Boundary Blow-Up Solutions of Cooperative Systems

Boundary blow-up solutions of cooperative systems

Juan Dávila

, Louis Dupaigne

, Olivier Goubet

, Salomé Martínez

aDepartamento de Ingeniería Matemática and CMM (CNRS UMI 2807), Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile bLAMFA, CNRS UMR 6140, Université Picardie Jules Verne, 33, rue St Leu, 80039 Amiens, France

Received 7 March 2008; received in revised form 9 December 2008; accepted 16 December 2008 Available online 8 January 2009

Abstract

We study the existence, uniqueness and boundary profile of nonnegative boundary blow-up solution to the cooperative system u=g(u−v) inΩ,

v=f (v−βu) inΩ, u=v= ∞ on∂Ω

in a smooth bounded domain ofR^N, wheref,gare nondecreasing, nonnegativeC¹functions vanishing in(−∞,0]andβ >0 is a parameter.

©2009 Elsevier Masson SAS. All rights reserved.

Keywords:Cooperative system; Boundary blow-up; Keller–Osserman condition

1. Introduction

For a single equation of the formu=f (u),f 0, the existence of solutions which blow up at the boundary of a smoothly bounded domainΩ is equivalent to a growth condition onf known as the Keller–Osserman condition (see [15,17] as well as [9]):

∞ dt

√F (t )<∞, (1)

whereF (t )=_t

0f (s) ds.

If in additionf is nondecreasing, some boundary blow-up solution (BBUS) is maximal: it dominates all other solutions of the equation. In particular, interior uniform estimates can be derived for anysolution of the original equation, independently of its boundary values.

The current paper is an attempt at generalizing the above theory to autonomous systems of semilinear elliptic equations. Since blow-up solutions are strongly related to the Maximum Principle, it is natural to consider the case of cooperative systems first. Failing short of a theory for general cooperative systems (see Remark 2.4 for further discussion), we study systems of the form

* Corresponding author.

E-mail addresses:jdavila@dim.uchile.cl (J. Dávila), louis.dupaigne@math.cnrs.fr (L. Dupaigne), olivier.goubet@u-picardie.fr (O. Goubet), samartin@dim.uchile.cl (S. Martínez).

0294-1449/$ – see front matter ©2009 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2008.12.003

u=g(u−v) inΩ, v=f (v−βu) inΩ, u=v= ∞ on∂Ω

(2) whereΩ is a smoothly bounded domain of Euclidean space,f, gare nondecreasing, nonnegativeC¹functions such thatf =g=0 onR⁻andβ >0 is a parameter. Solutions are sought in the classC²(Ω)and the boundary condition is to be understood as

xlim→x0

u(x)= lim

x→x0

v(x)= +∞ for allx₀∈∂Ω.

Boundary blow-up solutions of cooperative systems have been considered in [12] (with different nonlinearities than the ones treated here) and some examples of competitive systems have already been studied in [10,11]. Boundary blow-up solutions in different cooperative, competitive or predator–prey systems arise in problems with “refuge”, that is, in nonhomogeneous systems where one of the coefficients vanishes on a subset of the domain, see [5,7,8,16]. For yet another type of systems with large solutions, see [6].

We study existence, first order asymptotics and uniqueness of solutions of (2) respectively in Sections 2, 3, 4. In Section 5, we study in more detail a list of relevant examples. Here is a summary of our main results.

Theorem 1.1.LetΩ denote a smoothly bounded domain of Euclidean space,f, g:R→Rnondecreasing, nonneg- ativeC¹functions such thatf =g=0onR⁻andβ >0. There exists a solution of the system(2)if and only if the following three conditions hold

• f satisfies the Keller–Osserman condition(1),

• gsatisfies the Keller–Osserman condition(1),

• β <1.

The asymptotics of solutions is obtained at the price of a technical assumption on the nonlinearities commonly found in the literature (see e.g. [1]). More precisely, let

φ(u)= ∞ u

√dt 2F (t ), where

F (t )= t

0

f (s) ds.

We assume in what follows thatf satisfies lim inf

t→∞

φ(at )

φ(t) >1 ∀a∈(0,1). (3)

Examples are given byf (u)=e^uorf (u)=u^p,p >1. A counter-example is given byf (u)=u (ln(1+u))^2p,p >1.

For 0< β <1 andβ < θ1, we letw_θ>0 denote the minimal solution to w_θ=f (^θ−_θ^βw_θ) inΩ,

w_θ= +∞ on∂Ω (4)

(see e.g. [9] for the notion of minimal blow-up solution). Then, we have the following result.

Theorem 1.2.Make the same assumptions as in Theorem1.1. Assume in addition thatf satisfies(3).

(a) Iff is smaller thangat infinity in the sense that for anyε >0,

t→+∞lim f (t )

g( t )=0 (5)

then for any solution(u, v)of the system(2)

x→lim∂Ω

u

w₁ =1, lim

x→∂Ω

v

w₁=1, (6)

wherew₁is the minimal nonnegative solution to(4)withθ=1.

(b) Iff is of the order ofgat infinity in the sense that for someθ₀∈(β,1), ifθ∈(β, θ0), then lim inf

t→+∞θg((1−θ )t ) f ((θ−β)t )1, ifθ∈(θ₀,1), then lim sup

t→+∞θg((1−θ )t )

f ((θ−β)t )1 (7)

then for any solution(u, v)of the system(2)

x→lim∂Ω

u wθ₀ = 1

θ0

, lim

x→∂Ω

v

wθ₀ =1, (8)

wherew_θ0 is the minimal solution to(4)withθ=θ₀.

Remark 1.3.Condition (7) looks rather unpleasant. Nevertheless, its validity can be easily checked on examples. If e.g.f (t )=g(t )=t^p, by the Intermediate Value Theorem there existsθ0∈(β,1)such that limt→+∞θ0g((1−θ₀)t )

f ((θ₀−β)t )= θ0(1−θ₀)^p

(θ₀−β)^p =1. Since the quantity θ_{f ((θ}^g((1−₋^{θ )t )}_{β)t )} is nonincreasing in θ, (7) follows. If f (t )=g(t )=e^t, then letting θ₀=(1+β)/2, we have limt→+∞θ_{f ((θ}^g((1−₋^{θ )t )}_{β)t )}= +∞forθ < θ₀, while the limit is equal to 0 ifθ > θ₀. Observe that in this case, though (7) holds, there is no value ofθfor which the limit is equal to 1.

Remark 1.4.Note that when condition (5) holds, the first order asymptotics of bothuandvis independent of the nonlinearityg. The influence ofgcan be detected in the next terms of the asymptotic expansion of the solution. See Example 1 in Section 5.

On the contrary, when condition (7) holds,f andgalready interplay in the value of the constantθ0of the leading asymptotics ofuandv. See Example 2 in Section 5.

As a consequence of Theorem 1.2, we obtain under an extra convexity assumption:

Corollary 1.5.Make the same assumptions as in Theorem 1.1. Assume in addition that f satisfies (3) and either condition(5)or(7)holds.

(a) Iff andgare convex, then the system(2)has a unique solution.

(b) If Ω is a ball and ^{f (t )}_t ,^{g(t )}_t are nondecreasing in a neighborhood of +∞, then the system(2) has a unique solution.

Remark 1.6.Even in the case of the ball, we do not know whether uniqueness remains true under the sole assumption thatf andgare nondecreasing.

Notation.For functionsm, n:[0,∞)→ [0,∞)we say thatm∼nat infinity if

tlim→∞

m(t ) n(t ) =1

and use similar notation whenm, nare defined neart0and limt→t0

m(t ) n(t ) =1.

2. Existence

The proof of the existence of solutions in Theorem 1.1 follows a standard scheme where one first solves the system with a finite boundary conditionmand then letsm→ +∞. The former step can be carried out in a more general setting as described next. Consider the system

u=^g(u, v) inΩ, v=^f(v, u) inΩ, u=v= ∞ on∂Ω,

(9) wherefandgare two nonnegativeC¹functions such thatf(0,0)=^g(0,0)=0 and∂g/∂v0,∂f/∂u0 (the system is then called cooperative).

Proposition 2.1.Givenm >0the system u=^g(u, v) inΩ,

v=^f(u, v) inΩ, u=v=m on∂Ω

(10) admits a unique minimal nonnegative solution(u, v).

In the previous statement minimality refers to the following property: take any open setω⊆Ω andu,¯ v¯∈C²(ω)¯ satisfying

⎧⎪

⎨

⎪⎩

u¯^g(u,¯ v)¯ inω, v¯^f(u,¯ v)¯ inω,

¯

u0, v¯0 inω,

¯

uu, v¯v on∂ω.

(11)

Then,

uu,¯ vv¯ inω.

To solve the system with finite boundary values we use sub and supersolutions. A convenient reference for mono- tone methods for equations and systems is [18].

Proof. Choosea >0,b >0 sufficiently large such that the functions

u→^g(u, v)−au, v→^f(u, v)−bv are decreasing for 0u, vm. (12) Define

u0≡0, v0≡0 (13)

and fork1

u_k−au_k=^g(u_k−1, v_k−1)−au_k−1 inΩ, v_k−bv_k=^f(u_k−1, v_k−1)−bv_k−1 inΩ,

u_k=v_k=m on∂Ω.

(14) We claim that

0u_k−1u_km inΩ and

0v_k−1v_km inΩ.

Indeed, the property is straightforward ifk=1. Takek2 and assume by induction thatu_k−2u_k−1,v_k−2v_k−1 inΩ. Then,

(uk−uk−1)−a(uk−uk−1)=^g(uk−1, vk−1)−^g(uk−2, vk−2)−a(uk−1−uk−2) ^g(uk−1, vk−2)−^g(uk−2, vk−2)−a(uk−1−uk−2)

0 inΩ

and henceuk−uk−10 inΩ. The remaining inequalities are obtained similarly. In particular, the limits u= lim

k→∞uk, v= lim

k→∞vk

exist, solve (10) and satisfy

0um, 0vm inΩ.

Let us show now that the solution constructed in this way does not depend on a, b as long as these parameters satisfy (12). For this we argue as follows: suppose thatu, vare constructed usinga,bandu,˜ v˜are constructed witha˜,

˜

bsatisfying (12). Letu_k,v_kdenote the sequences defined by (13), (14). Arguing by induction we see that ifu˜u_k−1,

˜

vv_k−1then

(u˜−u_k)−a(u˜−u_k)=^g(u,˜ v)˜ −^g(u_k−1, v_k−1)−a(u˜−u_k−1) ^g(u, v˜ _k−1)−^g(u_k−1, v_k−1)−a(u˜−u_k−1)

0

and thenu˜−u_k0 inΩ. Note thatu→^g(u, v)−auandv→^f(u, v)−bvare monotone in the appropriate regions becauseu, vandu,˜ v˜are between 0 andm. Similarly,v˜−v_k0 inΩ and thusu˜u,v˜vinΩ. By symmetry we obtain the converse inequality and we deduce thatu˜=u,v˜=v.

Minimality.Letω⊂Ω be open andu,¯ v¯∈C(ω)¯ satisfy (11). Choosea,b large enough so thatg(u, v)−auis decreasing inuandf(u, v)−bvis decreasing invfor allu, vin the range 0u, vMwithMm,Mmaxω_¯u¯ andMmaxω_¯v.¯

Consideru_k,v_kdefined by (13), (14). Now we show thatu¯u_k,v¯v_kinωfor allk. By induction, ifu¯u_k−1,

¯

vv_k−1inωthen

(u¯−u_k)−a(u¯−u_k)^g(u,¯ v)¯ −^g(u_k−1, v_k−1)−a(u¯−u_k−1) ^g(u, v¯ _k−1)−^g(u_k−1, v_k−1)−a(u¯−u_k−1)

0 inω

and henceu¯−uk0 inω. 2

Proof of Theorem 1.1. Necessary conditions.Suppose that(u, v)is a solution to (2) and for givenγ >0 setw= min(γ u, v). LetχAdenote the characteristic function of a setA. By Kato’s inequality (see [14]),

wγ uχ_{[γ u<v]}+vχ_{[γ u>v]}

=γ g(u−v)χ_{[γ u<v]}+f (v−βu)χ_{[γ u>v]} γ g

(1−γ )u

χ_[γ u<v]+f

1−β γ

v

χ_[γ u>v]

=γ g 1−γ

γ w

χ_[γ u<v]+f

1−β γ

w

χ_[γ u>v]

max

γ g

1−γ

γ w

, f

1−β

γ

w

=:h1(w).

Hencewis a supersolution to the single equationu=h1(u) inΩ withu= +∞on∂Ω. Therefore this problem admits a solution and henceh1must satisfy the Keller–Osserman condition (1) (see e.g. [9]). Choosingγ=1 implies thatf satisfies (1) andβ <1. Then, choosingγ=β <1 implies thatgsatisfies (1).

Sufficient conditions.Consider the minimal solution(um, vm)to the truncated problem u=g(u−v) inΩ,

v=f (u−βv) inΩ, u=v=m on∂Ω

(15) wherem >0. Such a solution can easily be constructed by the method of sub and supersolutions, see Proposition 2.1.

Letγ∈(β,1)and set wm=max(γ um, vm).

Then,

w_mγ u_mχ_{[γ um>vm]}+v_mχ_{[γ um<vm]}

=γ g(u_m−v_m)χ_{[γ um>vm]}+f (v_m−βu_m)χ_{[γ um<vm]} γ g

1 γ −1

w_m

χ_{[γ um>vm]}+f

1−β γ

w_m

χ_{[γ um<vm]} h₂(w_m),

whereh₂(w)=min(γ g(¹⁻_γ^γw), f ((1−^β_γ)w)). Sinceh₂ satisfies (1) and is nondecreasing, the boundary blow up equation w=h2(w)inΩ,w= +∞on∂Ω has a maximal solutionw(obtained e.g. as the limit of(wn), where wndenotes the minimal blow-up solution on a subdomainΩnΩwith

nΩn=Ω). By comparison,wmwinΩ for allm >0. Hence(u_m),(v_m)remain bounded on compact sets ofΩ asm→ ∞, and by standard elliptic estimates they converge- up to a subsequence – inC_loc² (Ω)to a solution of (2). 2

Remark 2.2.The proof of Theorem 1.1 implies that whenever solutions exist, one of them is minimal in the class of nonnegative solutions. Moreover this solution(u, v)satisfies

βuvu inΩ. (16)

Indeed let us show that the minimal nonnegative solution(u_m, v_m)to (15) satisfiesv_mu_minΩ. For this let us recall thatu_m=limk→∞u_m,k,v_m=limk→∞v_m,kwhereu_m,k,v_m,kare defined recursively by (14) starting with the trivial solutions, withg(u, v)=g(u−v)andf(u, v)=f (v−βu). We chosea=blarge so that (12) is satisfied. We claim thatvm,kum,k. Proceeding inductively, assumevm,k−1um,k−1. Then

um,k−aum,k=g(um,k−1−vm,k−1)−a(um,k−1−vm,k−1)−avm,k−1−avm,k−1

while

v_m,k−av_m,k=f (v_m,k−1−βu_m,k−1)−av_m,k−1−av_m,k−1.

By the maximum principleu_m,kv_m,kinΩ. For the other inequality in (16) we may proceed similarly, but this time it is convenient to work withu˜_m,k,v˜_m,kdefined by (14) but with the boundary conditionsu˜_m,k=mandv˜_m,k=βm on∂Ω. The limit ofu˜_m,k,v˜_m,kask→ +∞and then asm→ +∞is the minimal nonnegative solution to the system, as can be seen by comparison.

Remark 2.3.For a general system (9) the same proof as that of Theorem 1.1 yields the following necessary condition for existence:

∀γ >0 max

γg w

γ, w

,f w

γ, w

satisfies (1). (17)

Similarly the next condition is sufficient for existence

∃γ >0 such that min

γg w

γ, w

,f w

γ, w

satisfies (1). (18)

However, these conditions are not equivalent in general, see Example 3.

Remark 2.4.For general cooperative systems the problem with finite boundary values (10) is always solvable and generates an increasing sequence of approximate solutions(u_m, v_m)_m∈N. Going back to (9), obtaining a sharp exis- tence criterion similar to the Keller–Osserman condition (1) is equivalent to characterizing the nonlinearities for which (u_m, v_m)_m∈Nremains bounded on compact subsets ofΩ. Such a result seems out of reach for general cooperative sys- tems. Still, it would be interesting to answer the following question: assume (9) admits a solution for any domain of the formΩ=(−R, R),R >0. Is it true that (2) is solvable in any smoothly bounded domainΩ⊂R^N,N2?

3. Asymptotics

Under the hypotheses onf stated in Theorem 1.1 and the Keller–Osserman condition (1) the problem

u=f (u) inΩ, u= +∞ on∂Ω (19)

admits a minimal solutionuand a maximal solutionU. The maximal solution can be constructed as the limitU= limδ→0u_δwhereu_δis the minimal solution of (19) in the domainΩ_δ= {x∈Ω: dist(x, ∂Ω) > δ}.

The next lemma is well known. It asserts that under hypothesis (3) all solutions to (19) have the same first order boundary behavior, see for instance [1,2] or [3].

Lemma 3.1.LetΩ be a bounded smooth domain inR^N. Assumef satisfies(3). Then for any solutionuof (19)we have

x→lim∂Ω

u(x) ψ (d(x))=1,

whereψ=φ⁻¹andφis the function appearing in(3).

Lemma 3.2.Supposef∼gat infinity and thatf satisfies(3). Letube any solution to(19)andvany solution to(19) with nonlinearity replaced byg. Then

x→lim∂Ω

u v =1.

Proof. LetG(t )=_t

0g(s) ds,φ_g(u)=_∞

u

√dt

2G(t ),ψ_g=φ_g⁻¹. Letφ_f andψ_f denote the corresponding functions associated tof. By Lemma 3.1 it suffices to prove that

δlim→0

ψ_f(δ) ψ_g(δ) =1.

Sincef ∼gat infinity we also haveF ∼Gat infinity and thereforeφ_f ∼φ_gat infinity. It follows from this and the fact thatφ_f satisfies the condition (3) thatφ_gsatisfies this condition too.

Letm >1. Condition (3) onφ_gimplies that there existsη >1 andδ0>0 such that

ψg(δ)mψg(ηδ) ∀0< δ < δ0. (20)

Sinceφ_f ∼φ_gat infinity we have

δlim→0

δ

φ_g(ψ_f(δ))=1.

Hence takingδ₁>0 small, δηφg

ψf(δ)

< δ0 ∀0< δ < δ1. Sinceψ_gis nonincreasing we deduce

ψ_g(δ)ψ_g ηφ_g

ψ_f(δ)

∀0< δ < δ₁ and by (20)

ψ_g(δ) 1

mψ_f(δ) ∀0< δ < δ1. It follows that

lim sup

δ→0

ψ_f(δ) ψ_g(δ) m

and sincem >1 was arbitrary, that lim sup

δ→0

ψ_f(δ) ψg(δ) 1.

The corresponding inequality for the liminf is proved by reversing the roles ofψ_f andψ_g. 2

The next lemma asserts that under the condition (3) the boundary behavior of solutions to (19) depends continu- ously on multiplicative perturbations of the nonlinearity.

Lemma 3.3.Assumef satisfies(3). Givenγ >0letu_γ denote any solution of u_γ=f (γ u_γ) inΩ, u_γ= +∞ on∂Ω.

Then lim sup

γ→1

lim sup

x→∂Ω

u_γ

u1 1lim inf

γ→1 lim inf

x→∂Ω

u_γ u1

.

Proof. Given γ >0 let f_γ(u)=f (γ u), F_γ(t )=_t

0f_γ(s) ds= _γ¹F (γ t ),φ_γ(u)=_∞

u

√dt

2Fγ(t ) = ^√1_γφ(γ u) and ψ_γ=(φ_γ)⁻¹. Note thatψ_γ(δ)=_γ¹ψ (√

γ δ).

By Lemma 3.1 it is enough to establish lim sup

γ→1

lim sup

δ→0

ψ_γ(δ)

ψ (δ) 1lim inf

γ→1 lim inf

δ→0

ψ_γ(δ) ψ (δ) .

Letm >1 andδ0>0,η >1 be such that (20) holds. Then if√γηit follows that 1

γψ (δ)m

γψ (ηδ)m γψ (√

γ δ)=mψγ(δ) ∀0< δ < δ0

and therefore lim inf

δ→0

ψ_γ(δ) ψ (δ) 1

γ m ∀0< γ < η². Asm >1 is arbitrary we deduce

lim inf

γ→1 lim inf

δ→0

ψ_γ(δ) ψ (δ) 1.

Similarly, letm >1 andδ₀>0,η >1 such that (20) holds. If√γ¹_η we have ψ_γ(δ)= 1

γψ (√ γ δ) 1

γψ (δ/η)m γψ (δ) forδ >0 small and therefore

lim sup

δ→0

ψ_γ(δ) ψ (δ) m

γ ∀γ 1 η². Hence

lim sup

γ→1

lim sup

δ→0

ψ_γ(δ)

ψ (δ) 1. 2

Proof of Theorem 1.2, part (a). Let(u, v)be any solution to (2) andw₁be the minimal nonnegative solution to (4) withθ=1. For simplicity we writew=w₁. First we note that we have

wvu. (21)

Indeed for the minimal solution(u, v), we always haveuvby (16). Consequently, v=f (v−βu)f

(1−β)v

sovis a supersolution of (4) and sincewis the minimal nonnegative solution it follows thatwv. Let z_θ=max(θ u, v),

where

β < θ <1.

By Kato’s inequality we have z_θh_θ(z)

withh_θ given by hθ(w)=min

θg

1−θ

θ w

, f

θ−β

θ w

. (22)

Letw_θ be the minimal solution to (4) andw˜_θ be the maximal solution to w˜θ=hθ(w˜θ) inΩ, w˜θ= +∞ on∂Ω.

Thenz_θw˜_θinΩ. Note that under condition (5), we haveh_θ(w)=f (^θ−_θ^βw)for largew. It follows from Lemma 3.2 that

x→lim∂Ω

˜ w_θ wθ =1 and therefore

lim sup

x→∂Ω

zθ

wθ 1

for anyθ∈(β,1). It follows from the previous inequality that lim sup

x→∂Ω

z_θ

w lim sup

x→∂Ω

z_θ

w_θ lim sup

x→∂Ω

w_θ

w lim sup

x→∂Ω

w_θ w . Letting nowθ→1 and using Lemma 3.3 we deduce that

lim sup

θ→1

lim sup

x→∂Ω

zθ

w 1.

This together with (21) yields the conclusion. 2

Proof of Theorem 1.2, part (b). We use Kato’s inequality with z_θ=max(θ u, v),

where

β < θ < θ0. We have

z_θh_θ(z)

withh_θ given by (22). By assumption (7), givenε >0,h_θ(t )(1−ε)f (^θ−_θ^βt )fortlarge. In particular, there exists a neighborhoodV of∂Ω,V ⊂Ω such that

z_θ(1−ε)f θ−β

θ z_θ

.

Letwε,θ denote the maximal solution of

wε,θ=(1−ε)f (^θ−_θ^βwε,θ) inV ,

w_ε,θ= +∞ on∂V .

Then,z_θw_ε,θinV. By Lemma 3.3 lim sup

ε→0,θ→θ0

lim sup

x→∂Ω

w_ε,θ

w 1, (23)

wherewis the minimal solution of (4) withθ=θ₀. Thus, lim sup

θ→θ₀

lim sup

x→∂Ω

z_θ

w 1. (24)

Letθ∈(θ₀,1)andz˜_θ=min(θ u, v). Then, as before,z˜_θw˜_ε,θ where noww˜_ε,θ is the minimal solution of

⎧⎨

⎩

w˜ε,θ=(1−ε)f (^θ−_θ^βw˜ε,θ) inV ,

˜

w_ε,θ= +∞ on∂Ω,

˜

w_ε,θ=τ on∂V \∂Ω

andτ >0 is a fixed small constant. Using Lemma 3.3 one proves that lim inf

ε→0,θ→θ0

lim inf

x→∂Ω

˜ w_ε,θ

w 1, (25)

whence lim inf

θ→θ0

lim inf

x→∂Ω

˜ zθ

w 1. (26)

Collecting (24) and (26), the theorem is proved. 2 4. Uniqueness

In this section, we prove Corollary 1.5, which states the uniqueness of the solutions of (2) provided thatf, g are nondecreasing, nonnegativeC¹functions such thatf=g=0 onR⁻that satisfy (3), that either condition (5) or (7) holds, and that

(a) eitherf, gare convex functions;

(b) orΩ is a ball and^{f (t )}_t ,^{g(t )}_t nondecreasing in a neighborhood of+∞, andf, gnondecreasing everywhere.

We begin with the proof of the uniqueness result assuming thatf, gare convex functions.

Proof of Corollary 1.5, part (a). Letε >0. Consider(u, v)the minimal BBUS solution and(u1, v1)another solution to (2). Actually, by (6) or (8) we have that(1+ε)u > u1u,(1+ε)v > v1vin a neighborhood of∂Ω.

Therefore, since by convexity ^{f (t )}_t , ^{g(t )}_t are increasing functions((1+ε)u, (1+ε)v)is a supersolution of (2):

(1+ε)ug((1+ε)u−(1+ε)v) inΩ, (1+ε)vf ((1+ε)v−β(1+ε)u) inΩ, (1+ε)u=(1+ε)v= ∞ on∂Ω.

Therefore,w:=u₁−(1+ε)u,z:=v₁−(1+ε)vsatisfyw0,z0 in a neighborhood of∂Ω, and w−g(ξ₁)w+g(ξ₁)z0 inΩ,

z−f(ξ₂)z+βf(ξ₂)w0 inΩ,

for someξ10,ξ20. Sinceg(ξ1)0,f(ξ2)0 and(−1+β)f(ξ2)0 we can apply the maximum principle for cooperative systems (see for example Appendix A of this paper or [19], Theorem 3.15 and its following remark) to conclude thatw0 andz0 inΩ. Lettingε→0, we obtain the desired inequality. 2

We now prove the uniqueness result relaxing the hypotheses onf, gifΩis a ball. This result compares with the uniqueness result in [9]. We begin with a lemma concerning the boundary behavior of the minimal solution(u, v)of our problem, and that is interesting in itself, since it is true for general domains and gives some insight of the boundary behavior of solutions.

Lemma 4.1.Letu, vdenote the minimal boundary blow-up solution of(2). Then,

x→lim∂Ω

u(x)−v(x)

= +∞, (27)

x→lim∂Ω

v(x)−βu(x)

= +∞. (28)

Proof. We establish (27), the other limit being similar. FixA >0. Consider the problem

u_A,m=g(u_A,m−v_A,m), (29)

v_A,m=f (v_A,m−βu_A,m), (30)

u_A,m=m+A, v_A,m=m on∂Ω. (31)

For a givenA >0, ifmis large enough then(m+A, m)is a supersolution to this problem and by the classical iterative method described in Proposition 2.1, we can construct a solution to this approximated problem. As we did in the proof of Theorem 1.1,(u_A,m, v_A,m)converges to the minimal BBUS solution whenm→ +∞.

In addition, we have that

(u_A,m−v_A,m)g(u_A,m−v_A,m), (32)

u_A,m−v_A,m=A on∂Ω. (33)

Thereforeu_A,m−v_A,mis a supersolution to a single equation problem. Setw_Afor the solution to

w_A=g(w_A) inΩ, (34)

w_A=A on∂Ω. (35)

Therefore, everywhere inΩ

wA(x)uA,m(x)−vA,m(x). (36)

We now letm→ +∞, thenA→ +∞that leads to

w(x)u(x)−v(x), (37)

wherewis the minimal BBUS solution to (34). 2 We now complete the proof of the uniqueness result.

Proof of Corollary 1.5, part (b). To fix ideas, assume thatΩ is the unit ball. Consider(u, v)the minimal solution to (2) and(u1, v1)the maximal solution (obtained as the limit(u1, v1)=limδ→0(uδ, vδ)where(uδ, vδ)is the minimal solution in the ball with radius 1−δ). Then it suffices to show thatu≡u1andv≡v1. It is worth to observe thatu, v,u₁,v₁are radial functions.

Considerr∈(0,1). LetΩ_r be the ball of radiusr. Then for eachx∈Ω_r there existξ, ξ∈Rsuch that

(u1−u)=g(u1−v1)−g(u−v)=g(ξ )(u1−u)−g(ξ )(v1−v), (38) (v1−v)=f (v1−βu1)−f (v−βu)=f(ξ)(v1−v)−βf(ξ)(u1−u). (39) By the maximum principle for cooperative systems, we have

sup

Ωr

(u1−u)max

u1(r)−u(r), v1(r)−v(r) ,

sup

Ω_r

(v₁−v)max

u₁(r)−u(r), v₁(r)−v(r) .

This ensures that the function r→M(r):=max

u1(r)−u(r), v1(r)−v(r) is nondecreasing in(0,1).

Assume that

u(0) < u1(0) or v(0) < v1(0) (40)

for if u(0)=u₁(0)andv₁(0)=v(0)by uniqueness for the system of ODEs we would haveu≡u₁andv≡v₁in (0,1)and the proof is over.

By Lemma 4.1 there isR0such that min(u−v, v−βu)(t )afor alltR0, whereais such that both ^{f (t )}_t ,^{g(t )}_t nondecreasing forta.

We argue in a slightly different way in the following cases:

there existsr∈(R₀,1)such thatu₁(r)−u(r) > v₁(r)−v(r), (41) there existsr∈(R₀,1)such thatu₁(r)−u(r) < v₁(r)−v(r), (42)

u₁(r)−u(r)=v₁(r)−v(r) for allr∈(R₀,1). (43)

To begin with we assume that (41) holds. In this case chooseR₁∈(R₀,1)such that

u₁(R₁)−u(R₁) > v₁(R₁)−v(R₁). (44)

Define

w:=u₁−(1+ε)u, z:=v₁−(1+ε)v and takeε >0 small enough such that

w(R₁) > z(R₁) and by (40)

w(R1) >0 or z(R1) >0.

Thus in particularw(R1) >0. We chooserε> R1close to 1, such thatw(rε) <0 andz(rε) <0. This is possible by (6) or (8).

In the annulus{r:R1< r < rε}we then have

((1+ε)u−u1)g((1+ε)u−(1+ε)v)−g(u1−v1), ((1+ε)v−v₁)f ((1+ε)v−β(1+ε)u)−f (v₁−βu₁).

Therefore,(w, z)satisfy in the annulus w−g(ξ₁)w+g(ξ₁)z0, z−f(ξ₂)z+βf(ξ₂)w0.

By the maximum principle in the annulusR₁rr_εwe have max(w, z)max

w(R₁), z(R₁)

=w(R₁) and hence

u1(r)−(1+ε)u(r)u1(R1)−(1+ε)u(R1).

Note that asε→0,rεcan be taken to approach 1. We then letε→0 to obtainu1(r)−u(r)u1(R1)−u(R1)for R1r <1. By continuityM(r)=u1(r)−u(r)in some interval aroundR1. Sincer→M(r)is nondecreasing we deduce thatM(r)=λ=const in some interval of the form[R₁, R₁+σ]withσ >0. From Eq. (38) we also have g(u₁−v₁)=g(u−v)in that interval. Butg(t )is strictly increasing fortaandu(r)−v(r)a,u₁(r)−v₁(r)a forrR₀if we chooseR₀close enough to 1. Hencev₁−v=λ=const in[R₁, R₁+σ]. This contradicts (44).

A similar argument rules out the case (42) and therefore we are in the situation (43). The previous argument with R₁replaced byR₀then yields

max(w, z)max

w(R₀), z(R₀)

, R₀rr_ε, which implies

u₁(r)−(1+ε)u(r)max

u₁(R₀)−(1+ε)u(R₀), v₁(R₀)−(1+ε)v(R₀)

forR₀rr_ε. Lettingε→0 we have

M(r)=u₁(r)−u(r)M(R₀), R₀r <1,

and sinceMis nondecreasing we conclude thatMis constant in[R₀,1). Thusu₁−u=λandv₁−v=μare constant in[R₀,1). Going back to the system we then have in the annulusR₀< r <1

0=(u1−u)=g(u1−v1)−g(u−v), 0=(v1−v)=f (v1−βu1)−f (v−βu).

Sincef, gare strictly increasing functions in the appropriate range we then haveu1−v1=u−vandv1−βu1= v−βuin R0< r <1. Henceλ=μ=0 and therefore u1−u=0 andv1−v=0 in[0,1). This completes the proof. 2

5. Examples

Example 1.The first example falls in case (a) of Theorem 1.2.

⎧⎨

⎩

u=e^u−v−1 inΩ, v=(v−βu)^p inΩ, u=v= ∞ on∂Ω,

(45) wherep >1, 0< β <1.

By Theorems 1.1, 1.2 and Corollary 1.5, problem (45) has a unique solution(u, v), which we know the leading order asymptotics of. We investigate here how the first equation affects the next terms in the asymptotic expansions ofu, v. We do this for simplicity in the casep >3.

Proposition 5.1.Assume0< β <1andp >3. Letd denote the function distance to the boundary. Then the unique solution(u, v)of (45)has the behavior

u=cd^−α+e₁logd+f₁+O d^ε

, v=cd^−α+e₂logd+f₂+O

d^ε

whereε >0is suitably small and the constants are uniquely determined by the equations α= 2

p−1, (46)

(1−β)^pc^p−1=α(α+1), (47)

e₁−e₂= −α−2, e₂−βe₁=0, (48)

e^f1−f2=cα(α+1), cα(α+1)pf2−βf1

(1−β)c = −e₂. (49)

Proof. The argument relies on constructing a sub- and a supersolution having a suitable boundary behavior. The reader is invited to check that the sub- and the supersolution that we construct depend continuously onΩ: this means that if a comparison principle is available (for systems with standard boundary conditions), then the solution(u, v)of (45) can be compared e.g. on an increasing sequence of domainsΩ_nΩ with the supersolutionu¯_n,v¯_nblowing-up on∂Ωn. Lettingn→ ∞and checking thatu¯n(x),v¯n(x)converge pointwise to the supersolutionu,¯ v¯blowing-up on

∂Ω, we obtain the desired inequality:uu,¯ vv. A similar approximation by outer domains enables us to compare¯ (u, v)with a given subsolution.

We finally note that the standard comparison principle for systems can be used, since forg(u, v)=e^u−v,f (u, v)= (v−βu)^pwe have^∂g_∂u+^∂g_∂v0,^∂g_∂v0 and ^∂f_∂u+^∂f_∂v0,^∂f_∂u 0. So it remains to construct the sub- and supersolution of (45). Letδ >0 be small and defineU_δ= {x∈Ω: d(x) < δ}.

We use as a supersolution

¯

u=cd^−α+e₁logd+f₁+g₁d^ε and v¯=cd^−α+e₂logd+f₂+g₂d^ε

where 0< ε < αandg₁, g₂>0 are to be fixed later on.

InU_δwe have

u¯=cα(α+1)d^−α−2−e1d⁻²−g1ε(1−ε)d^ε−2−cαd^−α−1d+e1d⁻¹d+g1εd^ε−1d, v¯=cα(α+1)d^−α−2−e2d⁻²−g2ε(1−ε)d^ε−2−cαd^−α−1d+e2d⁻¹d+g2εd^ε−1d and

e^u¯−¯v=cα(α+1)d^−α−2e^(g1−g2)dε

=cα(α+1)d^−α−2+cα(α+1)d^−α−2

e^(g1−g2)dε−1 . We takeg₁,g₂of the form

g1=t a1, g2=t a2

wheret1 anda₁,a₂>0 are fixed such that βa1< a2< a1.

Using convexity

e^u¯−¯vcα(α+1)d^−α−2+cα(α+1)(a1−a2)t d^ε−α−2 and hence

u¯−

e^u¯−¯v−1

−cα(α+1)(a1−a₂)t d^ε−α−2+Cd^−α−1+Ct d^ε−2 inU_δ whereCdepends only onp,β,Ω,a₁anda₂.

Again, using convexity

(v¯−βu)¯ ^pcα(α+1)d^−α−2+cα(α+1)pf₂−βf₁

(1−β)cd⁻²+cα(α+1)pa₂−βa₁ (1−β)ct d^ε−2. Sincep >3 we haveα∈(0,1). Hence, using (49) we find

v¯−(v¯−βu)¯ ^p−cα(α+1)pa₂−βa₁

(1−β)ct d^ε−2+Ct d^ε−1+Cd^−α−1 inU_δ.

Then there isδ >0 such thatu,¯ v¯is a supersolution of the system in the setU_δ= {x∈Ω: d(x) < δ}for anyt1.

Having fixedδwe now selecttlarge such that

¯

uu and v¯v ond(x)=δ.

It follows by comparison thatuu¯andvv¯inU_δ. The construction of a subsolutionu,vis similar. We take

u=cd^−α+e1logd+f1−a1t d^ε and v=cd^−α+e2logd+f2−a2t d^ε where 0< ε < α,a₁>0,a₂>0 are chosen such that

βa1< a2< a1

andt >0 is to be fixed later on. Let us introduce σ=a₂−βa₁>0.

Later on we will needσto be small.

Letδ >0 be small andU_δ= {x∈Ω: d(x) < δ}. Recall that the unique solutionu, vto (45) satisfiesu0,v0.

We takeClarge so that if

t=Cδ^−α−ε (50)

then

u0 and v0 atd(x)=δ.