On he uniqueness o posi i e solu ion
o an ellip ic equa ion1
M. Delgado and A. Su´
a ez2
Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico
Fac. Ma em´a icas, C/ Ta ia s/n
C.P. 41012, Uni . Se illa, Spain
e-mail: [email p o ec ed] and [email p o ec ed]
Abs ac
This wo k deals wi h he uniqueness o posi i e solu ion o an ellip ic equa ion
whose nonlinea i y sa is ies an speci ic mono ony p ope y. In o de o p o e he main
esul , we employ a change o a iable used in p e ious pape s and he maximum
p inciple.
Key Wo ds. Uniqueness o posi i e solu ion, maximum p inciple.
AMS Classi ica ion. 35J65, 35B50.
1 In oduc ion
Le Ω ⊂IRNbe a egula domain and : Ω ×IR 7→ IR a measu able unc ion. We a e
in e es ed in he classical and posi i e solu ions o he ellip ic p oblem









−∆u= (x, u) in Ω,
u= 0 on ∂Ω.
(1.1)
1Suppo ed by he Spanish Minis y o Science and Technology unde G an s BFM2000-0797 and
BFM2003-06446.
2Au ho who ecei es co espondence and p oo s o co ec ion
2M. Delgado and A. Su´a ez
One o he mo e di icul p oblem ela ed o (1.1) is p o ing he uniqueness o solu ion o
(1.1). I is well known ha i is dec easing in u hen he e exis s a mos one solu ion
o (1.1), see o ins ance [1] and [2]. When o a. e. x∈Ω he map
u7→ (x, u)
uis dec easing in (0,∞) (1.2)
hen, he e exis s a mos one posi i e solu ion o (1.1), see [3] and [4].
In his no e, we employ an app op ia e change o a iable (ye used in [5], [6], [7] and
[8]) and he s ong maximum p inciple o p o e ha i he e exis s a egula , posi i e and
conca e unc ion g(see Theo em 2.1 and P oposi ion 2.2 o he exac condi ions on g)
such ha
u7→ (x, u)
g(u)is non-inc easing in (0,∞) o a. e. x∈Ω (1.3)
hen, he e exis s a unique posi i e solu ion.
When (x, u) = a(x)g(u) wi h a∈L∞(Ω), he uniqueness was s udied in [5], [6], [7]
and [8]. We e e o [6] whe e a e iew o he uniqueness ques ion is made. We would like
o ema k ha al hough he condi ions (1.2) and (1.3) seem a he simila , he echniques
o he p oo s o uniqueness a e qui e di e en . In ac , he p oo s o he uniqueness esul
unde (1.2) use he mono onici y o he quo ien be ween (x, ) and exac ly he linea
unc ion g( ) = . Ou p oo , which allows us o use he mono onici y o he quo ien
be ween (x, ) and a conca e unc ion g( ), does no each he linea unc ion; whe eas
(x, )/g( ) is no necessa ily dec easing.
In he ollowing sec ion we p o e he main esul o his wo k. In he las sec ion we
employ a speci ic example om popula ion dynamics ha shows ha ou esul imp o es
and complemen s ha ob ained unde he condi ion (1.2).
2 Main esul
Ou main esul eads as ollows:
Theo em 2.1 Assume ha he e exis s a unc ion g∈C1(0,+∞)∩C0([0,+∞)),g( )>0
o > 0, such ha
a) g0is non-inc easing and
Z
0
1
g( )d < ∞, o > 0.(2.1)
Uniqueness o posi i e solu ion 3
b) The map
u7→ (x, u)
g(u)is non-inc easing in (0,∞) o a. e. x∈Ω.(2.2)
Then, he e exis s a mos one posi i e solu ion o (1.1).
P oo : Conside he change o a iable
=Zu
0
1
g( )d (2.3)
which ans o ms (1.1) in o









−∆ =g0(h( ))|∇ |2+ (x, h( ))
g(h( )) in Ω,
= 0 on ∂Ω,
(2.4)
whe e
u=h( ),(2.5)
and hsa is ies, om (2.3), h0( ) = g(h( )).
Assume ha he e exis s wo posi i e solu ions u16=u2o (1.1). Le Ω1:= {x∈Ω :
u1(x)> u2(x)}. Assume ha Ω1is no emp y. I is clea ha u1=u2on ∂Ω1. Thanks
o mono onici y o h, 1> 2in Ω1and 1= 2on ∂Ω1, whe e ui=h( i)i= 1,2.
Conside he unc ion
Φ := 1− 2,
which is posi i e in Ω1and Φ = 0 on ∂Ω1. A e some calcula ion, we ob ain ha Φ
e i ies
−∆Φ −g0(h( 1))|∇ 1|2+g0(h( 2))|∇ 2|2=µ (x, h( 1))
g(h( 1)) − (x, h( 2))
g(h( 2)) ¶.(2.6)
Since g0is non-inc easing, g0(h( 1)) ≤g0(h( 2)); and by (2.2), we ge ha
−∆Φ −g0(h( 1))∇( 1+ 2)· ∇Φ≤0,
which is a con adic ion by he maximum p inciple. This comple es he p oo . 2
I we look o posi i e solu ions in a mo e es ic i e se , we can weaken he condi ion
(2.1). Le de ine
P:= {u∈C1
0(Ω) : u(x)≥0, u 6= 0 in Ω},
4M. Delgado and A. Su´a ez
whose in e io is
in (P) = {u∈P:u(x)>0 o all x∈Ω, ∂u/∂n < 0 on ∂Ω},
whe e ndeno es he ou wa d no mal di ec ion.
P oposi ion 2.2 Assume ha he e exis s gas in Theo em 2.1 bu e i ying
lim
s→0
s
g(s)= 0,(2.7)
ins ead o (2.1). Then, he e exis s a unique solu ion in in (P)o (1.1).
P oo : Obse e i s ha i u∈in (P), he e exis posi i e cons an s 0 < k1≤k2such
ha
k1dis (x)≤u(x)≤k2dis (x),(2.8)
whe e dis (x) := dis (x, ∂Ω). Assume ha he e exis s wo posi i e u16=u2o (1.1) wi h
ui∈in (P), i= 1,2 . Le Ω1:= {x∈Ω : u1(x)> u2(x)}. We de ine now o x∈Ω1
Φ(x) := Zu1(x)
u2(x)
1
g( )d .
Fi s , obse e ha unc ion Φ is con inuous in Ω1and
Φ = 0 on ∂Ω1.
Indeed, o x∈∂Ω1∩Ω i is clea ha Φ(x) = 0. Fo each x∈Ω1 he e exis s ξ(x) wi h
u2(x)≤ξ(x)≤u1(x) such ha
Φ(x) = u1(x)−u2(x)
g(ξ(x)) ≤Cdis (x)
g(ξ(x)) →0,as dis (x)→0,
whe e we ha e used (2.7) and (2.8).
On he o he hand, as in he p oo o Theo em 2.1, we ge ha
−∆Φ −g0(u1)µ∇u1
g(u1)+∇u2
g(u2)¶· ∇Φ≤0.
This las inequali y leads o a con adic ion o he maximum p inciple in he same way as
in he p oo o Theo em 2.1. 2
Rema k 2.3 a) Obse e ha , o example, g(s) = slog2(s) e i ies (2.7) bu no (2.1).
Uniqueness o posi i e solu ion 5
b) Condi ions on can be imposed in o de ha e e y non-nega i e and non- i ial
solu ion o (1.1) belongs o in (P), see o ins ance [9].
c) The same esul s hold o second o de uni o mly ellip ic ope a o o he o m
L:= −
N
X
i,j=1
aij
∂2
∂xi∂xj
+
N
X
i=1
bi
∂
∂xi
wi h aij ∈C0(Ω), bi∈C0(Ω), aij =aji, see [12].
d) I gis posi i e only in (0, R) o some R > 0, and
ZR
0
1
g( )d < +∞(2.9)
hen, we deduce a uniqueness esul o posi i e solu ions, u, such ha kuk∞≤R.
3 Example and compa ison
In his sec ion we apply ou esul o he nonlinea i y
(x, u) = a(x)uq+b(x)up
wi h di e en alues o qand p, and a, b ∈L∞(Ω). This nonlinea i y a ises om he s udy
o he popula ion densi y o a species whose mobili y depends upon i s densi y, see [10]
and [11]. Some uniqueness esul s we e ob ained in [12] and [13]. Fo his unc ion, he
condi ion (1.2) is equi alen o
(q−1)a(x)+(p−1)b(x)up−q<0.(3.1)
Now, we dis inguish be ween he di e en cases:
Case q= 1, p < 1: In his case, (3.1) holds i b > 0. Theo em 2.1 complemen s his esul .
Indeed, aking g(u) = upwe ob ain uniqueness o posi i e solu ion o a≤0 and any
unc ion b.
Case q < 1, p > 1: (3.1) holds, o example, i ais posi i e and b≤0; aposi i e and b
posi i e o changes sign and kuk∞small, see [14] and [11]. By Theo em 2.1, he e exis
a mos one posi i e solu ion i b≤0 and any unc ion a.

6M. Delgado and A. Su´a ez
Case q < 1, p < 1: In his case (3.1) is sa is ied i , o example, aand ba e bo h posi i e.
In he pa icula case p=q, (3.1) is equi alen o a+b > 0.
By Theo em 2.1 we conside h ee cases:
a) I p < q, hen we ha e uniqueness o posi i e solu ion o any unc ion aand b≥0
( aking g(u) = uq) and o any unc ion band a≤0 ( aking g(u) = up).
b) I p > q, hen he esul is simila o case a) changing aby band bby a.
c) I p=q, hen he e exis s a mos one posi i e solu ion i a+bis non-nega i e o
changes sign. Obse e ha i a+bis non-posi i e, (1.1) does no posses non-nega i e
solu ion.
In he cases p= 1, q < 1 and p < 1, q > 1 simila esul s o he i s and hi d cases
espec i ely can be ob ained in e changing he oles o aand b.
Acknowledgemen s: We a e deligh ed o hank o he e e ee o his/he ca e ul
eading o he manusc ip and o se e al use ul ema ks and sugges ions imp o ing he
p esen a ion o he pape .
Re e ences
[1] H. Amann, On he exis ence o posi i e solu ions o nonlinea ellip ic bounda y alue
p oblems, Indiana Uni . Ma h. J.,21, 125–146 (1971).
[2] H. Amann, Fixed poin equa ions and nonlinea eigen alue p oblems in o de ed Ba-
nach spaces, SIAM Re .,18, 620–709 (1976).
[3] H. B ezis and L. Oswald, Rema ks on sublinea ellip ic equa ions, Nonlinea Anal.,
10, 55–64 (1986).
[4] P. Hess, On uniqueness o posi i e solu ions o nonlinea ellip ic bounda y alue
p oblems, Ma h. Z.,154, 17–18 (1977).
[5] C. Bandle, M. A. Pozio and A. Tesei, The asymp o ic beha iou o he solu ions o
degene a e pa abolic equa ions, T ans. Ame . Ma h. Soc.,303, 487–501 (1987).
[6] H. B ezis and S. Kamin, Sublinea ellip ic equa ions in IRN,Manusc ip a Ma h.,74,
87–106 (1992).
Uniqueness o posi i e solu ion 7
[7] T. Lae sch, Uniqueness o sublinea bounda y alue p oblems, J. Di e en ial Equa-
ions,13, 13–23 (1973).
[8] J. Sp uck, Uniqueness in a di usion model o popula ion biology, Comm. in Pa ial
Di . Eqns.,15, 1605–1620 (1983).
[9] P. Pucci and J. Se in, The s ong maximum p inciple e isi ed, J. Di e en ial Equa-
ions,196, 1–66 (2004).
[10] M. E. Gu in and R. C. MacCamy, On he di usion o biological popula ions, Ma h.
Biosci.,33, 35–49 (1977).
[11] M. Delgado and A. Su´a ez, Posi i e solu ions o he degene a e logis ic inde ini e
supe linea p oblem: he slow di usion case, Hous on J. o Ma h.,29, 801–820 (2003).
[12] M. Delgado and A. Su´a ez, On he s uc u e o he posi i e solu ions o he logis ic
equa ion wi h nonlinea di usion, J. Ma h. Anal. Appl.,268, 200–216 (2002).
[13] M. Delgado, J. L´opez-G´omez and A. Su´a ez, Cha ac e izing he exis ence o la ge
solu ions o a class o sublinea p oblems wi h nonlinea di usion, Ad . Di e en ial
Eqns.,7, 1235–1256 (2002).
[14] A. Amb ose i, H. B ezis and G. Ce ami, Combined e ec s o conca e and con ex
nonlinea i ies in some ellip ic p oblems, J. Func . Anal.,122, 519–543 (1994).