Extension of Díaz-Saá's inequality in RN and application to a system of p-Laplacian

Publ. Ma . 46 (2002), 473–488
EXTENSION OF D´
IAZ-SA´
A’S INEQUALITY IN RNAND
APPLICATION TO A SYSTEM OF p-LAPLACIAN
Ka im Cha¨
ıb
Abs ac
The pu pose o his pape is o ex end he D´ıaz-Sa´a’s inequali y
o he unbounded domains as RN:
RN−∆pu
up−1+∆p
p−1(up− p)dx ≥0
wi h ∆pu= di |∇u|p−2∇u.
The p oo is based on he Picone’s iden i y which is e y use ul
in p oblems in ol ing p-Laplacian. In a second pa , we s udy
some p ope ies o he fi s eigen alue o a sys em o p-Laplacian.
We use D´ıaz-Sa´a’s inequali y o p o e uniqueness and Ego o ’s
heo em o he isola ion. These esul s gene alize J. Fleckinge ,
R. F. Man´ase ich, N. M. S a akakis and F. de Th´elin’s wo k [9]
o he fi s p ope y and A. Anane’s one o he isola ion.
In a well-known pape [7]H.B ´ezis and L. Oswald ob ain some nec-
essa y and sufficien condi ions o he exis ence and uniqueness o a
posi i e solu ion o he equa ion:
−∆u= (x, u)inΩ,u=0on∂Ω
when Ω is a bounded open se in RN.
La e J. I. D´ıaz and J. E. Sa´a[8] ex end hese esul s o he case o an
equa ion in ol ing he p-Laplacian, ∆pu= di |∇u|p−2∇u(p>1). In
hei p oo a undamen al ing edien is he so called D´ıaz-Sa´a inequali y:
(1) Ω−∆pz1
zp−1
1
+∆pz2
zp−1
2(zp
1−zp
2)dx ≥0
whe e ∆pu= di (|∇u|p−2∇u).
2000 Ma hema ics Subjec Classifica ion. 35J65 (35P30, 35B05).
Key wo ds. D´ıaz-Sa´a inequali y, Picone iden i y, p-Laplacian, ellip ic sys em, nonlin-
ea eigen alue, unbounded domain.
474 K. Cha¨
ıb
This inequali y exp esses he mono ony o he ope a o w→ −∆pw
1
p
w
p−1
p
.I
has been p o ed in [8] unde he ollowing hypo heses:
Fo i=1,2, zi∈L∞(Ω) ∩W1,p(Ω) such ha zi≥0a.e.onΩand
∆pzi∈L∞(Ω);
Fo i=j,i, j =1,2, zi
zj∈L∞(Ω).
In he case o bounded domains, he Hop ’s Maximum P inciple gi es
hem he hypo hesis zi
zj∈L∞(Ω) which is necessa y in he o iginal
p oo o (1). Bu when we conside unbounded domains such as RN his
condi ion is no e ified in gene al.
He e, unde some adap ed hypo heses o solu ions o ellip ic p ob-
lems, we es ablish an inequali y o D´ıaz-Sa´a ype which emains ue in
whole RN. When we say he hypo heses a e well adap ed, i means ha
hey na u ally appea in weigh ed p oblems in ol ing he p-Laplacian
on RN.
In he second pa , we apply his D´ıaz-Sa´a ype inequali y o ob ain
an uniqueness esul o he fi s eigen alue o a sys em o p-Laplacian
in RN
(Sλ)






−∆pu=λb(x)|u|α| |β x∈RN
−∆q =λb(x)|u|α| |βux∈RN
lim|x|→+∞u(x)=0=lim
|x|→+∞ (x).
This wo k ollows an A. Anane’s pape [3] whe e he s udies he case
o one equa ion and J. Fleckinge , R. F. Man´ase ich, N. M. S a akakis
and F. de Th´elin [9] whe e a sys em qui e diffe en is conside ed. In [9],
a local me hod is p esen ed o con ol a possible blowing-up o he quo-
ien zi
zjin RN. Bu , in he in eg a ion by pa s, some in eg als on he
bounda y appea which make he demons a ion qui e bo ing.
Finally, ollowing A. Anane’s a icle [3], we will conclude his pape
by showing he isola ion o he fi s eigen alue o he sys em (Sλ). As
in his a icle, we will use he Ego o ’s heo em. This p ope y was no
p o ed in [9]; besides i seems ha he me hod used he e canno be
applied o hei sys em because in ui i ely he sign o umus depend
on he sign o and ice e sa.
Ex ension o D´
ıaz-Sa´
a’s Inequali y in RN475
1. D´ıaz-Sa´a inequali y
Deno e by D1,p(RN) he closu e o C∞
0(RN) o he Lp(RN) no m o
he g adien
||u||p
D1,p(RN)=RN
|∇u|pdx.
The ollowing esul s a e he consequences o he p incipal heo ems
which will be p esen ed la e .
P oposi ion 1 (D´ıaz-Sa´a inequali y on RNin he case 1<p<N).Fo
i=1,2, le zi∈D1,p(RN)such ha zi≥0(≡ 0) and diffe en iable.
Then we ha e
RN−∆pz1
zp−1
1
+∆pz2
zp−1
2(zp
1−zp
2)dx ≥0
i we assume ha ∆pzi
zp−1
i
∈LN
p(RN)∩L∞
loc(RN) o i=1,2.
I we ha e equali y hen he e exis s a cons an Csuch ha z1=Cz2.
P oposi ion 2 (D´ıaz-Sa´aonRNin he case p≥N).Fo i=1,2, le
zi∈W1,p(RN)such ha zi≥0(≡ 0) and diffe en iable. Then he
asse ions abo e emain ue i we assume ha o N=p, he e exis s
some s>1such ha
∆pzi
zp−1
i
∈Ls(RN)∩L∞
loc(RN)
o o N<p,
∆pzi
zp−1
i
∈L1(RN)∩L∞
loc(RN)
wi h i=1,2.
The p oo o he heo ems ha we will p esen he e needs Picone’s
iden i y o he p-Laplacian desc ibed o example in [2] by W. Alleg e o
and Y. X. Huang. Picone’s esul is an equali y ue almos e e ywhe e
which a oids some in eg a ion p oblems when we wo k in unbounded
open se s. Now, we p esen a Picone’s iden i y o he p-Laplacian qui e
mo e gene al han he W. Alleg e o and Y. X. Huang’s one because we
only assume a sign condi ion o one o he wo equa ions.
476 K. Cha¨
ıb
P oposi ion 3 (Gene alized Picone’s iden i y o he p-Laplacian).Le
u, diffe en iable and >0a. e. on Ωa subse o RN. No e
L(u, )=|∇u|p+(p−1)|u|p
p|∇ |p−p|u|p−2u
p−1∇u·∇ |∇ |p−2a. e. on Ω
R(u, )=|∇u|p−∇|u|p
p−1|∇ |p−2∇ a. e. on Ω.
Then L(u, )=R(u, )≥0.
Mo eo e , L(u, )=0a. e. on Ωi and only i ∇u
=0a. e. on Ω.
P oo o P oposi ion 3: By a simple calcula ion, we show ha R(u, )=
L(u, ), because
∇|u|p
p−1=p|u|p−2u∇u
p−1−(p−1)|u|p∇
p.
To p o e posi i i y, we obse e ha
|u|p−2u
p−1∇u·∇ |∇ |p−2≤|u|p−1
p−1|∇ |p−1|∇u|(2)
and by Young’s inequali y
p|u|p−1
p−1|∇ |p−1|∇u|≤(p−1)|u|p
p|∇ |p+|∇u|p.(3)
Hence by (2) and (3)
|∇u|p+(p−1)|u|p
p|∇ |p−p|u|p−2u
p−1∇u·∇ |∇ |p−2=L(u, )≥0.
Mo eo e , i L(u, ) = 0 hen by (2) and (3)
|∇u|p+(p−1)|u|p
p|∇ |p−p|u|p−1
p−1|∇ |p−1|∇u|= 0 a. e. on Ω.(4)
We define N:= x∈Ω such ha |u|
|∇ |=0
⊂Ω.
On N, by he equa ion (4) we ha e
|u|
|∇ |=|∇u|=0 ae. onN
and hence
u
∇ =∇u= 0 a. e. on N.(5)
Ex ension o D´
ıaz-Sa´
a’s Inequali y in RN477
On Nc, we no e Q:= |∇u|
|∇ ||u|
and subs i u ing in (4) we ob ain
Qp−pQ +p−1=0 iffQ= 1 because p>1,
i. e. |∇u|=|u|
|∇ |a. e. on Nc.(6)
Using (6) in L(u, ) = 0, i ollows ha
|∇u|p+(p−1)|∇u|p−2|∇u|2−p∇u·∇ u
|∇u|p−2= 0 a. e. on Nc,
so ∇u·∇u−∇ u
= 0 a. e. on Nc,
and u
∇ =∇uae.onNcbecause |∇u|=|u|
|∇ |.(7)
Indeed ∇ canno be pe pendicula o u
∇ −∇ubecause i would signi y
ha ∇uis he hypo enuse o a igh iangle wi h edges u
∇ and u
∇ −
∇u, ha is no possible because u
|∇ |=|∇u|.
By (5) and (7) we ha e u
∇ =∇ua. e. on Ω and finally,
∇u
= 0 a. e. on Ω.
We can easily ema k ha in Picone’s iden i y, we can ake any se Ω,
o example non connec ed and unbounded. Now, we can es ablish he
ollowing heo em whose p incipal a gumen o he p oo is he abo e
iden i y.
Theo em 1. Fo 1<p<N.Le Φin D1,p(RN)and z≥0(≡ 0) in
D1,p(RN)bo h diffe en iable.
Then we ha e
RN
|∇Φ|pdx ≥RN−∆pz
zp−1|Φ|pdx,
i we assume ha ∆pz
zp−1∈LN
p(RN)∩L∞
loc(RN).
Mo eo e , in he equali y case he e exis s Csuch ha z=CΦon RN.
P oo o Theo em 1: Fi s we ema k ha z∈D1,p(RN) is a non i ial
solu ion o he p oblem
−∆p =−∆pz
zp−1 p−1
≥0in RN.

478 K. Cha¨
ıb
Seeing ha ∆pz
zp−1∈L∞
loc(RN), we can apply V´azquez’s S ong Maximum
P inciple [16] o p o e ha z>0onRN. Mo eo e , we use P. Tolks-
do ’s egula i y heo em [15] o show ha o all >0, he e exis s
α( )>0 such ha z∈C1,α(B ). In pa icula o Ω0a bounded do-
main o RN, he e exis s α0>0 such ha z∈C1,α0(Ω0).
Le (Φn)n∈Na sequence o unc ions in C∞
0(RN) such ha (Φn)n∈N
con e ges o Φ in D1,p(RN). We apply Picone’s iden i y o he unc-
ions Φnand z
0≤RN
L(Φn,z)dx ≤RN
R(Φn,z)dx
≤RN
|∇Φn|pdx −RN
∇|Φn|p
zp−1|∇z|p−2∇z dx.
Bu Φn∈C∞
0(RN) and z>0 hen |Φn|p
zp−1is an admissible unc ion es ,
in eg a ing by pa s we ob ain
0≤RN
|∇Φn|pdx +RN
∆pz
zp−1|Φn|pdx.
(Φn)n∈Ncon e ges o Φ in D1,p(RN), so we ha e ha (Φn)n∈Ncon-
e ges o Φ in Lp∗(RN) and (|Φn|p)n∈Ncon e ges o |Φ|pin Lp∗
p(RN).
Consequen ly,
RN
∆pz
zp−1(|Φn|p−|Φ|p)dx≤



∆pz
zp−1


L
N
p(RN)
|Φn|p−|Φ|p
L
p∗
p(RN)
  
ends o 0
.(8)
And he esul ollows:
RN
|∇Φ|pdx ≥RN−∆pz
zp−1|Φ|pdx.(9)
We now conside he equali y case
RN
|∇Φ|pdx =RN−∆pz
zp−1|Φ|pdx.
Le Ω0a bounded domain in RNand (Φn)n∈Ndefined as be o e.
0≤Ω0
L(Φn,z)dx ≤RN
L(Φn,z)dx
≤RN
|∇Φn|pdx +RN
∆pz
zp−1|Φn|pdx ends o 0 when n→∞.
Ex ension o D´
ıaz-Sa´
a’s Inequali y in RN479
Bu Ω0L(Φn,z)dx con e ges o Ω0L(Φ,z)dx because Ω0is bounded
and z∈C1,α0(Ω0). So L(Φ,z)=0a.e.onΩ
0, he se Ω0being aken
a bi a y in RNwe can conclude ha L(Φ,z) = 0 a. e. on RN. And,
by Picone’s iden i y (P oposi ion 3), he e exis s C>0 such ha Φ =
Cz.
Theo em 2. Fo p≥N.Le Φin W1,p(RN)and z≥0(≡ 0) in
W1,p(RN)bo h diffe en iable.
Then we ha e
RN
|∇Φ|pdx ≥RN−∆pz
zp−1|Φ|pdx,
i we assume ha o p=N, he e exis s some s>1such ha
∆pz
zp−1∈Ls(RN)∩L∞
loc(RN)
o o p>N,
∆pz
zp−1∈L1(RN)∩L∞
loc(RN).
Mo eo e , i he abo e in eg al is ze o hen he e exis s Csuch ha
z=CΦon RN.
The p oo o his heo em is qui e simila o Theo em 1. We sol e
he p oblem o con e gence in (8) by he pa icula embeddings o he
space W1,p(RN) in he case p≥N(see [6]):
I p=N,W1,p(RN) is con inuously embedded in any Lq(RN)
whe e q∈[p, +∞);
I p>N,W1,p(RN) is con inuously embedded in L∞(RN).
The exis ence esul o a solu ion in W1,p(RN) o he p-Laplacian
whe e p≥Nhas been es ablished by W. Alleg e o and Y. X. Huang
in [1].
P oo s o P oposi ion 1 and P oposi ion 2 (D´ıaz-Sa´a inequali y): We only
ha e o apply he p eceding esul s o he couple o he unc ions (z1,z
2)
and we ha e
RN
|∇z1|pdx ≥RN−∆pz2
zp−1
2zp
1dx
and RN−∆pz1
zp−1
1
+∆pz2
zp−1
2zp
1dx ≥0.
480 K. Cha¨
ıb
Doing he same wo k o he couple o he unc ions (z2,z
1) and adding
he wo inequali ies ob ained we a i e o he expec ed esul . We no e
ha in D´ıaz-Sa´a inequali y, we assume unc ions a e posi i es so we
needn’ he absolu e alues as in he heo ems.
Following a discussion wi h P. Tak´aˇc, i appea s ha D´ıaz-Sa´a in-
equali y in RNand bo h heo ems can be p o ed using e y ca e ully
J. I. D´ıaz and J. E. Sa´a’s me hod. Fo his, we ha e o ema k fi s
ha i his inequali y is ue o posi i e unc ions Φ i emains ue o
unc ions Φ changing sign [11, “Chain ule”, Lemma 7.6]. A e ha ,
we jus ha e o p o e he con exi y o he applica ion w−→ |∇w1
p|p
and compu e he di ec ional de i a i e o J(w)=Ω|∇w1
p|pdx which is
o mally:
J(w) =1
pΩ
|∇w1
p|p−2∇w1
p∇(w1
p−1 )dx.
To ha e mo e de ails on his me hod we can see J. Fleckinge , J. He -
n´andez, P. Tak´aˇc and F. de Th´elin’s a icle [10]. Bu ha p oo needs
a good a en ion in he compu a ion because some e ms, in pa icula
he de i a i e o J, ha e o be defined co ec ly, and we hink ha ou
p oo using Picone’s iden i y is easie .
2. P ope y o fi s eigen alue o a sys em o
p-Laplacian
In his second pa , we s udy some p ope ies o he fi s eigen alue
o a po en ial sys em o p-Laplacian:
(Sλ)






−∆pu=λb(x)|u|α| |β x∈RN
−∆q =λb(x)|u|α| |βux∈RN
lim|x|→+∞u(x)=0=lim
|x|→+∞ (x).
And we assume:
(H1) N>p>1,N>q>1,α≥0,β≥0,α+1
p+β+1
q=1
and α+β+2<N.
(H2) b∈C0,γ
loc (RN) wi h γ∈(0,1),
b∈LN
α+β+2 (RN)∩L∞(RN) and b≥0(≡ 0).
Fo a s a , we es ablish exis ence o a fi s eigen alue o (Sλ) and
he egula i y o he associa ed eigen unc ions.
Ex ension o D´
ıaz-Sa´
a’s Inequali y in RN481
Theo em 3. We suppose ha Hypo heses (H1) and (H2) a e sa isfied.
(i) Sys em (Sλ)admi s a fi s eigen alue λ1which is posi i e and de-
fined by
λ1= in
Γα+1
pRN
|∇u|pdx +β+1
qRN
|∇ |qdx
whe e Γ=x∈RNsuch ha RN
b(x)|u|α| |βu dx =1
;
(ii) I (u, )is a couple o eigen unc ions solu ion o (Sλ1) hen o all
>0,u∈C1,ρ(B )and ∈C1,γ(B )whe e ρ=ρ( )>0and
γ=γ( )>0;
(iii) The e exis s a couple o eigen unc ions solu ion o (Sλ1)which a e
posi i e on RN.
The p oo o his heo em is mo e o less he same as J. Fleckinge ,
R. F. Man´ase ich, N. M. S a akakis and F. de Th´elin’s one in [9] o
he sys em:
−∆pu=λb(x)|u|α−1u| |β+1 x∈RN
−∆q =λb(x)|u|α+1| |β−1 x∈RN.
The app oach done in [9] is s anda d because hei p oblem as (Sλ)is
a ia ional.
We p esen now an uniqueness and isola ion esul o he fi s eigen-
alue. The p oo is an in e es ing applica ion o Theo em 1 and is simple
han in [9].
Theo em 4. We suppose ha Hypo heses (H1) and (H2) a e sa isfied.
(i) In he se o con inuous unc ions, he dimension o eigenspace
co esponding o p incipal eigen alue λ1is 1;
(ii) λ1is he only one eigen alue o (Sλ1)which co esponds o a con-
s an sign eigen ec o ;
(iii) λ1is isola ed i. e. he e exis s #>0such ha o all λ∈(λ1,λ
1+#]
he sys em (Sλ)has no solu ion.
488 K. Cha¨
ıb
[6] H. B ´
ezis,“Analyse onc ionnelle. Th´eo ie e applica ions”, Col-
lec ion Ma h´ema iques Appliqu´ees pou la Maˆı ise, Masson, Pa is,
1983.
[7] H. B ´
ezis and L. Oswald, Rema ks on sublinea ellip ic equa-
ions, Nonlinea Anal. 10(1) (1986), 55–64.
[8] J. I. D´
ıaz and J. E. Sa´
a, Exis ence e unici ´e de solu ions posi i es
pou ce aines ´equa ions ellip iques quasilin´eai es, C. R. Acad. Sci.
Pa is S´e . I Ma h. 305(12) (1987), 521–524.
[9] J. Fleckinge , R. F. Man´
ase ich, N. M. S a akakis and
F. de Th´
elin, P incipal eigen alues o some quasilinea ellip-
ic equa ions on RN,Ad . Diffe en ial Equa ions 2(6) (1997),
981–1003.
[10] J. Fleckinge , J. He n´
andez, P. Tak´
aˇ
c and F. de Th´
elin,
Uniqueness and posi i i y o solu ions o equa ions wi h he
p-Laplacian, in: “Reac ion diffusion sys ems” (T ies e, 1995), Lec-
u e No es in Pu e and Appl. Ma h. 194, Dekke , New Yo k, 1998,
pp. 141–155.
[11] D. Gilba g and N. S. T udinge ,“Ellip ic pa ial diffe en ial
equa ions o second o de ”, second edi ion, G undleh en de Ma h-
ema ischen Wissenscha en 224, Sp inge -Ve lag, Be lin, 1983.
[12] O. A. Ladyzhenskaya and N. N. U al’ se a,“Linea and
quasilinea ellip ic equa ions”, Academic P ess, New Yo k-London,
1968.
[13] J. Se in, Local beha io o solu ions o quasi-linea equa ions,
Ac a Ma h. 111 (1964), 247–302.
[14] F. de Th´
elin, P emi`e e aleu p op e d’un sys `eme ellip ique non
lin´eai e, Re . Ma . Apl. 13(1) (1992), 1–8.
[15] P. Tolksdo , Regula i y o a mo e gene al class o quasilinea
ellip ic equa ions, J. Diffe en ial Equa ions 51(1) (1984), 126–150.
[16] J. L. V´
azquez, A s ong maximum p inciple o some quasilinea
ellip ic equa ions, Appl. Ma h. Op im. 12(3) (1984), 191–202.
Ma h´ema iques pou l’Indus ie e la Physique
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