ
𝑎
𝑟𝑣𝑎,𝑟,1 =𝜇
𝑎,𝑟,1𝑚(𝑥)𝑣𝑟−1
𝑎,𝑟,1 in ℝ𝑁.
I is simple o calcula e 𝐼(𝑢𝑎,𝑟,1,𝑣
𝑎,𝑟,1)=0.Asa esul o Lemma3.1,weha e𝑢𝑎,𝑟,1 =𝑘𝑣
𝑎,𝑟,1 o
some 𝑘∈ℝ. This indica es ha 𝜇𝑎,𝑟,1 is simple.
A guing by con adic ion, we suppose 𝑢∈𝐷
1,𝜂(ℝ𝑁)is a non-nega i e solu ion o (3.1)
co esponding o 𝜇>𝜇
𝑎,𝑟,1. By he maximum p inciple, 𝑢>0. No ice
∫ℝ𝑁
𝑎(𝑥)|∇𝑢|𝑟𝑑𝑥 = 𝜇 ∫ℝ𝑁
𝑚(𝑥)|𝑢|𝑟𝑑𝑥 > 0.
In addi ion, we know ha i 𝑢∈𝐷
1,𝜂(ℝ𝑁)is a solu ion o (3.1), hen 𝑘𝑢 ∈ 𝐷1,𝜂(ℝ𝑁)is also a
solu ion o (3.1) o any 𝑘∈ℝ∖{0}. Then, by scaling, we may assume
0<∫ℝ𝑁
𝑚(𝑥)|𝑢|𝑟𝑑𝑥 < 1. (3.3)
Le 𝑢𝑎,𝑟,1 ∈and 𝑢𝑎,𝑟,1 >0be an eigen unc ion o(3.1) co esponding o 𝜇𝑎,𝑟,1.Then, 𝑢𝑎,𝑟,1 sol es
he equa ion
−Δ𝑎
𝑟𝑢𝑎,𝑟,1 =𝜇
𝑎,𝑟,1𝑚(𝑥)|𝑢𝑎,𝑟,1|𝑟−2𝑢𝑎,𝑟,1 in ℝ𝑁.
As a consequence o Lemma 3.1 and (3.3), we ha e
0⩽𝐼(𝑢,𝑢𝑎,𝑟,1)=𝜇∫ℝ𝑁
𝑚(𝑥)(𝑢𝑟−𝑢
𝑟
𝑎,𝑟,1)𝑑𝑥 + 𝜇𝑎,𝑟,1 ∫ℝ𝑁
𝑚(𝑥)(𝑢𝑟
𝑎,𝑟,1 −𝑢
𝑟)𝑑𝑥
=(𝜇−𝜇
𝑎,𝑟,1)∫ℝ𝑁
𝑚(𝑥)|𝑢|𝑟𝑑𝑥 − (𝜇 − 𝜇𝑎,𝑟,1)<0.
This is impossible, hence 𝑢is sign-changing and he p oo is comple ed. □
Theo em 3.2. Assume (𝐻) holds, 𝑁⩾2,1<𝑝,𝑞<𝑁,𝑝≠𝑞,𝑎∈𝐶
0,1(ℝ𝑁,[0,+∞))and 𝑎≢0.
Then, (1.1) has no non- i ial solu ions in 𝐷1,𝜉(ℝ𝑁) o any 0⩽𝜆⩽𝜇1,𝑞,1,whe e𝜇1,𝑞,1 >0is he i s
eigen alue o (3.1) wi h 𝑎≡1and 𝑟=𝑞.
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742 GOU and RĂDULESCU
P oo . Le 𝑢∈𝐷
1,𝜉(ℝ𝑁)be a solu ion o (1.1) o some0⩽𝜆⩽𝜇1,𝑞,1. Obse e i s ha
∫ℝ𝑁
𝑎(𝑥)|∇𝑢|𝑝𝑑𝑥 + ∫ℝ𝑁|∇𝑢|𝑞𝑑𝑥 = 𝜆 ∫ℝ𝑁
𝑚(𝑥)|𝑢|𝑞𝑑𝑥. (3.4)
This implies 𝑢=0i 𝜆=0. Le us assume 0<𝜆<𝜇
1,𝑞,1. Assume 𝑢≠0, i hen ollows om (3.4)
ha
∫ℝ𝑁
𝑚(𝑥)|𝑢|𝑞𝑑𝑥 > 0. (3.5)
In addi ion, since 𝜇1,𝑞,1 >0is he i s eigen alue o (3.1), hen
∫ℝ𝑁|∇𝑢|𝑞𝑑𝑥 ⩾𝜇1,𝑞,1 ∫ℝ𝑁
𝑚(𝑥)|𝑢|𝑞𝑑𝑥. (3.6)
This along wi h (3.4) leads o
𝜇1,𝑞,1 ∫ℝ𝑁
𝑚(𝑥)|𝑢|𝑞𝑑𝑥 ⩽𝜆∫ℝ𝑁
𝑚(𝑥)|𝑢|𝑞𝑑𝑥.
Using (3.5), we hen ge 𝑢=0. This is a con adic ion. Nex we assume 𝜆=𝜇
1. In his case, by
combining (3.4)and(3.6), we ob ain
∫ℝ𝑁
𝑎(𝑥)|∇𝑢|𝑝𝑑𝑥 ⩽0,
hence 𝑢=0. Thus, he p oo is comple ed. □
3.1 Case 𝒑<𝒒
In his case, o es ablish he exis ence o solu ions o (1.1), we shall adap some ideas om [1]. Le
us i s in oduce he weigh unc ion
𝜔(𝑥) = 1
(1 + |𝑥|)𝑞,𝑥∈ℝ𝑁.
Le 𝐸be he comple ion o 𝐶∞
0(ℝ𝑁)unde he no m
‖𝑢‖𝐸∶= ‖∇𝑢‖𝜉+(∫ℝ𝑁|𝑢|𝑞max{𝑚2,𝜔}𝑑𝑥)1
𝑞.
I is s anda d o conclude ha 𝐸is a sepa able and e lexi e Banach space. In o de o p o e he
exis ence o solu ions o (1.1), we shall de ine he associa ed ene gy unc ional 𝐽∶𝐸→ℝby
𝐽(𝑢) ∶= 1
𝑝∫ℝ𝑁
𝑎(𝑥)|∇𝑢|𝑝𝑑𝑥 + 1
𝑞∫ℝ𝑁|∇𝑢|𝑞𝑑𝑥 − 𝜆
𝑞∫ℝ𝑁
𝑚(𝑥)|𝑢|𝑞𝑑𝑥.
Theo em 3.3. Assume (𝐻) holds, 𝑁⩾2,1<𝑝<𝑞<𝑁,𝑎∈𝐶
0,1(ℝ𝑁,[0,+∞))and 𝑎≢0.Then,
he e exis posi i e solu ions o (1.1) o any𝜆>𝜇
1,𝑞,1.
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NON-AUTONOMOUS DOUBLE PHASE EIGENVALUE PROBLEMS 749
whe e he second ac holds because o 𝑚1∈𝐿
𝑁
𝑞(ℝ𝑁) om he assump ion (𝐻). Combining (3.23),
(3.24)and(3.25), by (3.22), we hen ob ain
∫ℝ𝑁(𝑎(𝑥)(|∇𝑢𝑛|𝑝−2∇𝑢𝑛−|∇𝑢|𝑝−2∇𝑢)+(|∇𝑢𝑛|𝑞−2∇𝑢𝑛−|∇𝑢|𝑞−2∇𝑢))⋅(∇𝑢𝑛−∇𝑢
)𝑑𝑥 = 𝑜𝑛(1).
Obse e ha
|𝑧1−𝑧
2|𝑟⩽𝐶((|𝑧1|𝑟−2𝑧1−|𝑧2|𝑟−2𝑧2)⋅(𝑧1−𝑧
2))𝜃
2(|𝑧1|𝑟+|𝑧2|𝑟)1− 𝜃
2,∀𝑧
1,𝑧
2∈ℝ𝑁,(3.26)
whe e 𝜃=𝑟i 1<𝑟<2and 𝜃=2i 𝑟⩾2. Then, we see
∫ℝ𝑁
𝑎(𝑥)(|∇𝑢𝑛−∇𝑢|𝑝)𝑑𝑥 + ∫ℝ𝑁|∇𝑢𝑛−∇𝑢|𝑞𝑑𝑥
⩽𝐶(∫ℝ𝑁
𝑎(𝑥)(|∇𝑢𝑛|𝑝−2∇𝑢𝑛−|∇𝑢|𝑝−2∇𝑢)⋅(∇𝑢𝑛−∇𝑢
)𝑑𝑥)𝜃
2(∫ℝ𝑁
𝑎(𝑥)(|∇𝑢𝑛|𝑝+|∇𝑢|𝑝)𝑑𝑥)1− 𝜃
2
+𝐶
(∫ℝ𝑁(|∇𝑢𝑛|𝑞−2∇𝑢𝑛−|∇𝑢|𝑞−2∇𝑢)⋅(∇𝑢𝑛−∇𝑢
)𝑑𝑥)𝜃
2(∫ℝ𝑁|∇𝑢𝑛|𝑞+|∇𝑢|𝑞𝑑𝑥)1− 𝜃
2=𝑜
𝑛(1).
This immedia ely indica es ha 𝑢𝑛→𝑢in 𝐷1,𝜉(ℝ𝑁)as 𝑛→∞. Taking ad an age o (3.20)and
(3.21), we hen ge
∫ℝ𝑁
𝑚(𝑥)|𝑢𝑛|𝑞𝑑𝑥 = ∫ℝ𝑁
𝑚(𝑥)|𝑢|𝑞𝑑𝑥 + 𝑜𝑛(1),
because o 𝜆𝑛=𝜆+𝑜
𝑛(1) and 𝜆≠0. In iew o (3.18),
∫ℝ𝑁
𝑚2(𝑥)|𝑢𝑛|𝑞𝑑𝑥 = ∫ℝ𝑁
𝑚2(𝑥)|𝑢|𝑞𝑑𝑥 + 𝑜𝑛(1).
Since 𝑢𝑛→𝑢in 𝐷1,𝑞(ℝ𝑁)as 𝑛→∞, by Ha dy’s inequali y,
∫ℝ𝑁|𝑢𝑛−𝑢|𝑞
(1 + |𝑥|)𝑞𝑑𝑥 ⩽(𝑝
𝑁−𝑝)𝑝∫ℝ𝑁|∇𝑢𝑛−∇𝑢|𝑞𝑑𝑥 = 𝑜𝑛(1).
Consequen ly, we de i e ha 𝑢𝑛→𝑢in 𝐸as 𝑛→∞. Thus, he p oo is comple ed. □
Theo em 3.4. Assume (𝐻) holds, 𝑁⩾2,1<𝑞<𝑝<𝑁,𝑎∈𝐶
0,1(ℝ𝑁,[0,+∞))and 𝑎≢0.Then,
he e exis s a sequence o solu ions (𝜆𝑘,𝑢
𝑘)∈ℝ×𝐸wi h 𝑢𝑘∈and
0<𝜆
1<𝜆
2⩽⋯⩽𝜆𝑘⩽⋯,lim
𝑘→∞ 𝜆𝑘→+∞as 𝑘 → +∞.
P oo . To es ablish he exis ence o a sequence o eigen alues o (1.1), we shall ake in o accoun
Ljus e nik–Schni elman heo y in [2]. De ine
Σ∶={𝐴⊂∶𝐴is compac and 𝐴=−𝐴
}.
14692120, 2024, 2, Downloaded om h ps://londma hsoc.onlinelib a y.wiley.com/doi/10.1112/blms.12961 by B no Uni e si y O Technology, Wiley Online Lib a y on [13/05/2024]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License

750 GOU and RĂDULESCU
Fo a se 𝐴∈Σ, he genus o 𝐴is de ined by
𝛾(𝐴) ∶= min{𝑛∈ℕ∶exis s a unc ion 𝜑∈𝐶(𝐴,ℝ𝑛∖{0}) sa is ying 𝜑(−𝑥) = −𝜑(𝑥)}.
I such a minimum does no exis , we se 𝛾(𝐴) = +∞.
Le us now de ine
Σ𝑘∶= {𝐴∈Σ∶𝛾(𝐴)⩾𝑘},∀𝑘∈ℕ+.
Fi s we see ha , o any 𝑘∈ℕ+,Σ𝑘≠∅. Indeed, le 𝑋𝑘be a 𝑘-dimensional subspace o 𝐸,by
Bo suk–Ulam’s heo em, hen 𝛾(∩𝑋
𝑘)⩾𝑘.De ine

𝜆𝑘∶= in
𝐴∈Σ𝑘
sup
𝑢∈𝐴
Φ(𝑢).
Since Σ𝑘+1 ⊂Σ
𝑘,
𝜆𝑘⩽
𝜆𝑘+1 o any 𝑘∈ℕ+. F om Lemma 3.2,
𝜆𝑘is a c i ical poin o 𝐽 es ic ed
on  o any 𝑘∈ℕ+. Then, we de i e ha
0< 
𝜆1<
𝜆2⩽⋯⩽
𝜆𝑘⩽
𝜆𝑘+1 ⩽⋯.
Nex we p o e 
𝜆𝑘→+∞as 𝑘→+∞.Le {𝑒𝑖}⊂𝐸be such ha 𝐸=span{𝑒1,𝑒
2,…,𝑒
𝑖,…}.Le {𝑒′
𝑖}⊂
𝐸be such ha 𝐸′=span{𝑒′
1,𝑒′
2,…,𝑒′
𝑖,…}, whe e 𝐸′deno es he dual space o 𝐸.De ine𝑋𝑖∶=
span{𝑒𝑖}and
𝑌𝑘∶=
𝑘
⨁
𝑖=1
𝑋𝑖,𝑍
𝑘∶=
∞
⨁
𝑖=𝑘
𝑋𝑖,∀𝑘∈ℕ+.
Le 𝐴∈Σ
𝑘sa is y 𝛾(𝐴) ⩾𝑘. By basic p ope ies o he genus, we ha e 𝐴∩𝑍
𝑘≠∅.De ine
𝛽𝑘∶= in
𝐴∈Σ𝑘
sup
𝑢∈𝐴∩𝑍𝑘
𝐽(𝑢), ∀ 𝑘 ∈ ℕ+.
Then, 𝛽𝑘→+∞as 𝑘→∞. O he wise, we may assume {𝛽𝑘}⊂ℝis bounded. Thus, he e exis s a
sequence {𝑢𝑘}⊂𝐴∩𝑍
𝑘such ha {Φ(𝑢𝑘)} ⊂ ℝis bounded. I hen ollows ha {𝑢𝑘}is bounded in
𝐸. Fu he , he e exis s 𝑢∈𝐸such ha 𝑢𝑘⇀𝑢in 𝐸as 𝑛→∞. Obse e ha ⟨𝑒′
𝑖,𝑢⟩=⟨𝑒′
𝑖,𝑢
𝑘⟩+
𝑜𝑘(1) = 𝑜𝑘(1), because o 𝑢𝑘∈𝑍
𝑘. The e o e, we ha e 𝑢=0and 𝑢𝑘⇀0in 𝐸as 𝑘→∞.This
along wi h he assump ion ha 𝑚1∈𝐿
𝑁
𝑞(ℝ𝑁) om he assump ion (𝐻) leads o
∫ℝ𝑁
𝑚1(𝑥)|𝑢𝑘|𝑞𝑑𝑥 = 𝑜𝑘(1).
Since 𝑚2⩾0 om he assump ion (𝐻),
∫ℝ𝑁
𝑚(𝑥)|𝑢𝑘|𝑞𝑑𝑥 = ∫ℝ𝑁
𝑚1(𝑥)|𝑢𝑘|𝑞𝑑𝑥 − ∫ℝ𝑁
𝑚2(𝑥)|𝑢𝑘|𝑞𝑑𝑥 ⩽𝑜𝑘(1),
which is impossible due o 𝑢𝑘∈. Consequen ly, we ge ha 𝛽𝑘→+∞as 𝑘→∞.Thanks o

𝜆𝑘⩾𝛽𝑘 o any 𝑘∈ℕ+,
𝜆𝑘→+∞as 𝑘→∞.Since𝑢𝑘∈𝐸is a c i ical poin o 𝐸 es ic ed on ,
he e exis s 𝜆𝑘∈ℝsuch ha
−Δ𝑎
𝑝𝑢𝑘−Δ
𝑞𝑢𝑘=𝜆
𝑘𝑚(𝑥)|𝑢𝑘|𝑞−2𝑢𝑘in ℝ𝑁,
14692120, 2024, 2, Downloaded om h ps://londma hsoc.onlinelib a y.wiley.com/doi/10.1112/blms.12961 by B no Uni e si y O Technology, Wiley Online Lib a y on [13/05/2024]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
NON-AUTONOMOUS DOUBLE PHASE EIGENVALUE PROBLEMS 751
whe e
𝜆𝑘=1
𝑞∫ℝ𝑁
𝑎(𝑥)|∇𝑢𝑘|𝑝𝑑𝑥 + 1
𝑞∫ℝ𝑁|∇𝑢𝑘|𝑞𝑑𝑥 > Φ(𝑢𝑘)= 
𝜆𝑘,∀𝑘∈ℕ+.
Thus, he p oo is comple ed. □
Lemma 3.3. De ine
𝐼(𝑢,𝑣) ∶ = −∫ℝ𝑁(Δ𝑎
𝑝𝑢)𝑢𝑞−𝑣
𝑞
𝑢𝑞−1 𝑑𝑥 − ∫ℝ𝑁(Δ𝑞𝑢)𝑢𝑞−𝑣
𝑞
𝑢𝑞−1 𝑑𝑥
−∫ℝ𝑁(Δ𝑎
𝑝𝑣)𝑣𝑞−𝑢
𝑞
𝑣𝑞−1 𝑑𝑥 − ∫ℝ𝑁(Δ𝑞𝑣)𝑣𝑞−𝑢
𝑞
𝑣𝑞−1 𝑑𝑥,
(3.27)
whe e 𝑢,𝑣 ∈ 𝐷1,𝜉(ℝ𝑁),𝑢, 𝑣 > 0 and 1<𝑞<𝑝.Then,𝐼(𝑢,𝑣) ⩾0. Mo eo e , 𝐼(𝑢,𝑣) = 0 i and only
i 𝑢=𝑘𝑣 o some 𝑘∈ℝ.
P oo . Le us i s show
𝐼1(𝑢,𝑣)∶=−∫ℝ𝑁(Δ𝑎
𝑝𝑢)𝑢𝑞−𝑣
𝑞
𝑢𝑞−1 𝑑𝑥 − ∫ℝ𝑁(Δ𝑎
𝑝𝑣)𝑣𝑞−𝑢
𝑞
𝑣𝑞−1 𝑑𝑥 ⩾0, 𝑢, 𝑣 ∈ 𝐷1,𝜉(ℝ𝑁), 𝑢, 𝑣 > 0.
I is s aigh o wa d o compu e
∇(𝑢𝑞−𝑣
𝑞
𝑢𝑞−1 )=(1+(𝑞−1)
(𝑣
𝑢)𝑞)∇𝑢 − 𝑞(𝑣
𝑢)𝑞−1∇𝑣, (3.28)
∇(𝑣𝑞−𝑢
𝑞
𝑣𝑞−1 )=(1+(𝑞−1)
(𝑢
𝑣)𝑞)∇𝑣 − 𝑞(𝑢
𝑣)𝑞−1∇𝑢. (3.29)
The e o e, by he di e gence heo em, we de i e ha
𝐼1(𝑢, 𝑣) = ∫ℝ𝑁
𝑎(𝑥)((1+(𝑞−1)
(𝑣
𝑢)𝑞)|∇𝑢|𝑝−𝑞
(𝑣
𝑢)𝑞−1|∇𝑢|𝑝−2∇𝑢 ⋅∇𝑣)𝑑𝑥
+∫ℝ𝑁
𝑎(𝑥)((1+(𝑞−1)
(𝑢
𝑣)𝑞)|∇𝑣|𝑝−𝑞
(𝑢
𝑣)𝑞−1|∇𝑣|𝑝−2∇𝑣 ⋅∇𝑢)𝑑𝑥.
Using Young’s inequali y, we know ha
𝑞(𝑣
𝑢)𝑞−1|∇𝑢|𝑝−2|∇𝑢 ⋅∇𝑣|⩽𝑞(𝑣
𝑢)𝑞−1|∇𝑢|𝑝−1|∇𝑣|
⩽𝑞(𝑝 − 1)
𝑝(𝑣
𝑢)𝑝(𝑞−1)
𝑝−1 |∇𝑢|𝑝+𝑞
𝑝|∇𝑣|𝑝
=𝑞(𝑝 − 1)
𝑝(𝑣
𝑢)𝑝(𝑞−1)
𝑝−1 |∇𝑢|𝑝2(𝑞−1)
𝑞(𝑝−1) |∇𝑢|𝑝(𝑝−𝑞)
𝑞(𝑝−1) +𝑞
𝑝|∇𝑣|𝑝
⩽(𝑞 − 1)(𝑣
𝑢)𝑞|∇𝑢|𝑝+𝑝−𝑞
𝑝|∇𝑢|𝑝+𝑞
𝑝|∇𝑣|𝑝.
14692120, 2024, 2, Downloaded om h ps://londma hsoc.onlinelib a y.wiley.com/doi/10.1112/blms.12961 by B no Uni e si y O Technology, Wiley Online Lib a y on [13/05/2024]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
752 GOU and RĂDULESCU
Simila ly, we can ge
𝑞(𝑢
𝑣)𝑞−1|∇𝑣|𝑝−2|∇𝑣 ⋅∇𝑢|⩽𝑞(𝑢
𝑣)𝑞−1|∇𝑣|𝑝−1|∇𝑢|⩽(𝑞 − 1)(𝑢
𝑣)𝑞|∇𝑣|𝑝+𝑝−𝑞
𝑝|∇𝑣|𝑝+𝑞
𝑝|∇𝑢|𝑝.
I hen ollows ha 𝐼1(𝑢, 𝑣) ⩾0. Nex , we p o e ha
𝐼2(𝑢,𝑣)∶=−∫ℝ𝑁(Δ𝑞𝑢)𝑢𝑞−𝑣
𝑞
𝑢𝑞−1 𝑑𝑥 − ∫ℝ𝑁(Δ𝑞𝑣)𝑣𝑞−𝑢
𝑞
𝑣𝑞−1 𝑑𝑥 ⩾0, 𝑢, 𝑣 ∈ 𝐷1,𝜉(ℝ𝑁), 𝑢, 𝑣 > 0.
In iew o (3.28)and(3.29), by he di e gence heo em,
𝐼2(𝑢, 𝑣) = ∫ℝ𝑁(1+(𝑞−1)
(𝑣
𝑢)𝑞)|∇𝑢|𝑞−𝑞
(𝑣
𝑢)𝑞−1|∇𝑢|𝑞−2∇𝑢 ⋅∇𝑣 𝑑𝑥
+∫ℝ𝑁(1+(𝑞−1)
(𝑢
𝑣)𝑞)|∇𝑣|𝑞−𝑞
(𝑢
𝑣)𝑞−1|∇𝑣|𝑞−2∇𝑣 ⋅∇𝑢 𝑑𝑥.
Using again Young’s inequali y, we ob ain
𝑞(𝑣
𝑢)𝑞−1|∇𝑢|𝑞−2|∇𝑢 ⋅∇𝑣|⩽𝑞(𝑣
𝑢)𝑞−1|∇𝑢|𝑞−1|∇𝑣|⩽(𝑞 − 1)(𝑣
𝑢)𝑞|∇𝑢|𝑞+|∇𝑣|𝑞,
𝑞(𝑢
𝑣)𝑞−1|∇𝑣|𝑞−2|∇𝑣 ⋅∇𝑢|⩽𝑞(𝑢
𝑣)𝑞−1|∇𝑣|𝑞−1|∇𝑢|⩽(𝑞 − 1)(𝑢
𝑣)𝑞|∇𝑣|𝑞+|∇𝑢|𝑞.
The e o e, we ha e 𝐼2(𝑢, 𝑣) = 0. Acco dingly, he e holds ha 𝐼(𝑢,𝑣) ⩾0 o any 𝑢,𝑣 ∈ 𝐷1,𝜉(ℝ𝑁)
and 𝑢,𝑣 > 0.I 𝐼(𝑢,𝑣)=0, hen 𝐼2(𝑢, 𝑣) = 0. This leads o
∇𝑢 ⋅∇𝑣 = |∇𝑢||∇𝑣|,(𝑣
𝑢)𝑞|∇𝑢|𝑞=|∇𝑣|𝑞,(𝑢
𝑣)𝑞|∇𝑣|𝑞=|∇𝑢|𝑞,
As a consequence, we see ha
|𝑢∇𝑣 − 𝑣∇𝑢|=0.
This implies ha he e exis s 𝑘∈ℝsuch ha 𝑢=𝑘𝑣and he p oo is comple ed. □
Rema k 3.1. In ac , Lemma 3.3 is es ablished o he double phase ope a o unde he assump ion
𝑞<𝑝, which is no a di ec consequence o Lemma 3.1. I is unknown o us i Lemma 3.3 emains
alid o he case 𝑝<𝑞. F om he p oo o Lemma 3.3, one can see ha he assump ion 𝑞<𝑝is
c ucial, which is he p emise o he use o Young’s inequali y.
P oposi ion 3.2. Assume (𝐻) holds, 𝑁⩾2,1<𝑞<𝑝<𝑁,𝑝
𝑞<1+ 1
𝑁,𝑎∈𝐶
0,1(ℝ𝑁,[0,+∞))
and 𝑎≢0. Assume ha any eigen unc ion o (1.1) co esponding o 𝜆is non-nega i e. Then, 𝜆
is simple.
P oo . Le 𝑢∈𝐸be a non-nega i e eigen unc ion o (1.1) co esponding o 𝜆. I ollows om
[8]and[23, P oposi ion 3] o [24, P oposi ion 2.3] ha 𝑢>0.Le 𝑢>0 and 𝑣>0 be wo
eigen unc ions o (1.1) co esponding o 𝜆. Then, we see ha
−Δ𝑎
𝑝𝑢−Δ
𝑞𝑢 = 𝜆𝑚(𝑥)𝑢𝑞−1,−Δ
𝑎
𝑝𝑣−Δ
𝑞𝑣 = 𝜆𝑚(𝑥)𝑣𝑞−1 in ℝ𝑁.
14692120, 2024, 2, Downloaded om h ps://londma hsoc.onlinelib a y.wiley.com/doi/10.1112/blms.12961 by B no Uni e si y O Technology, Wiley Online Lib a y on [13/05/2024]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
NON-AUTONOMOUS DOUBLE PHASE EIGENVALUE PROBLEMS 753
As a esul , he e holds ha
𝐼(𝑢,𝑣)=𝜆∫ℝ𝑁
𝑚(𝑥)(𝑢𝑞−𝑣
𝑞)𝑑𝑥 + 𝜆 ∫ℝ𝑁
𝑚(𝑥)(𝑣𝑞−𝑢
𝑞)𝑑𝑥 = 0.
I hen ollows om Lemma 3.3 ha he desi ed conclusion holds. This comple es he p oo . □
P oposi ion 3.3. Assume (𝐻) holds, 𝑁⩾2,1<𝑝,𝑞<𝑁,𝑝≠𝑞,𝑎∈𝐶
0,1(ℝ𝑁,[0,+∞))and 𝑎≢0.
Then,
𝜇1,𝑞,1 =in ⎧
⎪
⎨
⎪
⎩
1
𝑝∫ℝ𝑁𝑎(𝑥)|∇𝑢|𝑝𝑑𝑥 + 1
𝑞∫ℝ𝑁|∇𝑢|𝑞𝑑𝑥
1
𝑞∫ℝ𝑁𝑚(𝑥)|𝑢|𝑞𝑑𝑥 ∶𝑢∈𝐸∖{0},∫ℝ𝑁
𝑚(𝑥)|𝑢|𝑞𝑑𝑥 > 0⎫
⎪
⎬
⎪
⎭
.
P oo . Since 𝜇1,𝑞,1 is he i s eigen alue o (3.1)and𝐸⊂𝐷
1,𝑞(ℝ𝑁),
𝜇1,𝑞,1 =in {∫ℝ𝑁|∇𝑢|𝑞𝑑𝑥
∫ℝ𝑁𝑚(𝑥)|𝑢|𝑞𝑑𝑥 ∶𝑢∈𝐷
1,𝑞(ℝ𝑁)∖{0}, ∫ℝ𝑁
𝑚(𝑥)|𝑢|𝑞𝑑𝑥 > 0}
⩽in ⎧
⎪
⎨
⎪
⎩
1
𝑝∫ℝ𝑁𝑎(𝑥)|∇𝑢1,𝑞,1|𝑝𝑑𝑥 + 1
𝑞∫ℝ𝑁|∇𝑢|𝑞𝑑𝑥
1
𝑞∫ℝ𝑁𝑚(𝑥)|𝑢|𝑞𝑑𝑥 ∶𝑢∈𝐸∖{0},∫ℝ𝑁
𝑚(𝑥)|𝑢|𝑞𝑑𝑥 > 0⎫
⎪
⎬
⎪
⎭
.
Le 𝑢1,𝑞,1 ∈𝐸be an eigen unc ion o (3.1) co esponding o 𝜇1,𝑞,1 and 𝑝<𝑞, hen
in ⎧
⎪
⎨
⎪
⎩
1
𝑝∫ℝ𝑁𝑎(𝑥)|∇𝑢|𝑝𝑑𝑥 + 1
𝑞∫ℝ𝑁|∇𝑢|𝑞𝑑𝑥
1
𝑞∫ℝ𝑁𝑚(𝑥)|𝑢|𝑞𝑑𝑥 ∶𝑢∈𝐸∖{0},∫ℝ𝑁
𝑚(𝑥)|𝑢|𝑞𝑑𝑥 > 0⎫
⎪
⎬
⎪
⎭
⩽
𝑛𝑝
𝑝∫ℝ𝑁𝑎(𝑥)|∇𝑢1,𝑞,1|𝑝𝑑𝑥 + 𝑛𝑞
𝑞∫ℝ𝑁|∇𝑢1,𝑞,1|𝑞𝑑𝑥
𝑛𝑞
𝑞∫ℝ𝑁𝑚(𝑥)|𝑢1,𝑞,1|𝑞𝑑𝑥 =𝜇
1,𝑞,1 +𝑜
𝑛(1) as 𝑛→∞.
Simila ly, i 𝑞<𝑝, hen
in ⎧
⎪
⎨
⎪
⎩
1
𝑝∫ℝ𝑁𝑎(𝑥)|∇𝑢|𝑝𝑑𝑥 + 1
𝑞∫ℝ𝑁|∇𝑢|𝑞𝑑𝑥
1
𝑞∫ℝ𝑁𝑚(𝑥)|𝑢|𝑞𝑑𝑥 ∶𝑢∈𝐸∖{0},∫ℝ𝑁
𝑚(𝑥)|𝑢|𝑞𝑑𝑥 > 0⎫
⎪
⎬
⎪
⎭
⩽
1
𝑝𝑛𝑝∫ℝ𝑁𝑎(𝑥)|∇𝑢1,𝑞,1|𝑝𝑑𝑥 + 1
𝑞𝑛𝑞∫ℝ𝑁|∇𝑢1,𝑞,1|𝑞𝑑𝑥
1
𝑞𝑛𝑞∫ℝ𝑁𝑚(𝑥)|𝑢1,𝑞,1|𝑞𝑑𝑥 =𝜇
1,𝑞,1 +𝑜
𝑛(1) as 𝑛→∞.
Thus, he desi ed esul ollows and he p oo is comple ed. □
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754 GOU and RĂDULESCU
Rema k 3.2. Unde he assump ions o Theo em 3.4, by Theo em 3.2 and P oposi ion 3.3,weha e
𝜆1>in ⎧
⎪
⎨
⎪
⎩
1
𝑝∫ℝ𝑁𝑎(𝑥)|∇𝑢|𝑝𝑑𝑥 + 1
𝑞∫ℝ𝑁|∇𝑢|𝑞𝑑𝑥
1
𝑞∫ℝ𝑁𝑚(𝑥)|𝑢|𝑞𝑑𝑥 ∶𝑢∈𝐸∖{0},∫ℝ𝑁
𝑚(𝑥)|𝑢|𝑞𝑑𝑥 > 0⎫
⎪
⎬
⎪
⎭
.
Rema k 3.3. The a gumen sde elopedin hispape allow oob ainsimila esul si hehypo hesis
(𝐻) is eplaced by he ollowing condi ion in oduced by Szulkin and Willem [27],
()𝑚∈𝐿
1
𝑙𝑜𝑐(ℝ𝑁),𝑚+=𝑚
1+𝑚
2≠0,𝑚1∈𝐿
𝑁
𝑞(ℝ𝑁), o e e y 𝑦∈ℝ𝑁,lim𝑥→𝑦 |𝑥−
𝑦|𝑞𝑚2(𝑥) = 0 and lim|𝑥|→∞ |𝑥|𝑞𝑚2(𝑥) = 0, whe e 𝑚+∶= max{𝑚(𝑥), 0}.
ACKNOWLEDGEMENTS
T. Gou was suppo ed by he Na ional Na u al Science Founda ion o China (No. 12101483) and
he Pos doc o al Science Founda ion o China (No. 2021M702620). V.D. Rădulescu was suppo ed
by he g an “Nonlinea Di e en ial Sys ems in Applied Sciences” o he Romanian Minis y o
Resea ch, Inno a ion and Digi iza ion, wi hin PNRR-III-C9-2022-I8/22. The au ho s would like
o hank wa mly he anonymous e e ees o hei e y p ecise eading o ou pape and o gi ing
cons uc i e commen s and sugges ions.
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ORCID
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REFERENCES
1. W. Alleg e o and Y. X. Huang, Eigen alues o he inde ini e-weigh p-Laplacian in weigh ed spaces,Funkcial.
Ek ac 38 (1995), no. 2, 233–242.
2. A. Amb ose i and A. Malchiodi, Nonlinea analysis and semilinea ellip ic p oblems, Camb idge Uni e si y
P ess, Camb idge, 2007.
3. P. Ba oni, M. Colombo, and G. Mingione, Regula i y o gene al unc ionals wi h double phase, Calc. Va . Pa ial
Di e en ial Equa ions 57 (2018), no. 2, Pape No. 62, 48 pp.
4. N. Benouhiba and Z. Belyacine, A class o eigen alue p oblems o he (p, q)-Laplacian in ℝ𝑁, In . J. Pu e Appl.
Ma h. 80 (2012), no. 5, 727–737.
5. N. Benouhiba and Z. Belyacine, On he solu ions o he (𝑝, 𝑞)-Laplacian p oblem a esonance, Nonlinea Anal.
77 (2013), 74–81.
6. M. Boche , The smalles cha ac e is ic numbe s in a ce ain excep ional case, Bull. Ame . Ma h. Soc. 21 (1914),
6–9.
7. L. Che ils and Y. Il’yaso , On he s a iona y solu ions o gene alized eac ion di usion equa ions wi h 𝑝&𝑞-
Laplacian, Commun. Pu e Appl. Anal. 4(2005), no. 1, 9–22.
8. F. Colasuonno and M. Squassina, Eigen alues o double phase a ia ional in eg als, Ann. Ma . Pu a Appl. (4)
(6) 195 (2016), 1917–1959.
9. M. Colombo and G. Mingione, Regula i y o double phase a ia ional p oblems, A ch. Ra ion. Mech. Anal. 215
(2015), no. 2, 443–496.
14692120, 2024, 2, Downloaded om h ps://londma hsoc.onlinelib a y.wiley.com/doi/10.1112/blms.12961 by B no Uni e si y O Technology, Wiley Online Lib a y on [13/05/2024]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License

NON-AUTONOMOUS DOUBLE PHASE EIGENVALUE PROBLEMS 755
10. G. Cupini, P. Ma cellini, and E. Mascolo, Local boundedness o weak solu ions o ellip ic equa ions wi h 𝑝,𝑞-
g ow h, Ma h. Eng. 5(2023), no. 3, Pape No. 065, 28 pp.
11. L. Diening, P. Ha juleh o, P. Häs ö, and M. Ruzicka, Lebesgue and Sobole spaces wi h a iable exponen s,
Sp inge , Heidelbe g, 2011.
12. A. Fiscella, A double phase p oblem in ol ing Ha dy po en ials, Appl. Ma h. Op im. 85 (2022), no. 3, Pape No.
32, 16 pp.
13. L. Gasiński and N. S. Papageo giou, Cons an sign and nodal solu ions o supe linea double phase p oblems
Ad . Calc. Va . 14 (2021), no. 4, 613–626.
14. L. Gasiński and P. Winke , Cons an sign solu ions o double phase p oblems wi h supe linea nonlinea i y,
Nonlinea Anal. 195 (2020), 111739, pp. 9.
15. B. Ge, D. L , and J. Lu, Mul iple solu ions o a class o double phase p oblem wi hou he Amb ose i-Rabinowi z
condi ion, Nonlinea Anal. 188 (2019), 294–315.
16. P. Ha juleh o and P. Häs o, O licz spaces and gene alized O licz spaces, Lec u es No es in Ma hema ics, ol.
2236, Sp inge , Tu ku, 2019.
17. P. Hess and T. Ka o, On some linea and nonlinea eigen alue p oblems wi h inde ini e weigh unc ion, Comm.
Pa ial Di e en ial Equa ions 5(1980), 999–1030.
18. P. Ma cellini, Regula i y and exis ence o solu ions o ellip ic equa ions wi h 𝑝,𝑞-g ow h condi ions, J. Di e en ial
Equa ions 90 (1991), 1–30.
19. P. Ma cellini, E e ywhe e egula i y o a class o ellip ic sys ems wi hou g ow h condi ions, Ann. Sc. No m.
Supe . Pisa Cl. Sci. (4) 23 (1996), 1–25.
20. P. Ma cellini, G ow h condi ions and egula i y o weak solu ions o nonlinea ellip ic PDEs, J. Ma h. Anal.
Appl. 501 (2021), no. 1, Pape No. 124408, 32 pp.
21. P. Ma cellini, Local Lipschi z con inui y o 𝑝,𝑞-PDEs wi h explici 𝑢-dependence Nonlinea Anal. 226 (2023),
Pape No. 113066, 26 pp.
22. N. S. Papageo giou, A. Pudełko, and V. D. Rˇ
adulescu, Non-au onomous (𝑝, 𝑞)-equa ions wi h unbalanced
g ow h, Ma h. Ann. 385 (2023), no. 3-4, 1707–1745.
23. N. S. Papageo giou, V. D. Rˇ
adulescu, and Y. Zhang, Resonan double phase equa ions, Nonlinea Anal. Real
Wo ld Appl. 64 (2022), Pape No. 103454, 20 pp.
24. N. S. Papageo giou, C. Ve o, and F. Ve o, Mul iple solu ions o pa ame ic double phase Di ichle p oblems,
Commun. Con emp. Ma h. 23 (2021), no. 4, Pape No. 2050006, 18 pp.
25. A. Pleijel, On he eigen alues and eigen unc ions o elas ic pla es, Comm. Pu e Appl. Ma h. 3(1950), 1–10.
26. T. Singe , Exis ence o weak solu ions o pa abolic sys ems wi h 𝑝,𝑞-g ow h, Manusc ip a Ma h. 151 (2016), 87–
112.
27. A. Szulkin and M. Willem, Eigen alue p oblems wi h inde ini e weigh , S udia Ma h. 135 (1999), 191–201.
14692120, 2024, 2, Downloaded om h ps://londma hsoc.onlinelib a y.wiley.com/doi/10.1112/blms.12961 by B no Uni e si y O Technology, Wiley Online Lib a y on [13/05/2024]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License