A SUB-SUPERSOLUTION METHOD FOR NONLINEAR
ELLIPTIC SINGULAR SYSTEMS WITH NATURAL GROWTH
AND SOME APPLICATIONS
JOS´
E CARMONA, PEDRO J. MART´
INEZ-APARICIO, AND ANTONIO SU´
AREZ
Abs ac . In his pape we gi e a sub-supe solu ion me hod o nonlinea
ellip ic singula sys ems wi h quad a ic g adien whose model sys em is he
ollowing











−∆u+ β|∇u|2
uα= 1(x, u, ) in Ω,
−∆ +uµ|∇ |2
γ= 2(x, u, ) in Ω,
u= = 0 on ∂Ω,
whe e Ω is a smoo h bounded domain o RN(N≥3), β, µ ≥0, 0 < α, γ < 1
and egula 1, 2 unc ions. Mo eo e , we apply i o p o e exis ence o solu-
ion o some sys ems, including he classical Lo ka-Vol e a models wi h g a-
dien e ms. Speci ically, we s udy he compe i ion and he symbio ic Lok a-
Vol e a sys ems.
1. In oduc ion
The aim o his pape is o p o ide a sub-supe solu ion me hod o he ollowing
nonlinea ellip ic singula sys em wi h na u al g ow h
(1.1) 












−∆u+g1( )|∇u|2
uα= 1(x, u, ) in Ω,
−∆ +g2(u)|∇ |2
γ= 2(x, u, ) in Ω,
u= = 0 on ∂Ω,
whe e Ω is a smoo h bounded domain o RN(N≥3) he unc ions g1, g2∈
C([0,+∞)) and 1, 2∈C(Ω ×[0,+∞)×[0,+∞)) e i ying some gene al con-
di ions de ailed below.
Rega ding he li e a u e he e a e se e al pape s abou equa ions wi h quad a ic
g adien e ms. The exis ence o solu ions o he equa ion
(1.2) −∆u+g(u)|∇u|2=a(x) in Ω, u= 0 on ∂Ω,
o e e y unc ion a(x) in a gi en Lebesgue space has been sys ema ically s ud-
ied in [4, 5, 6] and e e ences he ein (in ac , o a mo e gene al nonlinea e m
H(x, u, ∇u) ins ead o g(u)|∇u|2). They conside in he lowe o de e m a con in-
uous gin Rwhich does no sa is y any g ow h es ic ion and he sign condi ion
g(s)s≥0 o e e y s∈Ris assumed. Thanks o he p esence o he lowe o -
de e m he Di ichle p oblem associa ed o he equa ion is allowed o ha e ini e
ene gy weak solu ions.
1
2 J. CARMONA, P. J. MART´
INEZ-APARICIO, AND A. SU´
AREZ
In [15] and [7] some o he abo e esul s we e ex ended o he case o sys ems.
Speci ically, in [7] he au ho s s udy sys ems o ellip ic equa ions wi h quad a ic
g adien . They conside a gene al sys em
(1.3) −∆ui+Hi(x, u, ∇u) = ai(x) in Ω, u= 0 on ∂Ω, i= 1, . . . , n
whe e u= (u1, . . . , un), ai∈H−1(Ω) and he quad a ic e ms Hi(x, u, ∇u) sa -
is y a mo e gene al one-side condi ion han he sign condi ion, bu in he case
Hi(x, u, ∇u) = gi(u)|∇u|2 his one-side hypo hesis is equi alen o he sign condi-
ion. In hei case giis con inuous in Rnand hey p o e he exis ence o solu ion
in he Sobole space.
In he las yea s, equa ion (1.2) has a ac ed much a en ion by he p esence o
singula e ms in on o he g adien , see [1, 2, 8] and e e ences he ein.
In [11] we p o e ha a sub-supe solu ion me hod wo ks o equa ions o he o m
−∆u+|∇u|2
uα= (λ, u)
and we apply i o di e en models.
In his pape we ocus ou a en ion in sys ems wi h quad a ic g adien and
singula e ms as (1.1).
Le us men ion ha he sub-supe solu ion me hod is alid o semilinea sys ems,
see o ins ance [13] and [20]. In his case, when g1≡g2≡0, he na u al ex ension
o he scala de ini ion o sub-supe solu ion depends on he mono onici y o he
unc ions 1and 2wi h espec o and u, espec i ely. A gene al de ini ion was
gi en in [13] and [20] whe e a pai o unc ions (u, ), (u, ), u, u, , ∈H1(Ω) ∩
L∞(Ω) is called a sub-supe solu ion i
u≤u, ≤ in Ω,
u≤0≤u, ≤0≤ on ∂Ω,
and
−∆u≤ 1(x, u, ),−∆u≥ 1(x, u, ),∀ ∈[ , ],
−∆ ≤ 2(x, u, ),−∆ ≥ 2(x, u, ),∀u∈[u, u],
whe e, gi en wo o de ed unc ions z≤w, we ha e deno ed
[z, w] := {q∈L∞(Ω) : z(x)≤q(x)≤w(x)}
(see also [19] whe e i is p o ed he alidi y o he me hod o singula semilinea
sys ems). Assuming he exis ence o a sub-supe solu ion, (u, ), (u, ), he e exis s
a solu ion (u, )∈I≡[u, u]×[ , ] o he semilinea sys em (i.e. (1.1) wi h
g1≡g2≡0).
When he eac ion e ms depend on he g adien , i.e.,
−∆u= 1(x, u, , ∇u, ∇ ),−∆ = 2(x, u, , ∇u, ∇ )
and he unc ions 1and 2a e egula e i ying some hypo heses, he de ini ion is
(see [21])
−∆u≤ 1(x, u, , ∇u, ∇ ),−∆u≥ 1(x, u, , ∇u, ∇ ),∀ ∈[ , ],
−∆ ≤ 2(x, u, , ∇u, ∇ ),−∆ ≥ 2(x, u, , ∇u, ∇ ),∀u∈[u, u].
Assuming again he exis ence o a sub-supe solu ion, he exis ence o a solu ion
(u, )∈I ollows.
NONLINEAR ELLIPTIC SINGULAR SYSTEMS WITH NATURAL GROWTH 3
In his pape , we use he abo e de ini ion o sub-supe solu ion, and aking ad an-
age o he o m o equa ion, we o e come he singula i ies di icul y o he sys em
(1.1) wi h espec o uand . Hence, o ou sys em (1.1) we de ine a couple o
sub-supe solu ion as ollows
−∆u+g1( )|∇u|2
uα≤ 1(x, u, ),−∆u+g1( )|∇u|2
uα≥ 1(x, u, ),∀ ∈[ , ],
−∆ +g2(u)|∇ |2
γ≤ 2(x, u, ),−∆ +g2(u)|∇ |2
γ≥ 2(x, u, ),∀u∈[u, u].
Mo eo e , we apply his me hod o p o e exis ence o posi i e solu ion o some
sys ems, including he classical Lo ka-Vol e a models con on ed wi h he Lapla-
cian ope a o pe u bed by a singula g adien e m, ha is, he ollowing sys ems
(1.4) 








−∆u+g1( )|∇u|2
uα=u(λ−u−b ) in Ω,
−∆ +g2(u)|∇ |2
γ= (µ− −cu) in Ω,
u= = 0 on ∂Ω,
whe e λ, µ ∈IR and b·c > 0. He e, u(x) and (x) deno e he densi ies o wo
species, λand µ ep esen he g ow h a es o he species, band cmeasu e he
in e ac ion a es be ween bo h species; i b, c > 0 hey a e compe ing and i b, c < 0
coope a ing. Mo eo e , in (1.4) a nonlinea con ec i e e m is included wi h a
singula e m. This e m is accompanied by a nonlinea unc ion depending on
he o he species. We gi e condi ions on λand µ ha assu e he exis ence o a
coexis ence s a e o (1.4), ha is, a solu ion wi h bo h componen s posi i e.
The s uc u e o he a icle is: in Sec ion 2 we s udy an auxilia y scala equa ion
ha we use in Sec ion 3 o p o e he alidi y o he sub-supe solu ion me hod o
(1.1). Sec ion 4 is de o ed o applica ions o he me hod.
2. An auxilia y scala equa ion
F izzing one o he unknown in each equa ion o (1.1) we a e led o conside he
scala bounda y alue p oblem
(2.1) 


−∆w+m(x)|∇w|2
wθ= (x, w) in Ω,
w= 0 on ∂Ω,
o con enien unc ions mand and a pa ame e 0 < θ < 1. In o de o use
mono one me hods o (2.1), as was poin ed ou in [11] o m(x) cons an , i is use ul
o conside a posi i e unc ion g∈C(0,+∞) such ha , deno ing G(u) = Ru
1g(s)ds,
he unc ion e−G(s)belongs o L1(0,1) and we de ine also Ψ by
Ψ(s) := Zs
0
e−G( )d , s > 0.
Obse e ha i he e exis s M≥0 such ha (x, s) + Ms is nondec easing o a.e.
x∈Ω hen (x, s)+MΨ(s)eG(s)is also nondec easing o a.e. x∈Ω. Thus, adding
he e m MΨ(w)eG(w)in (2.1) i becomes



−∆w+m(x)|∇w|2
wθ+MΨ(w)eG(w)= (x, w) + MΨ(w)eG(w)in Ω,
w= 0 on ∂Ω.
4 J. CARMONA, P. J. MART´
INEZ-APARICIO, AND A. SU´
AREZ
The e o e, in o de o s udy (2.1) using sub-supe solu ion, we need o es ablish a
compa ison p inciple o he p oblem
(2.2) 


−∆w+m(x)|∇w|2
wθ+MΨ(w)eG(w)= 0(x) in Ω,
w= 0 on ∂Ω,
whe e 0 < m(x)∈L∞(Ω), 0 < θ < 1 and 0 ≤ 0(x)∈L2N/(N+2)(Ω), 06≡ 0.
Obse e ha his p oblem is simila o ha s udied in [11] bu he e he unc ion g,
om which a e de ined Gand Ψ, is a bi a y and non necessa y ela ed wi h he
g adiend lowe o de e m. When M= 0, ha is,



−∆w+m(x)|∇w|2
wθ= 0(x) in Ω,
u= 0 on ∂Ω,
his p oblem has solu ion (see [8]) and i is unique (see [3]).
The concep o sub and supe -solu ion o he p oblem (2.2) is he ollowing:
De ini ion 2.1. A sub-solu ion o (2.2) is a unc ion u∈H1
0(Ω) such ha 0 < u
a.e. in Ω, |∇u|2
uγ, Ψ(u)eG(u)∈L1(Ω) and o e e y φ∈H1
0(Ω) ∩L∞(Ω), φ ≥0,
ZΩ
∇u· ∇φ+ZΩ
m(x)|∇u|2
uθφ+MZΩ
Ψ(u)eG(u)φ≤ZΩ
0(x)φ.
Simila ly u∈H1(Ω) such ha 0 < u a.e. in Ω, |∇u|2
uθ,Ψ(u)eG(u)∈L1(Ω) and o
e e y φ∈H1
0(Ω) ∩L∞(Ω), φ ≥0,
ZΩ
∇u· ∇φ+ZΩ
m(x)|∇u|2
uθφ+MZΩ
Ψ(u)eG(u)φ≥ZΩ
0(x)φ,
is called a supe -solu ion o (2.2). We say ha u∈H1
0(Ω) is a solu ion o (2.2) i
i is a sub and supe -solu ion o (2.2).
We ecall some classical esul s abou he egula i y o he equa ion (2.2) wi h
0(x) = (x, w(x)), ha is he non-linea equa ion
(2.3) 


−∆w+m(x)|∇w|2
wθ+MΨ(w)eG(w)= (x, w) in Ω,
w= 0 on ∂Ω.
Conc e ely, i can be deduced om [22] he ollowing wo lemmas, summa izing
some known L∞(Ω)-es ima es o sub-solu ions o (2.3). The i s one deals wi h a
subc i ical unc ion , he e he L∞(Ω)-es ima e ollows om a s anda d boo s ap
a gumen .
Lemma 2.2. Assume ha he e exis s C > 0such ha | (x, s)| ≤ C(1 + |s|q)
(q < (N+ 2)/(N−2)) o e e y s≥0, a.e. x∈Ω, and ha uis a sub-solu ion o
(2.3), hen u∈L∞(Ω).
Rema k 2.3. Once we ha e p o ed ha i is bounded, unde condi ions o he
p e ious lemma, we ha e ha any solu ion uis con inuous in Ω a guing as in
[14] (see Rema k 2.6 in [1] o a de ailed p oo ). Mo eo e , since ∂Ω is smoo h,
u∈C0,α(Ω) o some α∈(0,1).
NONLINEAR ELLIPTIC SINGULAR SYSTEMS WITH NATURAL GROWTH 5
Lemma 2.4. Assume ha he e exis s s0such ha (x, s)≤0a.e. x∈Ωand o
e e y s > s0. Assume also ha uis a sub-solu ion o (2.3), hen u∈L∞(Ω) and
kuk∞≤s0.
We can p o e ha he p oblem (2.2) has solu ion a guing as in Lemma 3.3 o
[11]. We include he e a ske ch o he p oo in o de o show how o deal wi h he
e m m(x)∈L∞(Ω).
Lemma 2.5. Assume ha g∈L1(0,1). Then he e exis s a solu ion o (2.2).
P oo . We use an app oxima i e scheme, namely
(2.4) −∆un+Bn(x, un,∇un) = min{ 0(x), n}in Ω,
un= 0 on ∂Ω,
whe e he unc ion Bn(x, s, p) is gi en, o e e y (x, s, p)∈Ω×R×RNand n∈N,
by
Bn(x, s, p) = m(x)s+|p|2
(1
n+s+)θ+1(1 + 1
n|p|2)+
+MeRs+
1g( +1
n)d Rs+
0e−R
1g(σ+1
n)dσd
1 + 1
neRs+
1g( +1
n)d Rs+
0e−R
1g(σ+1
n)dσd
.
Since Bn(x, s, p)s≥0 and Bn(x, s, p)≤ kmkL∞(Ω)n(nθ+M), he exis ence o
solu ion un∈H1
0(Ω) o (2.4) is deduced om [16]. Mo eo e , since Bn(x, s, p)≥0
hen −∆un≤ 0(x) and, using [22], un∈L∞(Ω) and he sequence unis bounded
in L∞(Ω), ha is, he e exis s R > 0 such ha
kunkL∞(Ω) ≤R.
Mo eo e , aking u−
nas es unc ion we ob ain ha un≥0. Simila ly, aking un
as es unc ion and using he posi i i y o he lowe o de e m we ge ha un
is bounded in H1
0(Ω). E en mo e, aking min{un, ε}/ε as es unc ion and using
Fa ou Lemma as ε→0 yields ha
ZΩ
Bn(x, un,∇un)≤ k 0k1.
The e o e unweakly con e ges o u∈H1
0(Ω), ∇un→ ∇ua.e. (see [9, Theo em
2.1]) and using Fa ou Lemma as n→ ∞,
m(x)|∇u|2
uθχ{u>0}∈L1(Ω) and Ψ(u)eG(u)χ{u>0}∈L1(Ω).
In pa icula , since gis in eg able a ze o, we ha e ha Ψ(u)eG(u)is bounded a
ze o and hus, Ψ(u)eG(u)∈L1(Ω).
In o de o pass o he limi and o p o e ha uis he solu ion o (2.2) i is
essen ial o p o e ha u > 0. In o de o do ha we ollow he ideas in [8]. Gi en
˜m≥ kmkL∞(Ω) we ake e−˜mRun
1
1
θd φ, wi h 0 ≤φ∈C∞
0(Ω), as es unc ion in
(2.4) and we ob ain
ZΩ
e−˜mRun
1
1
θd ∇un· ∇φ+ZΩBn(x, un,∇un)−˜m
uθ
ne−˜mRun
1
1
θd φ=
=ZΩ
min{ 0(x), n}e−˜mRun
1
1
θd φ≥ZΩ
min{ 0(x),1}e−˜mRun
1
1
θd φ.(2.5)

6 J. CARMONA, P. J. MART´
INEZ-APARICIO, AND A. SU´
AREZ
We can use now ha o 0 < s < R
Bn(x, s, p)−˜m
sθe−Rs
1
1
θd ≤m(x)−˜m
sθ|p|2e−Rs
1
1
θd +(2.6)
+MeRs
1(g( +1
n)−˜m
θ)d Zs
0
e−R
1g(σ+1
n)dσd ≤
≤MeRs
1(g( +1
n)−˜m
θ)d Zs
0
e−R
1g(σ+1
n)dσd ≤
≤CZs
0
e−˜mR
1
1
σθdσd .
The las inequali y is due o he ac ha
eRs
1(g( +1
n)−˜m
θ)d =eRs+1
n
1+ 1
n
g(σ)dσ−Rs
1
˜m
θd ≤eRR+1
1g(σ)dσ+R1
0
˜m
θd
and
Zs
0
e−R
1g(σ+1
n)dσd =Zs
0
eR
1(˜m
σθ−g(σ+1
n))dσe−˜mR
1
1
σθdσd ≤
≤eRR+1
1
˜m
σθdσ+R1
0g(σ+1
n)dσ Zs
0
e−˜mR
1
1
σθdσd ≤
≤eRR+1
1
˜m
σθdσ+R2
0g(σ)dσ Zs
0
e−˜mR
1
1
σθdσd .
Thus, we can ake C=MeRR+1
1g(σ)dσ+R1
0
˜m
θd eRR+1
1
˜m
σθdσ+R2
0g(σ)dσ, using (2.6) in
(2.5) and deno ing ˜
Ψ(s) = Rs
0e−˜mR
1
1
σθdσd we ge
ZΩ
∇˜
Ψ(un)· ∇φ+˜
CZΩ
˜
Ψ(un)φ≥ZΩ
min{ 0(x),1}e−˜mRun
1
1
θd φ.
F om now on he p oo deals exac ly as in [11]. Passing o he limi in he
p e ious inequali y i ollows ha
ZΩ
∇˜
Ψ(u)· ∇φ+˜
CZΩ
˜
Ψ(u)φ≥ZΩ
min{ 0(x),1}e−˜mRu
1
1
θd φ.
Thus, he s ong maximum p inciple allows us o assu e ha 0 <˜
Ψ(u)≤
eR1
0
˜m
σθdσu, in pa icula u > 0, and we can o pass o he limi in he app ox-
ima ed p oblem o deduce ha u∈H1
0(Ω) is a solu ion o (2.2) a guing as in
[8]. 
Respec o he uniqueness we p o e below a compa ison p inciple o his equa-
ion ha assu es ha his solu ion is unique. This compa ison p inciple is one o
he keys ones o he p oo o ou me hod. In he case M= 0 i co espond o he
compa ison p inciple in [3, Co olla y 3.5]. We include he e, o he con enience o
he eade , he p oo o ha esul wi h he new e m MΨ(u)eG(u)a he le -hand
side o he equa ion, is ha o say, we p o e a compa ison p inciple o he p oblem
(2.2) al hough he p oo ollows wi h no signi ican change ha o Theo em 1.1 in
[3].
P oposi ion 2.6. Assume ha 0< θ < 1and 0< m(x)∈L∞(Ω). Le u, u ∈
C(Ω) be, espec i ely, a sub and a supe -solu ion o (2.2). Suppose also ha g∈
NONLINEAR ELLIPTIC SINGULAR SYSTEMS WITH NATURAL GROWTH 7
C1(0,+∞),e−G( )∈L1(0,1),g≥0and he e exis s τ≥0such ha o a.e. x∈Ω
and o e e y 0< s < max{kukL∞(Ω),kukL∞(Ω)}we ha e
(2.7) τ−g0(s)−m(x)θ
sθ+1 +−g(s) + m(x)
sθg(s)≥−g(s) + m(x)
sθ2
.
Then u≤u.
P oo . We will use he usual unc ion Gε(s) = (s−ε)+ o e e y s∈R. We also
de ine w= Ψ(u)−Ψ(u) and obse e ha Gε(w) is bounded and has compac
suppo in Ω. In pa icula , e−G(u), e−G(u), g(u), g(u) a e bounded in he suppo
o Gε(w). Thus, o nequal o he in ege pa o τ+ 1, we can ake e−G(u)Gε(w)n
as es unc ion in he inequali y sa is ied by uand e−G(u)Gε(w)nin he inequali y
sa is ied by u. Sub ac ing we ha e
0≥ZΩ−g(u) + m(x)
uθe−G(u)|∇u|2Gε(w)n−
−ZΩ−g(u) + m(x)
uθe−G(u)|∇u|2Gε(w)n+
+nZΩ
Gε(w)n−1(e−G(u)∇u−e−G(u)∇u)· ∇w,
whe e we ha e used ha
ZΩ
(Ψ(u)−Ψ(u))Gε(w)n≥0.
We deno e s= Ψ−1( Ψ(u) + (1 − )Ψ(u)) and ξ= ∇Ψ(u) + (1 − )∇Ψ(u), his
means ha
0≥Z{w>ε}
Gε(w)nZ1
0
d
d −g(s) + m(x)
sθeG(s)|ξ|2d +
+nZ{w>ε}
Gε(w)n−1|∇w|2.
Now we pe o m he de i a i e and we ge
0≥Z{w>ε}
wGε(w)nZ1
0−g0(s)−m(x)θ
sθ+1 e2G(s)|ξ|2d +
+Z{w>ε}
wGε(w)nZ1
0−g(s) + m(x)
sθg(s)e2G(s)|ξ|2d +
+Z{w>ε}
Gε(w)nZ1
0−g(s) + m(x)
sθeG(s)2ξ· ∇wd +
+nZ{w>ε}
Gε(w)n−1|∇w|2.
Mul iplying by τ
nand aking in o accoun ha , by Young’s inequali y,
τ
n
Gε(w)n−g(s) + m(x)
sθeG(s)2ξ· ∇w
≤
≤τ2
nGε(w)n−1|∇w|2+Gε(w)n+1
n−g(s) + m(x)
sθ2
e2G(s)|ξ|2,
8 J. CARMONA, P. J. MART´
INEZ-APARICIO, AND A. SU´
AREZ
i ollows ha
0≥τ1−τ
nZ{w>ε}
Gε(w)n−1|∇w|2+
+Z{w>ε}Z1
0
wGε(w)nτe2G(s)
nh−g0(s)−m(x)θ
sθ+1 |ξ|2+
+−g(s) + m(x)
sθg(s)|ξ|2−Gε(w)
τw −g(s) + m(x)
sθ2
|ξ|2id ≥0.
The las inequali y due o he ac ha Gε(w)/w ≤1, M≥0 and (2.7). We deduce
ha he in eg ands a e ze o, which implies ha Gε(w) = 0 o e e y ε > 0, i.e.,
w+≡0, concluding he p oo . 
The ollowing echnical esul plays an essen ial ole in he u he wo k, and i
was p o ed in [3, Co olla y 3.5] (see condi ion (3.6) o ha pape ).
Lemma 2.7. Fix m∈L∞(Ω),m > 0in Ωand ν > 0. Then, he e exis g∈
C1(0,∞)∩L1(0,1)1and τsuch ha i uand ua e a sub and a supe solu ion o
(2.2) such ha max{kukL∞(Ω),kukL∞(Ω)} ≤ ν,gsa is ies condi ion (2.7), and as
consequence
u≤u.
In ac , gdepends on kmkL∞(Ω) and θ, bu nei he νno τ. Speci ically, ixed
d, C, m1wi h 0<θ<d<1,C > 0and
m1≤min (dC, C d−θ
1−θ1−θ),
o any m∈L∞(Ω) wi h kmk∞≤m1we can choose g(s)≡gθ,d,C (s)gi en by
gθ,d,C(s) = 






dC
sθ, s < θ
C1
1−θ,
dθ
θs +θ
C1
1−θ(1 −θ)
, s ≥θ
C1
1−θ,
o e e y s > 0. Mo eo e , τis such ha
τ > max 






dC +m1
C(1 −d),2dm2
1ν2(1−θ)+θ2
(1 −d)θ2,
2m2
11−θ
d−θ2(1−θ)+ 2d2C2
d(1 −d)C21−m1
C1−θ
d−θ1−θ






.
3. The sub-supe solu ion me hod
Now, we a e eady o s a e he me hod o sub and supe -solu ions in o de o
ge exis ence o solu ion o (1.1). In iew o he esul s o he p e ious sec ion he
concep o sub and supe -solu ion o (1.1) is he ollowing.
De ini ion 3.1. A pai (u, u),( , ) is a sub-supe solu ion o (1.1) i u, u, , ∈
H1(Ω) ∩C(Ω), u, ∈H1
0(Ω) such ha
(1) 0 < u ≤u, 0 < ≤ almos e e ywhe e in Ω,
(2) |∇u|2
uα,|∇u|2
uα,|∇ |2
γ,|∇ |2
γ∈L1(Ω),
1in pa icula e−G( )∈L1(0,1)
NONLINEAR ELLIPTIC SINGULAR SYSTEMS WITH NATURAL GROWTH 9
(3) o e e y φ∈H1
0(Ω) ∩L∞(Ω), φ > 0,
(3.1) ZΩ
∇u· ∇φ+ZΩ
g1( )|∇u|2
uαφ−ZΩ
1(x, u, )φ≤0≤
≤ZΩ
∇u· ∇φ+ZΩ
g1( )|∇u|2
uαφ−ZΩ
1(x, u, )φ∀ ∈[ , ],
(4) o e e y φ∈H1
0(Ω) ∩L∞(Ω), φ > 0,
(3.2) ZΩ
∇ · ∇φ+ZΩ
g2(u)|∇ |2
γφ−ZΩ
2(x, u, )φ≤0≤
≤ZΩ
∇ · ∇φ+ZΩ
g2(u)|∇ |2
γφ−ZΩ
2(x, u, )φ∀u∈[u, u].
Rema k 3.2. Obse e ha any o he ou inequali ies in (3.1) and (3.2) is e i ied
i i is sa is ied in he classical sense, o ins ance, he i s inequali y in (3.1) is
sa is ied i uis wice di e en iable and
−∆u+g1( )|∇u|2
uα− 1(x, u, )≤0 a.e. x∈Ω,∀ ∈[ , ].
Theo em 3.3. Assume ha (u, u),( , )is pai o sub-supe solu ion o (1.1) and
deno e I:= [u, u]×[ , ]⊂C(Ω) ×C(Ω). Assume also he ollowing condi ions on
1, 2, g1and g2:
(F) The e exis s a cons an M≥0such ha he maps s7→ 1(x, s, )+Ms and
7→ 2(x, u, ) + M a e posi i e and inc easing o (s, )∈[0,supΩu]×
[0,supΩ ]and o all (u, )∈I.
(G) g1,g2a e non-nega i e unc ions and gi(s)=0i and only i s= 0 o
i= 1,2.
Then, he e exis s a solu ion (u, )o (1.1) such ha (u, )∈I.
P oo . Taking in o accoun (G), he e exi s a posi i e numbe 0 < m1such ha
0< g1(z(x)), g2(w(x)) ≤m1,∀(w, z)∈I.
Then, aking ν= max{k kL∞(Ω),kukL∞(Ω)}, by Lemma 2.7 he e exis h1, h2∈
C1(0,+∞)∩L1(0,1) and τ1, τ2≥0 such ha o a.e. x∈Ω, o e e y 0 < s < ν
and o e e y (w, z)∈Iwe ha e
τ1−h0
1(s)−g1(z(x))α
sα+1 +−h1(s) + g1(z(x))
sαh1(s)≥
≥−h1(s) + g1(z(x))
sα2
and
τ2−h0
2(s)−g2(w(x))γ
sγ+1 +−h2(s) + g2(w(x))
sγh2(s)≥
≥−h2(s) + g2(w(x))
sγ2
.
We would like o ema k again ha nei he hino τidepend on (w, z), see Lemma
2.7.
16 J. CARMONA, P. J. MART´
INEZ-APARICIO, AND A. SU´
AREZ
sa is ied. In o de o e i y he second inequali y in (3.1) and in (3.2) i is enough
o ake K1−p−q
1≥λkekp+q
∞and
K2(−∆E) + g2(u)K2−γ
2
|∇E|2
Eγ≥µKm+n
2emEn,∀u∈[u, u],
o which i su ices ha
g2(u)K2−γ−m−n
2|σ|2≥µkekm
∞kEkn+γ
∞,∀u∈[u, u].
I g2is inc easing, hen g2(u)≥g2(0) and so we need ha 2 −γ > m +nand
g2(0) >0. Thus, in his case, K1and K2depend only on λand µ, espec i ely.
Howe e , i g2is dec easing hen g2(u)≥g2(K1kek∞)>0 and, using ha
2−γ > m +nwe can choose K2depending on K1(which depends on λ) and µ.
On he o he hand, he i s inequali y in (3.1) is sa is ied i εis small enough
and (3.2) is sa is ied i
aϕa(1−(m+n))−2
1|∇ϕ1|2h(1 −a) + g2(u)aϕa(1−α)
1i+ϕ1−(m+n)
1≤µ,
∀u∈[u, u].
I g2is inc easing, hen g2(u)≤g2(K1(λ)e)≤K3(λ), and so he abo e inequali y
is ue o µ > K(λ) o some cons an K(λ)>0.
I g2is dec easing, hen g2(u)≤g2(0) and so he abo e inequali y is ue o µ
la ge and independen o λ. This comple es he p oo . 
Rema k 4.5. (1) Simila esul o he case 2 −α > p +q≥1 and m+n < 1.
(2) We could ob ain esul s o any posi i e unc ion g2imposing mo e es ic-
i e condi ions in λand µ.
Using simila a gumen s o he p oo s o he abo e esul s, we can show he
ollowing esul .
Theo em 4.6. Assume ha 1≤p+q < 2−αand 1≤m+n < 2−γ.
(1) Assume ha g1and g2a e inc easing and g1(0), g2(0) >0. Then, he e
exis K1(µ)and K2(λ)such ha i λ>K1(µ)and µ>K2(λ)sys em (4.6)
possesses a leas a posi i e solu ion.
(2) Assume ha g1is inc easing, g2dec easing and g1(0) >0. Then, he e
exis K1(µ)and K2such ha i λ > K1(µ)and µ > K2, sys em (4.6)
possesses a leas a posi i e solu ion.
(3) Assume ha g1and g2a e dec easing. Then, he e exis K1, K2>0such
ha i λ > K1and µ > K2, sys em (4.6) possesses a leas a posi i e
solu ion.
4.2. Example 2: compe i ion Lo ka-Vol e a sys em. We conside he sys-
em
(4.7) 








−∆u+g1( )|∇u|2
uα=u(λ−u−b ) in Ω,
−∆ +g2(u)|∇ |2
γ= (µ− −cu) in Ω,
u= = 0 on ∂Ω,
whe e λ, µ ∈R,b, c ≥0, and g1and g2 e i y (G). When g1≡g2≡0 sys em (4.7)
is he classical compe i ion Lo ka-Vol e a model, s udied ex ensi ely in he las
yea s, see o ins ance [10].

NONLINEAR ELLIPTIC SINGULAR SYSTEMS WITH NATURAL GROWTH 17
Fi s , i is clea ha i λ≤λ1o µ≤λ1 hen (4.7) does no ha e posi i e solu ion.
So, assume ha λ, µ > λ1. Again, i is clea ha 1(x, u, ) = u(λ−u−b ) and
2(x, u, ) = (µ− −cu) e i y (F) o any pai o sub-supe solu ion o (4.7).
Theo em 4.7. Assume ha one o he ollowing condi ions holds:
(1) g1and g2a e inc easing and (λ, µ)sa is ies
(4.8) λ>λ1(bΘ[µ,γ,g2(0)])and µ > λ1(cΘ[λ,α,g1(0)]);
(2) g1and g2a e dec easing and (λ, µ)sa is ies
(4.9) λ>λ1(bΘ[µ,γ,g2(λ)])and µ>λ1(cΘ[λ,α,g1(µ)]);
(3) g1is inc easing, g2is dec easing and (λ, µ)sa is ies
(4.10) λ>λ1(bΘ[µ,γ,g2(0)])and µ>λ1(cΘ[λ,α,g1(µ)]).
Then (4.7) possesses a leas a posi i e solu ion.
Rema k 4.8. Obse e ha condi ions (4.8), (4.9) and (4.10) de ine egions in
he plane (λ, µ) which could e en ually be emp y. Fo he semilinea case, ha is
g1≡g2≡0, i can be shown, see o example [18] and [17], ha hese egions a e
no emp y, imposing some condi ions (bo csmall). The s udy o hese egions a e
ou o he scope o his pape , bu le us ema k some aspec s. Obse e ha he
map
λ∈[λ1,∞)7→ λ1(cΘ[λ,α,g1(0)])
is inc easing. Hence, o example, he egion de ined by (4.8) in no emp y i bo
cis small.
P oo . (1) Assume ha g1and g2a e inc easing. Then, ake
(u, ) = (Θ[λ,α,g1(0)],Θ[µ,γ,g2(0)]),
(u, ) = (Θ[λ−bΘ[µ,γ,g2(0)],α,R],Θ[µ−cΘ[λ,α,g1(0)],γ,S]),
o some posi i e cons an s Rand S o be chosen. Since u, , u and a e solu ions
o logis ic equa ions as (4.3), hen i ems (1) and (2) o De ini ion 3.1 a e sa is ied.
Using he equa ion o u, i can be shown ha usa is ies he second inequali y
in (3.1) i
u(λ−u)−g1(0)|∇u|2
|u|α≥u(λ−u−b )−g1( )|∇u|2
|u|α,
o equi alen ly,
bu + (g1( )−g1(0))|∇u|2
|u|α≥0,
which is ue because g1in inc easing and > 0.
Fo u, we need ha
|∇u|2
|u|α(g1( )−R)≤0,∀ ∈[ , ].
Take R≥g1( ).
Obse e ha by he inc ease o he map λ+m7→ Θ[λ+m,α,k], i ollows ha
u≤Θ[λ,α,R]≤Θ[λ,α,g1(0)] =u,
his las inequali y because R≥g1(0).
18 J. CARMONA, P. J. MART´
INEZ-APARICIO, AND A. SU´
AREZ
(2) Assume ha g1and g2a e dec easing. Then, ake
(u, ) = (Θ[λ,α,g1(µ)],Θ[µ,γ,g2(λ)]),
(u, ) = (Θ[λ−bΘ[µ,γ,g2(λ)],α,g1(0)],Θ[µ−cΘ[λ,α,g1(µ)],γ,g2(0)]).
Indeed, obse e ha , wi h a simila a gumen o he used in he i s pa ag aph, u
sa is ies he second inequali y in (3.1) i
u(λ−u)−g1(µ)|∇u|2
|u|α≥u(λ−u−b )−g1( )|∇u|2
|u|α,
o wha i is su icien ha
g1( )≥g1(µ).
Bu , om (4.2) we ha e ha ≤θµ≤µ, and since g1is dec easing, i ollows ha
g1( )≥g1(µ).
Wi h espec o u, i can be p o ed ha usa is ies he i s inequali y in (3.1)
because g1( )≤g1(0).
Again, i can shown ha u≤u.
(3) Assume ha g1is inc easing and g2is dec easing. Then, ake in his case
(u, ) = (Θ[λ,α,g1(0)],Θ[µ,γ,g2(λ)]),
(u, ) = (Θ[λ−bΘ[µ,γ,g2(λ)],α,R],Θ[µ−cΘ[λ,α,g1(0)],γ,g2(0)]).

4.3. Example 3: symbio ic Lo ka-Vol e a sys em. We conside he sys em
(4.11) 








−∆u+g1( )|∇u|2
uα=u(λ−u+b ) in Ω,
−∆ +g2(u)|∇ |2
γ= (µ− +cu) in Ω,
u= = 0 on ∂Ω,
whe e λ, µ ∈R,b, c > 0, g1and g2 e i y (G).
Theo em 4.9. Assume ha bc < 1and (λ, µ)sa is ies
(4.12) λ>λ1(−bΘ[µ,γ,g2])and µ > λ1(−cΘ[λ,α,g1]),
whe e gi=gi(0) when giis dec easing and
g1=g1µ+cλ
1−bc when g1is inc easing,
and
g2=g2λ+bµ
1−bc when g2is inc easing.
Then (4.11) possesses a leas a posi i e solu ion.
P oo . Fi s , ecall ha Θ[µ,γ,g2]≤µ, and hen i λand µ e i y (4.12), we ha e
ha
λ>λ1(−bΘ[µ,γ,g2])≥λ1(−bµ) = λ1−bµ,
and so λ+bµ > 0. Analogously, µ+cλ > 0.
Now, ake
(u, )=(R, S)
NONLINEAR ELLIPTIC SINGULAR SYSTEMS WITH NATURAL GROWTH 19
whe e Rand Sa e la ge posi i e cons an s and
(u, ) = (Θ[λ+bΘ[µ,γ,g2],α,g1],Θ[µ+cΘ[λ,α,g1],γ,g2]).
Indeed, Rand Smus e i y
λ−R+bS ≤0 and µ−S+cR ≤0.
Since bc < 1, we can ake
R=λ+bµ
1−bc , S =µ+cλ
1−bc .
On he o he hand, uis subsolu ion p o ided o
g1( )≤g1,∀ ∈[ , ].
Then, i g1is dec easing ( espec i ely inc easing) we can ake g1=g1(0) ( espec-
i ely g1=g1( ) = g1(µ+cλ
1−bc )).
Finally, obse e ha
u= Θ[λ+bΘ[µ,γ,g2],α,g1]≤λ+b(Θ[µ,γ,g2])M≤λ+bµ ≤λ+bµ
1−bc =u.

Rema k 4.10. Obse e again ha condi ion (4.12) could de ine an emp y egion
in he plane (λ, µ). As in Rema k 4.8 we poin ou ha he maps
λ∈[λ1,∞)7→ λ1(−cΘ[λ,α,g1(0)])
and
µ∈[λ1,∞)7→ λ1(−bΘ[µ,γ,g2(0)])
a e dec easing, and so he egion de ined by (4.12) is non emp y when g1and g2
a e dec easing, see also [12] o he semilinea case g1≡g2≡0.
Acknowledgemen s.
Resea ch suppo ed by MICINN Minis e io de Ciencia e Inno aci´on, Spain unde
g an s MTM2012-31799 (JC and PJMA) and MTM2012-31304 (AS) and Jun a de
Andaluc´ıa FQM-116 (PJMA), FQM-194 (JC) and FQM-131 (AS).
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(JC) Depa amen o de Ma em´
a icas, Uni e sidad de Alme ´
ıa, C a. Sac amen o s/n,
La Ca˜
nada de San U bano, 04120 - Alme ´
ıa, Spain. [email p o ec ed]
(PJMA) Depa amen o de Ma em´
a ica Aplicada y Es ad´
ıs ica, Uni e sidad Poli ´
ecnica
de Ca agena, 30202 - Mu cia, Spain. ped oj.ma [email p o ec ed]
(AS) Depa amen o de Ecuaciones Di e enciales y An´
alisis Num´
e ico, Facul ad de
Ma em´
a icas, Calle Ta ia s/n, 41012-Se illa, Spain. [email p o ec ed]