A comparison result for perturbed radial p-laplacians

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1 A comaison esult fo etubed adial -Lalacians Raul Manásevich and Guido Swees Diectoy Table of Contents Begin Aticle Coyight c 23 Last Revision Date: Ail 1, 23

2 Table of Contents 1. Intoduction and main esult 1.1.A non-quasimonotone system 1.2.The main condition and the theoem Otimal bounds in the linea case that use one aamete 1.3.Othe examles 2. On the solution oeato 2.1.Elementay oeties of G 2.2.Comaing with a sot of fundamental solution 2.3.Two-sided estimates fo G 2.4.Estimates fo G alied to a singula function 3. Veification of the main condition 4. Main oofs 4.1.Comaison esults fo G. 4.2.A fixed oint agument Poof of Theoem 1.2

3 Section 1: Intoduction and main esult 3 A comaison esult fo etubed adial -Lalacians Raul Manásevich & Guido Swees Abstact Conside the adially symmetic -Lalacian fo 2 unde zeo Diichlet bounday conditions. The main esult of the esent ae is that unde aoiate conditions a solution of a etubed adially symmetic -Lalacian can be comaed with the solution of the unetubed one. As a consequence one obtains a sign eseving esult fo a system of -Lalacians which ae couled in a non-quasimonotone way. 1. Intoduction and main esult One of the goals when studying the Schödinge equation u + V u = f is to find comaison esults, that is, when consideing the oblem fo V 1 and V 2, what ae the conditions such that fo the same f the coesonding solutions u 1 and u 2 can be comaed see [7]. Zhao and collaboatos see [11], [2] and the efeences theein obtained such comaison esults on bounded domains Ω R n, fo u satisfying zeo Diichlet bounday conditions, by estimating the iteated Geen function with the Geen function itself. The difficulties aise both by the singulaity on the diagonal of the solution oeato the Geen function G Ω x, y in dimension n 2 has a singulaity when x = y and as well as by the zeo bounday condition. The main tool in thei oofs ae the Hanack inequalities both in the inteio and at the bounday. With the estimates of Zhao one may even show that fo ε > but small the nonlocal V defined by V u x = ε Ω G Ω x, y u y dy is in this class. As a consequence one obtains a maximum incile fo a system of ellitic equations with a noncooeative couling. In this ae we will show a fist ste in tansfeing such a comaison esult to a nonlinea equation, namely one containing the -Lalacian with the assumtion of adial symmety. Of couse the otential should have the same tye of nonlineaity and that leads us to conside comaison inciles fo { u + λ V u = f in B, 1 u = on B, with B = {x R n ; x < 1}, n 2, 2,, λ > some small aamete and whee V is an oeato having the same homogeneity as the -Lalacian. This oeato may be nonlocal but is assumed to eseve adial symmety. Fo λ = it is well known that f imlies u even in a much moe geneal setting see [1]. Since we will estict ouselves to the adial symmetic case we have following exession fo the -Lalacian: u = 1 n n 1 u 2 u. As a consequence we will find a sign eseving esult fo a system of -Lalace oeatos which ae couled in a noncooeative way. We ecall that the bounday value oblem 1 is called sign eseving if evey solution u is ositive wheneve the souce tem f is ositive. In contay to the oiginal maximum incile, that is, u cannot have a negative minimum, such a sign eseving oety may deend on nonlocal aguments. In the esent ae thee will not be a maximum incile in this oiginal sense but we suoted by Fonda M.A. and Milenio gant-p1-34 Keywods: -lalacian system, ositivity, noncooeative couling AMS-MSC: 35B5, 35J6, 34C1

4 Section 1: Intoduction and main esult 4 will show that fo ositive f solutions u of the etubed λ small and unetubed λ = oblem can be comaed. Hence a sign eseving oety will hold fo the etubed oblem, wheneve λ is small enough, fo all f >. Fo easy efeence we fix the following: Notation 1.1 φ u = u 2 u and its invese is being denoted by φ inv φ inv u = u 2 u ; The solution oeato G fo the adial -Lalacian with Diichlet bounday condition: t s n 1 G f = fsds dt; 2 t f > denotes f fo all [, 1] and f ; f means that thee is c > such that f c1 fo all [, 1] A non-quasimonotone system A secial case that we conside is the following non-quasimonotone if λ > nonlinea ellitic system u = f λφ v in B, v = φ u in B, 3 u = v = on B. Remembe that the system u = F 1 u, v, v = F 2 u, v is quasimonotone iff u F 2 and v F 1, and that in a quasimonotone setting the maximum incile can be used simila as fo one equation. In a linea setting quasimonotone is also known as cooeative. Using the solution oeato G, the Geen oeato defined in 2 fo the adial case, the system coincides with { u + λ φ G φ u = f in B, 4 u = on B, Notice that φ G φ tu = φ t φ G φ u fo all t R and hence satisfies the aoiate homogeneity condition. The 1-dimensional case has been studied in [5]. In a eaction to that ae W. Walte aised the question what would haen in the highe dimensional case. This ae is a fist ste in that diection. The linea case, = 2, of 3 was studied in [6] even fo geneal nonadial functions on smooth domains. A ealie esult fo the ball can be found in [8]. The cucial esult that was used in that ae was the so-called 3G-theoem which oiginates fom Zhao [11]. The nonlinea natue of 3 makes the geneal system much hade. By esticting ouselves to the adial case we ae able to ove a ositivity eseving oety fo this noncooeative system and in doing so we encounte some citical dimensions. Fo the linea case the Geen function becomes unbounded fo n 2. Similaly, the k th iteated Geen function is bounded if and only if 2k > n. Fo the -Lalacian ointwise boundedness of the k th - iteated homogenized Geen oeato, defined by G φ k = G φ G φ k 1 fo k 1, is elated to k > n. These numbes eaea as a estiction in the esults down below. Fo sign eseving esults fo cooeative systems with the -Lalacian we efe to [3]. Positivity eseving oeties of 1 fo = 2 and linea, ossibly nonlocal V, have been studied in [4].

5 Section 1: Intoduction and main esult The main condition and the theoem The basic conditions that we will use to show that a etubation by V does not destoy the ositivity eseving oety fo λ sufficiently small is the following. Condition 1.1 The oeato V is as follows: i. V t u = t V u fo t ; ii. V is continuous fom C 1 [, 1] to C [, 1] and moeove thee is C V > such that V u C[,1] C V u C 1 [,1] fo all u C 1 [, 1]; iii. thee is C V,,n > such that G V G f C V,,n G f fo [, 1] and fo all f C[, 1]. Remak Fo V = φ the thid item in the condition above imlies a nonlinea 3G-tye esult: G φ G f C V,,n G f fo [, 1] and < f C[, 1]. 5 Fo V = φ we ae able to show that iii. is satisfied when both 2 and > 1 2 n hold. See Lemma 3.1 below. Remak Notice that if V and Ṽ satisfy Condition 1.1 then so does V + Ṽ. Only the thid condition needs some eflection. Set v = V G f and ṽ = Ṽ G f and one obtains by φ inv a + b φ inv a + φ inv b fo a, b that t s n 1 G v + ṽ v s + ṽ s ds dt t φ inv G v + G ṽ. Theoem 1.2 the main esult Fix > 2 and suose that the oeato V satisfies Condition 1.1 above. Then thee exists λ such that fo all f C[, 1] with f > and λ [, λ ] the following holds: i. thee exists a solution u C 1, 1 [, 1] of 1 with φ u C 1 [, 1]; ii. evey solution u of 1 satisfies and hence evey solution is ositive. 1 2 G f u 3 2 G f fo [, 1], The oof will be ostoned to the following sections. Remak Notice that we do not state uniqueness of the solution fo the etubed oblem. As can be seen fom the case n = 1 in [5] uniqueness is not obvious in geneal. Fo the non-quasimonotone system 3 we have the following esult.

6 Section 1: Intoduction and main esult 6 Coollay 1.3 If > 1 3n then the adially symmetic case of the non-quasimonotone system in 3 is ositivity eseving fo λ sufficiently small. That is, thee exists λ > such that fo evey λ [, λ ] and f C B with f = f x and f > thee exists a adially symmetic solution u of 3 and evey adially symmetic solution is stictly ositive. Poof. It is sufficient to show that V satisfies Condition 1.1 whee V u = φ G φ u. Indeed Coollay 3.5 imlies that this holds wheneve 2 and > 1 3 n. The aoach of this ae is to get estimates fom above fo the etubation in tems of a function that itself gives a unifom estimate fom below fo the Geen oeato. A stong estiction of this aoach is that one needs to catch the ositive function f in one numbe α f such that, fo some unifom constant C, the following holds: G V f C α f 1 and α f 1 G f. Fo > n we will use α f = G f and fo < n the numbe α f = su [,1] n G f. Needless to say that we do not exect this to give the best ossible esult. The next aagah contains an exlanation fo the case = 2. Otimal bounds in the linea case that use one aamete Fo = 2 a unifom estimate using only one aamete would coincide with obtaining an estimate fom below fo the Geen function G, s by a multile of the oduct g 1 g 2 s whee g 1, g 2 ae ositive functions. Fo the linea Diichlet oblem such an estimate is almost neve otimal since this would mean that the Geen function could be estimated fom below and fom above by multiles of the same oduct. Only fo the 1-dimensional Neumann oblem this is ossible. In the linea case the adial symmetic Geen oeato educes to an integal oeato G 2 f = G, sfsds with the following kenel: G, s = { 1 n 2 sn 1 s 2 n 1 if s, 1 n 2 sn 1 2 n 1 if s <, fo n > 2, G, s = s log max, s fo n = 2. Fo n > 2 these can be estimated in tems of owes of s and and distances to the bounday 1 and 1 s, by 1 c 1 s n 1 min n 2, 1 s 1 s n 2 G, s c 2 s n 1 min n 2, 1 s s n 2, 6 Hence otimal estimates in oduct fom ae fom below and fo the estimate fom above one cannot go beyond G, s c n s n 1 1 s 1, 7 G, s C n s n 1 1 s s n 2 θ 1 1 θ n 2 with θ [, 1]. Otimal two-sided estimates fo the Geen function on geneal domains ae due to Zhao [11]. See also [6] o [9]. As just exlained, the sha exessions that ae used in these aes cannot be of the fom g 1 g 2 s.

7 Section 2: On the solution oeato Othe examles Ou main inteest focuses on the non-quasimonotone system in 3. Othe examles of oeatos V satisfying the main condition Condition 1.1 ae V = φ A with A as follows: i. Au = au + bu, fo > n and a, b C[, 1]. If b = then we may allow > 1 2n. This esult follows fom Lemma s 3.1 and 3.3 and the emak following Condition 1.1. Fo a this examle is not so inteesting since with this local etubation one may oceed by the local aguments of the maximum incile. ii. Au = a, s us sn 1 ds, with aoiate kenel a. Fo the ecise condition see Lemma 3.6. Also kenels like Au = a, s us ds 1 α α o Au = a, t, s us ds 1 α α dt satisfy Condition 1.1 fo aoiate estictions elating α with n and. If we set α = q 1 and a, t, s = χ [t>] χ [s<t] s/t n 1 we find that u is a solution of the non-quasimonotone nonlinea ellitic system 3. Fo = q and n = 1, but with f not necessaily symmetic, this system was studied in [5]. iii. A u = G q φ q. Fo this oeato A, which coesonds with the system u = f λφ v in B, q v = φ q u in B, u = v = on B we find that Condition 1.1 is satisfied when n,, q 2 ae such that n < 2 + q q 1. See Lemma 3.4. A simila condition can be found when using fou diffeent owes as long as the -Lalacians have 2 and the homogeneity fits. 2. On the solution oeato 2.1. Elementay oeties of G The solution oeato fo 1 with λ = is G. Fist note that f C[, 1] imlies that G f C 1, 1 [, 1]. Moeove, if f > set t = inf {t [, 1] ; f t > } and we find that s n 1 φ inv fsds C 1 [, t t, 1] C 1 [, 1]. 9 By integating we find that G f C 2 [, 1] \ {t } C 1, 1 [, 1]. Also9 immediately shows that G f = fo t and G f < fo t < Comaing with a sot of fundamental solution We stat by studying the outcome of this oeato alied on some secial distibutions fo the ight hand side, namely d s = s 1 n δ s, with δ the Diac delta function at s, 1 with weight s 1 n. This weight is the aoiate nomalization fo the adial symmetic natue of the oblem. We will see that = n is citical in the following sense. If n < then and only then the functions G d s ae unifomly bounded with esect to s. 8

8 Section 2: On the solution oeato 8 Lemma 2.1 Set d s = s 1 n δ s. i. If > n, then G d s = n 1 max, s n. ii. If = n, then G d s = log max, s. iii. If < n, then G d s = n max, s n 1. Poof. The esult follows fom a diect comutation. G d s G d s n s s Figue 1: > n esectively n With the solutions fo the delta function we will have the following vesion of a comaison incile. Lemma 2.2 Let u = G f with < f C[, 1]. Then fo evey [, 1] and s, 1 one has u s u G d s s G d s. 1 t σ n 1 t fσdσ dt it is immediate fo f that Poof. Since u = φinv imlies u 1 u 2 and hence 1 holds on [, s]. Fo t [s, 1] we oceed by contadiction. Set v = us G G d ss d s and suose that v τ > u τ fo some τ s, 1. Then thee exist τ 1, τ 2 [s, 1] such that s τ 1 τ τ 2 1 with and with eithe u τ 1 v τ 1 = 1 and u v < 1 fo τ 1, τ 2, τ 2 = 1 o u τ 2 v τ 2 = 1.

9 Section 2: On the solution oeato 9 It follows by an elementay agument that u τ 1 v τ 1 1 u τ 2 v τ The diffeential equations fo u and v on s, 1 give, using φ v <, that u φ n 1 φ u n 1 φ u v = n 1 φ v = n 1 φ v. It follows, afte integating and alying φ inv and by using 11, that fo any τ τ 1, τ 2 1 u τ 2 v τ 2 u τ v τ u τ 1 v τ 1 1. Hence u v on [τ 1, τ 2 ] which imlies u v on [τ 1, τ 2 ], a contadiction Two-sided estimates fo G In the next thee lemmata we will ove a elation between the ue and lowe estimates of G f. Lemma 2.3 If > n, then fo evey f C[, 1] with f > one has n 1 1 G f G f G f fo all [, 1]. 12 Poof. The estimate fom above is obvious by the definition of G. Fo the estimate fom below note that Lemma 2.2 imlies that fo evey ε > Letting ε one finds G f G f G f ε G d ε ε G d ε fo [, 1]. 1 n G f n 1 G f. Lemma 2.4 Suose that = n and let f C[, 1] with f >. Set Then one has Poof. Let be the numbe such that G f θ f = su < 1 1 log. θ f 1 G f θ f 1 log. 13 θ f 1 log = G f. Then G f G f G δ G δ = θ f 1 log log log max, θ 1 log f min log, 1 1 log = θ 1 f min 1 log, log θ f 1.

10 Section 2: On the solution oeato 1 Lemma 2.5 Suose that < n and let f C[, 1] with f >. Set θ f = su n G f. < 1 Then one has θ f min 1, n 1 G f θ f n Remak The numbe θ f is a weighted L -nom fo the function G f. Poof. The estimate fom above follows by the definition of θ f. Fo the estimate fom below conside g = n G f. Since G f C[, 1] we find g = g 1 =. Hence g has an global maximum inside, say in, and θ f = g. By Lemma 2.2 we have G f G f G d G d = θ f n n 1 max, n 1 15 and using that n 1 n 1, we continue 15 by n θ f n 1 = θ f min min n 1 n 1, n 1 n, n n = θ f min θ f min 1, n 1 1 1, n 1. 1 Remak It is cucial in this oof that we ae able to give an estimate indeendent of. Note that any lage exonent, say α > n and θ f = su < 1 α G f fails to give a unifom estimate fom below Estimates fo G alied to a singula function Finally we will show the following two estimates that will be used late. Lemma 2.6 Let α [, n and set g α = α if α > and g = 1 log, then 1 if α <, G g α c n,,α log if α =, α 1 if α >. Poof. Let α, n. Since α < n holds, a staightfowad comutation yields G g α = t s 1 n 1 s α ds dt = t 16

11 Section 3: Veification of the main condition 11 = n α 1 t 1 α dt = c n,,α 1 α if α <, log if α =, α 1 if α >, imlying 16. Fo α = a simila comutation shows that fo some c n, > it holds that G g c n, Veification of the main condition Fist we will show a comaison between the solution oeato G and the iteated G φ G. Lemma 3.1 Suose that 2 satisfies > 1 2n. Let a C[, 1] with a. Then thee is a constant C a,,n such that fo all f C[, 1] : 17 G φ a G f C a,,n G f. 18 Remak The majo estiction hee is > 1 2n. Although we do exect this lemma to hold fo all 2 a oof will be much moe involved. The eason is the following. Fo > 1 2 n we ae able to chaacteize G f by one numbe G f fo > n, see 12, θ f fo = n, see 13 o θ f fo < n, see 14. Indeed this numbe is used to find unifom estimates fom below fo G f and fom above fo G φ G f. Such estimates in tems of one numbe follow fom Lemma 2.6 only if > 1 2 n. Wheneve 2, 1 2 n] such chaacteization by one numbe does not seem to be sufficient and consequently it will be necessay to catue the behavio of G f in a moe elaboate way. Poof. Since G φ a.g f a G φ G f it will be sufficient to conside H f := G φ G f fo < f C[, 1]. Let us denote by 1 the function 1x 1. If > n then the estimates in Lemma 2.3 imly that Fo 1 2 n, n we have H f G f G φ 1 19 t s 1 n 1 G f 1 ds dt t H f G φ G f 1 1 n G f. θ f n t s 1 n 1 θ f s n ds dt t = 1 1 θ f 1 2 n t n+1 dt = max max 1 2 n min 1, 2 n 1, 2 n 1, n 1 2 n θ f 1 G f. θ f 1 2 n

12 Section 3: Veification of the main condition 12 Finally the case = n. It follows that H f G φ θ f 1 log t s 1 n 1 θ f 1 log s n 1 n 1 ds dt t c n θ f t 1 n 1 1 log t dt c n θ f 1 c n G f. Fo such multilication by a it is obvious that it is sufficient to conside the ositive oeato whee a is elaced by a. Fo moe geneal oeatos let us intoduce a slitting in a ositive and a negative at. Lemma 3.2 Suose that V satisfies Condition 1.1. Defining fo f C[, 1] V + G f = max {, V G f } 2 V G f = min {, V G f } 21 we find that V ± G ae continuous fom C[, 1] to C[, 1]. Moeove, thee exist c V, C V > such that V ± G f C[,1] c V f C[,1] fo all f C[, 1]; 22 G V ± G f C V G f fo all f C[, 1]. 23 Poof. The continuity is staightfowad. By Condition 1.1.ii one finds V ± G f V G f c V G f C 1 [,1] c V 2 f. By Condition 1.1.iii G V ± G f G V G f C V,,n G f. Next we addess the etubation by a deivative. Lemma 3.3 Suose that 2 satisfies > n. Let b C[, 1]. Then thee is a constant C b,,n such that fo all f C [, 1] : G φ b. d d G f C b,,n G f. Hence fo all, n, b as above V defined by V u = φ b u satisfies Condition 1.1. Poof. Note that d s n 1 d G f = φ inv f s ds

13 Section 3: Veification of the main condition 13 imlies that V ± G f = φinv φ inv b s n 1 ± f s ds + φ inv b ± s n 1 f s ds s n 1 b f s ds Since > n we may oceed simila as 19 fo f > stating with G V + G f b G f G V + G 1. Fo the couled system we have to deal with G φ G φ G. It will not be much moe touble to have a -Lalacian with anothe exonent say q in the second equation as long as the homogeneity fits. In that case we would have to conside G φ G q φ q G, with G q and φ q defined in Notation 1.1 with the obvious elacement of by q. Lemma 3.4 Suose that n,, q 2 ae such that n < 2+q q 1. Then thee is a constant C,q,n such that fo all f C[, 1] with f : G φ G q φ q G f C,n G f. In othe wods, fo n,, q as above Condition 1.1 is satisfied fo V = φ G q φ q G φ. Coollay 3.5 Suose that > 2 and n < 3. Then V = φ G φ G φ satisfies Condition 1.1. Poof. of Lemma 3.4 Let us denote H,q f = G φ G q φ q G f. If n < then the estimates in Lemma 2.3 imly that Fo n = Lemma 2.6 imlies that H,q f G f G φ G q φ q 1 G f 1 1 n G f. H,q f θ f G φ G q φ q 1 log c n,q θf G φ 1 c n,,q 1. Fo n > we have by Lemma 2.4 and Lemma 2.6 that with H f as in a Lemma 3.1 H f G q φ q c,q,n θ f θ f n = θ f G q q 1 n 1 if q 1 + n >, log if q 1 + n =, 1+ 1 q 1 + n 1 if q 1 + n <. and hence again by Lemma 2.6, deending on the sign of q 1 + n γ,q,n = = q 1 + n 1

14 Section 4: Main oofs 14 then H,q f c,q,n θ f 1 if γ,q,n >, log if γ,q,n =, γ,q,n 1 if γ,q,n <. Since G f c,n θ f 1 we find the esult of the lemma wheneve γ,q,n >, that is, fo n < 2 + q q 1. Lemma 3.6 Let, q 2 and suose that Au = a, s u s sn 1 ds is such that fo some γ < if > n then a, s s n 1 ds C γ 1 if = n then a, s 1 log s s n 1 ds C γ 24 if < n then 1 Then oeato V = φ A satisfies Condition 1.1. n 1 n a, s s ds C γ Poof. Set a, s = a +, s a, s with a +, a and denote g α = α Fist suose that > n. We find by Lemma 2.6 if 1 α < that 1 G V + G f G f G φ a +, s s n 1 ds C G f G φ g α = C G f G g α C,n,α G f 1 The condition 1 α < coincides with α < Fo < n we oceed fo n α < by 1 G V + G f G φ a +, s G f s s n 1 ds θ f G φ a +, s s n s n 1 ds C,n,α θ f Main oofs 4.1. Comaison esults fo G. In this section we comae the Geen oeato fo the etubed and the unetubed ight hand side. Fist we need an elementay estimate: Lemma 4.1 Suose that 2. Fo a, b it holds that φ inv a b φ inv Poof. Fo a b we may use Minkowski s inequality: φ inv a b φ inv a = φ inv a 2 φ inv b. a b φ inv a b + b a b 1 a b 1 + b 1 = b 1.

15 Section 4: Main oofs 15 If a b we oceed by: φ inv a b φ inv a = b a 1 + a 1 b 1 + b 1 = 2 b 1. We will also need the following ode esult: Lemma 4.2 Let g 1, g 2 C[, 1] with g 1 g 2. Then fo all [, 1] : G g 1 G g 2, 25 G g 1 G g Poof. Diectly fom 2: G f = 1 n φ inv sn 1 fsds. Lemma 4.3 Let 2 and let f, g C [, 1] with f. Then fo all [, 1] one finds: G f + g G f 2 G g. 27 If moeove g s θ α fo some α [, n and θ >, then the following estimate holds with C = 2 n α 1 : G f + g G f C θ 1 1 α. 28 Remak Note that G g 1 n Poof. By Lemma 4.2 we find Using that 1 that 1 g L 1 whee g L1 = g s sn 1 ds. G f + g + G f G f + g + G f. Lemma 4.1 shows, 1] imlies a + b 1 a 1 + b 1, we find with a = s n 1 fsds and b = s n 1 gs ds G f + g + G f G g G f + g + G f G f g + G f 2 G g. The estimate of 28 follows by 1 n s α s n 1 ds n = 1 n α n α. Coollay 4.4 Let 2 and let f, g C[, 1] with f. Then fo all [, 1] one finds: G f 2 G g G f + g G f + 2 G g. 29 Poof. The esult follows by 27 and an integation fom = 1.

16 Section 4: Main oofs A fixed oint agument Fo 1 one might obtain a solution when λ is small by the following iteation ocedue. Defining S λ, : C[, 1] C 1 [, 1] C 1 [, 1] by one consides the iteation u = G f, and S λ, f; u := G f λv u. 3 u n+1 = G f λv u n fo n N. Since the esent oblem does not satisfy an ode esevation such an iteation might esult in a sequence that does have a conveging subsequence, but that is not conveging itself. Fo examle it could haen that u 2n u and u 2n+1 u u. The functions u and u do satisfy an 4 th -ode system but do not necessaily satisfy 1. Instead of using such an iteation we will use a fixed oint agument fo existence of a solution to Fo a suvey on fixed oint methods see [1]. u S λ, f; u. Poosition 4.5 Let 2 and λ >. Suose that V satisfies Condition 1.1. Then, with C V > as in Lemma 3.2, the following holds fo all f, g C[, 1] with f > S λ, f; G g G f 2C V,,n λ 1 G g. Poof. By Lemma 4.3 and Lemma 4.2 one finds S λ, f; G g G f = G f λv G g G f and similaly G f + λv G g G f 2 G λv G g S λ, f; G g G f G f λv + G g G f 2 G λv + G g. Condition 1.1 and Lemma 3.2 imly that fo all [, 1] comleting the estimate. G λv G g 2C V λ 1 G g, G λv + G g 2C V λ 1 G g, Coollay 4.6 With, λ, V and C V,,n as in Poosition 4.5, setting it follows that fo all λ that S λ, f; G g satisfies ˆλ = 1 8 V,,n C, 31 [, ˆλ ], f, g C[, 1] with f > and G g 2G f fo all [, 1], < 1 2 G f S λ, f; G g 3 2 G f fo all [, 1].

17 Section 4: Main oofs 17 Poof of Theoem 1.2 An aioi bound. We will fist show that fo λ sufficiently small evey fixed oint of u = S λ, f; u will necessaily lie in [ 1 2 G f, 3 2 G f ]. Indeed, using Coollay 4.4 twice and Condition 1.1 we find that Hence G V u = G V S λ, f; u and fo λ [, λ ] with we get Again using Coollay 4.4 we have = G V G f λv u 2λ 1 C V,,n G f λv u 2λ 1 C V,,n G f + λ V u 2λ 1 C V,,n G f + 4λ 2 C V,,n G V u G V u 2λ 1 C V,,n 1 4λ 2 C V,,n G f λ = 8C V,,n G V u 4λ 1 C V,,n G f. 33 u G f = S λ, f; u G f 2λ 1 G V u. 34 Combining 33 and 34 shows u [ 1 2 G f, 3 2 G f ] fo λ [, λ ]. Existence. Fix f C[, 1] and let us conside D = { u C 1 [, 1] with u C 1 2 G f C 1}. The maing u S λ, f; u fom C 1 [, 1] to itself is comletely continuous and mas D into D. Indeed by Lemma 4.3, the oeties of G and the assumtion on V one finds that Sλ, f; u G f 2 G λv u 4λ 1 V u 1/ 4 c V λ 1 u C 1 [,1], 35 and since S λ, f; u and G f ae zeo in = 1 it follows fo λ [, λ ], whee that λ := 8c V 1, 36 S λ, f; u C 1 [,1] S λ, f; u G f C 1 [,1] + G f C 1 [,1] = S λ, f; u G f + Sλ, f; u G f + G f C 1 [,1] 1 2 G f + 4 c V λ 1 u C 1 [,1] + G f C 1 [,1] 2 G f C 1 [,1]. By Schaude s fixed oint Theoem thee exists u D such that u = S λ, f; u. Conclusion. Thee exists a solution u [ 1 2 G f, 3 2 G f ] wheneve λ [, λ ] with λ = min { λ, λ } defined by 32 and 36.

18 Section 4: Main oofs 18 Refeences [1] H. Amann, Fixed oint equations and nonlinea eigenvalue oblems in odeed Banach saces, SIAM Rev , [2] Kai Lai Chung, and Zhong Xin Zhao, Fom Bownian motion to Schödinge s equation. Gundlehen de Mathematischen Wissenschaften [Fundamental Pinciles of Mathematical Sciences], 312. Singe-Velag, Belin, [3] J. Fleckinge, J. Henandez and F. de Thélin, On maximum inciles and existence of ositive solutions fo some cooeative ellitic systems. Diffeential Integal Equations , no. 1, [4] P. Feitas and G. Swees, Positivity esults fo a nonlocal ellitic equation. Poc. Roy. Soc. Edinbugh Sect. A , no. 4, [5] R. Manásevich and G. Swees, A noncooeative ellitic system with -Lalacians that eseves ositivity. Nonlinea Anal , no. 4, Se. A: Theoy Methods, , 5, 7 [6] E. Mitidiei and G. Swees, Weakly couled ellitic systems and ositivity, Math. Nach , , 6 [7] R. G. Pinsky, Positive hamonic functions and diffusion. Cambidge Studies in Advanced Mathematics, 45. Cambidge Univesity Pess, Cambidge, [8] G. Swees, A stong maximum incile fo a noncooeative ellitic system, SIAM J. Math. Anal , [9] G. Swees, Positivity fo a stongly couled ellitic system by Geen function estimates. J. Geom. Anal , no. 1, [1] J.L. Vázquez, A stong maximum incile fo some quasilinea ellitic equations, Al. Math.Otim , [11] Z. Zhao, Geen functions fo Schödinge oeato and conditioned Feynman-Kac gauge, J. Math. Anal. Al , , 4, 6 Raul Manásevich: Cento de Modelamiento Matemático Univesidad de Chile Av. Blanco Encalada 212 Piso 7 Santiago Chile Guido Swees: Det. of Alied Mathematical Analysis Delft Univesity of Technology P.O. Box GA Delft The Nethelands

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