on the existence of boundary blow-up solutions for a ...psauvy.perso.univ-pau.fr/documents/article...

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On the existence of boundary blow-up solutions fora general class of quasilinear elliptic systems

Sonia Ben Othman, Rym Chemmam and Paul Sauvy

Abstract. In this paper, we investigate the following quasilinear elliptic system (P) withexplosive boundary conditions:

∆pu = f1(x, u, v) in Ω ; u|∂Ω = +∞, u > 0 in Ω,

∆qv = f2(x, u, v) in Ω ; v|∂Ω = +∞, v > 0 in Ω,

where Ω is a smooth bounded domain of RN , N ≥ 2, 1 < p, q < +∞ and f1, f2 are twoCarathéodory functions in Ω× (R∗+ × R∗+). Under rather general conditions on f1 and f2and assuming the existence of a sub and supersolutions pair, we prove the existence of alarge solution to (P). Next, we give some applications in Biology.

1 Introduction

1.1 Presentation of the problem

In this paper we are interested in the following boundary blow-up quasilinear el-liptic system:

(P)

∆pu = f1(x, u, v) in Ω ; u|∂Ω = +∞, u > 0 in Ω,

∆qv = f2(x, u, v) in Ω ; v|∂Ω = +∞, v > 0 in Ω.

Here, Ω is a bounded domain of RN , N ≥ 2, with a C 1,α boundary ∂Ω, forsome 0 < α < 1. In the left-hand sides of the two partial differential equations,1 < p, q < +∞ and

∆rwdef= div(|∇w|r−2∇w)

denotes the r-Laplace operator. And in the right-hand sides, f1, f2 : Ω×(R∗+ × R∗+

)→

R satisfy several assumptions we detail later. We define a boundary blow-up solu-tion to (P) in the following sense:

Definition 1.1. A couple (u?, v?) ∈ [W1,ploc (Ω)∩C (Ω)]× [W1,q

loc(Ω)∩C (Ω)] is saidto be a weak positive boundary blow-up solutions pair (or just a large solution for

2 S.Ben Othman, R. Chemmam and P. Sauvy

short) to problem (P) if both u? and v? are positive in Ω, if

limd(x)→0+

u?(x) = limd(x)→0+

v?(x) = +∞,

where d(x) is the distance from any point x ∈ Ω to the boundary ∂Ω, and ifequations

∆pu? = f1(x, u

?, v?) and ∆qv? = f2(x, u

?, v?) in Ω

are satisfied in the distribution sense; i.e., for any couple of test functions (ϕ,ψ) ∈C∞c (Ω)× C∞c (Ω),∫

Ω

|∇u?|p−2∇u? · ∇ϕ dx =

∫Ω

f1(x, u?, v?)ϕ dx

and ∫Ω

|∇v?|q−2∇v? · ∇ψ dx =

∫Ω

f2(x, u?, v?)ψ dx.

Boundary blow-up solutions in different competitive, cooperative or predator/preysystems may arise in population dynamics problems with "spacial heterogeneity".Namely, in these type of non-homogeneous systems, one of the equations containsan absorption term with a multiplying weight which vanishes in a subset ω of thedomain Ω. This traduces that the growth of the associated specie, may be "out ofcontrol" in ω and a blow-up phenomenon may occur at the boundary ∂ω. We referto [5, 9, 10, 29] for more details about the modelling. We may also cite here [7]where an other type of boundary blow-up elliptic system, related to a fluid dynam-ics model, is investigated. But, this problem involves a gradient dependency in thenon-linearity of one of the equations, which is not exactly the subject of this paper.

One of the main difficulties of boundary blow-up problems relies on the a pri-ori lack of knowledge of the precise blow-up behaviour of the solutions near theboundary. Then, the direct use of comparison principles or Sobolev spaces (evenweighted), in the whole domain Ω, is inappropriate in this context. Despite this,sub and supersolution methods may be employed to study some particular bound-ary blow-up elliptic systems. We may cite here for instance [6,12,15,16,22,27,37],for the semilinear case (i.e. p = q = 2) and most recently [3,13,23,36,38], wherethe quasilinear case is investigated.

In the above cited papers, the authors take advantage of the particular nature(competitive, cooperative or predator/prey) of the system they considered to con-struct large solutions. Namely, the nature of the system is related to the mono-tonicity of the right-hand sides f1 with respect to v and f2 with respect to u. Then,

On general quasilinear systems with boundary blow-up 3

from an explicit sub and supersolutions pair, they use a monotone iterative schemetechnique which provides the existence of a large solution to the system.

However, in these previous mentioned studies, the definition of sub and super-solutions used and the monotonicity method employed seem different with respectto the competitive, the cooperative or the predator/prey configuration. Neverthe-less, to our knowledge, no general existence theorem, i.e. which does not take intoaccount of the nature of the system, has been written yet for boundary blow-upsystems. Moreover, it is also obvious that all biological or chemical interactionscannot be model only by one of the three above configurations. For instance, someGierer-Meinhardt elliptic systems (see [19]) contain both a competitive and a co-operative contribution in each right-hand side, which results in a sign-changingbehaviour of f1 and f2. Nevertheless, the previous sub and supersolutions meth-ods used cannot easily be generalized for this type of systems.

For these two reasons, in this paper, our goal is to develop a new sub and su-persolution method allowing to study the most general class of boundary blow-upsystems shaped as (P) as possible; in particular encompassing all the above men-tioned type of systems (competitive, cooperative, predator/prey and with "sign-changing" right-hand sides). For this, we want to get read of any monotonicityor sign assumption on f1 and f2. Then, we only make in this study the followingquite non-restrictive hypotheses on the right-hand sides:

(H1) f1 and f2 are two Carathéodory functions on Ω×(R∗+ × R∗+

).

(H2) For any (t1, t2) ∈ R∗+ × R∗+, f1(·, t1, t2) and f2(·, t1, t2) are L∞loc(Ω).

(H3) For a.e. x ∈ Ω, f1(x, ·, ·) and f2(x, ·, ·) are of class C 1(R∗+ × R∗+).

1.2 Contents of the paper

The main result of this paper, Theorem 2.1, is given in section 2. From the knowl-edge of a sub and supersolutions pair, it establishes the existence of a large solutionto (P) under an additional growth assumption on f1 and f2. To overcome the dif-ficulties generated by the lack of monotonicity of the right-hand sides, we firstgeneralise the definition of sub and supersolutions pairs to (P) (see Definition 1.2below). Then, we combine a fixed point approach based on an idea developed in[17, 21] with some "interior considerations" extracted from [13, 15].

In a second time, we illustrate the contribution of our main result. For that, wegive in section 3 a first example of application considering a boundary blow-up


system shaped as ∆pu = K1(x)ua1vb1 in Ω ; u|∂Ω = +∞, u > 0 in Ω,

∆qv = K2(x)va2ub2 in Ω ; v|∂Ω = +∞, v > 0 in Ω.

(1.1)

The choice to focus on this particular class of quasilinear systems is motivatedby several previous publications (see [3, 13, 23, 36, 38]) where (1.1) is studiedthroughout different configurations. Namely, under the standard super-homogeneitycondition

(a1 − p+ 1)(a2 − q + 1)− b1b2 > 0, (1.2)

the authors give the same kind of results on the existence/non-existence, the unique-ness and the asymptotic behaviour of solutions to (1.1) in the competitive or inthe cooperative case. More precisely, in [13, 38] the competitive case (i.e. withb1, b2 > 0) is investigated. Whereas in [3, 36] the authors considered the coop-erative case (i.e. with b1, b2 < 0). Accurately, [3] starts to study (1.1) withoutany weight Ki, for i = 1, 2 and [36] gives a generalization considering weightsbehaving like a power of distance function. Namely, for some λ1 < p and λ2 < q,

limd(x)→0+

d(x)λ1K1(x) = 1 and limd(x)→0+

d(x)λ2K2(x) = 1. (1.3)

Concerning problem (1.1), we bring here two contributions:

(i) The first one is that, we do not impose any sign assumption on b1 and b2, thuswe study the competitive, cooperative and predator/prey (i.e. with b1b2 < 0)cases in the same way. For that, condition (1.2) is naturally replaced by

(a1 − p+ 1)(a2 − q + 1)− |b1b2| > 0.

To our knowledge, this is the first time that the predator/prey case is investi-gated for problem (1.1).

(ii) The second one relies on the adding of Karamata perturbation in assumption(1.3) on the weights Ki(x). This makes the computations harder but allowsto widen the class of systems which may be considered.

Then, under appropriate assumptions on K1,K2 and exponents a1, a2, b1 and b2,we prove in Theorem 3.5 both the existence and the precise behaviour of largesolutions to (1.1).

Actually, Theorem 3.5 is a direct application of Theorem 2.1 which relies onthe validity of the existence of a suitable sub and supersolutions pair to (1.1). For


this, we will first make the study of a connected boundary blow-up scalar ellipticequation shaped as ∆rw = K(x)wδ, w > 0 in Ω,

w|∂Ω = +∞.(1.4)

Precisely, under suitable assumptions on the parameters, we establish in Propo-sition 3.3 (whose proof is given in Appendix A) the existence and the precisebehaviour of a large solution to (1.4).

Concerning the boundary blow-up problem (1.4), we refer to [14] where a de-tailed history of this equation, containing a list of the main references on the topic,is presented. We can add to this list [30] (and references therein), where an al-ternative method to the sub and super solutions method is developed via Keller-Osserman methods (see [26,32]); and also [1], where one of the first studies (with[2]) on a boundary blow-up equation containing Karamata perturbation is made.

Finally, in section 4, we although give in Theorems 4.1 and 4.2 two otherillustrations of Theorem 2.1 throughout the studies of systems presenting sign-changing right-hand sides.

Note that in this paper, we only focus on the existence of a large solution to (P).For this reason, the (although interesting) questions of non-existence or uniquenessof solutions to problems (1.1) and (1.4) will not be raised here (see points (iii) inRemarks 3.4 and 3.6 for more details).

1.3 Notations

Throughout this paper, we will use the following notations and definitions:

(i) For x ∈ Ω, we denote the distance from x to ∂Ω by

d(x)def= d(x, ∂Ω) = inf

y∈∂Ω

d(x, y).

(ii) We denote by D def= sup

x,y∈Ω

d(x, y), the diameter of the domain Ω.

(iii) Let f, g ∈ C (Ω) be two positive functions in Ω. Then, we write

f(x) ≈ g(x) in Ω

if there exist C1, C2 > 0 such that C1g ≤ f ≤ C2g in Ω.


(iv) Let 1 < r < +∞. We denote by ϕ1,r ∈ W1,r0 (Ω), the (unique) positive and

Lr-renormalized eigenfunction corresponding to the first eigenvalue of ther-Laplace operator:

λ1,rdef= inf

‖∇v‖Lr(Ω) ∈ R, v ∈W1,r

0 (Ω) and ‖v‖Lr(Ω) = 1

(see [33, §1.4 - p. 20]). By definition, it is the unique weak solution to thefollowing eigenvalue problem:

−∆rw = λ1,rwr−1, w ≥ 0 in Ω,

w|∂Ω = 0, ‖w‖Lr(Ω) = 1.

We just recall that, from Moser iterations [31] and [28, Theorem 1], ϕ1,r ∈C 1,β

(Ω), for some 0 < β < 1, and from the strong maximum principle for

quasilinear operators (see [35, Theorem 10]), ϕ1,r satisfies

ϕ1,r(x) ≈ d(x) in Ω (1.5)

andϕ1,r(x)

r + |∇ϕ1,r(x)|r ≈ 1 in Ω. (1.6)

(v) Let w,w ∈ W1,rloc(Ω) ∩ C (Ω) be two positive functions such that w ≤ w in

Ω. We define the convex set

[w,w]def=w ∈W1,r

loc(Ω) ∩ C (Ω), w ≤ w ≤ w in Ω

.

Definition 1.2. Let u, u ∈ W1,ploc (Ω) ∩ C (Ω) and v, v ∈ W1,q

loc(Ω) ∩ C (Ω) be fourpositive functions such that u ≤ u in Ω, v ≤ v in Ω and

limd(x)→0+

u(x) = limd(x)→0+

v(x) = +∞.

The couple (u, v) and (u, v) is said to be a sub and supersolutions pair to (P) ifthe following inequalities are satisfied in the distribution sense:

∆pu ≥ f1 (x, u, v) in Ω, for any v ∈ [v, v] , (1.7)

∆qv ≥ f2 (x, u, v) in Ω, for any u ∈ [u, u] , (1.8)

∆pu ≤ f1 (x, u, v) in Ω, for any v ∈ [v, v] , (1.9)

∆qv ≤ f2 (x, u, v) in Ω, for any u ∈ [u, u] . (1.10)

Then, the conical shell [u, u]× [v, v] of sub and supersolutions is denoted by C.


2 Main result

Theorem 2.1. Let (u, v), (u, v) ∈ [W1,ploc (Ω)∩C (Ω)]× [W1,q

loc(Ω)∩C (Ω)] be a suband supersolutions pair to (P). Assume in addition that there exist g1, g2 ∈ L∞loc(Ω)such that: ∀ a.e.x ∈ Ω, ∀(u, v) ∈ C,

∂f1

∂u(x, u, v) ≤ g1(x) and

∂f2

∂v(x, u, v) ≤ g2(x). (2.1)

Then, there exists a large solution (u?, v?) ∈ C to problem (P). Moreover, (u?, v?) ∈C 1,β

loc (Ω)× C 1,βloc (Ω), for some 0 < β < 1.

Remark 2.2.

(i) In the semilinear case, if we assume that f1, f2 are locally β-Hölder continu-ous with respect to x ∈ Ω, for some 0 < β < 1, the Schauder’s estimates (seefor instance [20, Theorem 6.14]) ensure us that (u, v) ∈ C 2,β

loc (Ω)×C 2,βloc (Ω).

Thus, (u, v) is a large solution to (P) in the classical sense.

(ii) Instead of assumptions (H3) and (2.1), we can rather suppose that there existg1, g2 ∈ L∞loc(Ω) and %1, %2 > 1 such that : ∀ a.e.x ∈ Ω, ∀(u, v) ∈ C,

w 7→ f1(x,w, v)− g1(x)w%1−1 is non increasing on [u, u],

w 7→ f2(x, u, w)− g2(x)w%2−1 is non increasing on [v, v].

In this case, we get the same result and this condition is sharper if %1, %2 > 2.For that, it suffices to replace the first equation of problem (Pn1 ), given below,by

∆pun + h1(x, un) = f1(x, u, v) + g1(x)u%1−1 in Ωn,

with h1 : Ωn × R→ R+ the cut-off function defined as follows:

h1(x, z)def=

g1(x)u

%1−1 if z ≥ u(x),g1(x)z

%1−1 if z ∈ [u(x), u(x)] ,

0 if z ≤ u(x)(2.2)

and proceed similarly for (Pn2 ).

Proof. (of Theorem 2.1) We follow here the idea of [15, Theorem A.2] and [13,Theorem A.2] considering at first the system at the interior of Ω. For this, weconsider (Ωn)n∈N∗ an increasing sequence (for the inclusion) of sub-domains of


Ω with smooth boundaries (at least C 1,α) such that Ωn −−−−→n→+∞

Ω in the Hausdorff

topology with : ∀n ∈ N∗ (large enough), ∀x ∈ Ωn,

D

n+ 1≤ d(x, ∂Ω) <

D

n.

And for any n ∈ N∗, we look for a weak solution to problem:

(Pn)

∆pun = f1(x, un, vn) in Ωn ; un|∂Ωn = u, u > 0 in Ωn,

∆qvn = f2(x, un, vn) in Ωn ; vn|∂Ωn = v, vn > 0 in Ωn.

Remark 2.3. It is always possible to construct a such sequence (Ωn)n∈N∗ becauseof the regularity we assume on the domain Ω (see for instance [4, Theorem A.2p.27]).

As stated in the introduction, we use here a fixed-point argument. Then, letn ∈ N∗ (large enough), let (u, v) be a fixed couple of the conical shell C of suband supersolutions and let us consider problems (Pn1 ) and (Pn2 ) defined as follows:

(Pn1 )

∆pun − h1(x, un) = f1(x, u, v)− g1(x)u, un > 0 in Ωn,

un|∂Ω = u,

(Pn2 )

∆qvn − h2(x, vn) = f2(x, u, v)− g2(x)v, vn > 0 in Ωn,

vn|∂Ω = v

with g1, g2 defined in assumption (2.1) and h1, h2 : Ω × R → R+ the followingcut-off functions:

h1(x, z)def=

g1(x)u if z ≥ u(x),g1(x)z if z ∈ [0, u(x)] ,

0 if z ≤ 0,

h2(x, z)def=

g2(x)v if z ≥ v(x),g2(x)z if z ∈ [0, v(x)] ,

0 if z ≤ 0.

From now on and without loss of generality, we will only study problem (Pn1 ) andconclude in the same way for problem (Pn2 ).


In problem (Pn1 ), h1 is a Carathéodory function on Ω×R. Thus, for a.e.(x, s) ∈Ω× R, setting

H1(x, s)def=

∫ s

0h1(x, z)dz,

we consider the following functional: ∀w ∈W1,p0 (Ωn),

E n1 (w)

def=

1p

∫Ωn

|∇(w + u)|p dx+

∫Ωn

H1(x,w + u) dx

+

∫Ωn

[f1(x, u, v)− g1(x)u] (w + u) dx.

Then, E n1 is well defined in W1,p

0 (Ωn) and, taking into account that

f1(x, u, v)− g1(x)u ∈ L∞(Ωn),

from Hölder’s and Poincaré’s inequalities, we get : for any w ∈W1,p0 (Ωn),

E n1 (w) ≥ 1

p

(‖∇w‖Lp(Ωn) − ‖∇u‖Lp(Ωn)

)p− ‖f1(x, u, v) + g1(x)u‖Lp′ (Ωn)

((λ1,p)

− 1p ‖∇w‖Lp(Ωn) + ‖u‖Lp(Ωn)

)(2.3)

and E n1 is bounded from below in W1,p

0 (Ωn). So, let us define

In1def= inf

w∈W1,p0 (Ωn)

E n1 (w) (2.4)

and let (wk)k∈N ⊂W1,p0 (Ωn) be a minimizing sequence of E n

1 , i.e.

limk→+∞

E n1 (wk) = In1 .

Using (2.3) and Poincaré’s inequality, (wk)k∈N is bounded in W1,p0 (Ωn). There-

fore, there exist a subsequence (wk(m))m∈N and an element u∗n ∈ W1,p0 (Ωn) such

that wk(m) −−−−−→m→+∞

u∗n, weakly in W1,p0 (Ωn) and a.e. in Ωn. In particular,

lim infm→+∞

∥∥wk(m)

∥∥W1,p

0 (Ωn)≥ ‖u∗n‖W1,p

0 (Ωn)

and using Fatou’s lemma,

lim infm→+∞

∫Ωn

H1(x,wk(m)

)dx ≥

∫Ωn

H1 (x, u∗n) dx.


Hence, E n1 (u∗n) = In1 and u∗n is a solution to the following Euler-Lagrange equa-

tion:∫Ωn

|∇(u∗n + u)|p−2∇(u∗n + u) · ∇w dx+

∫Ωn

h1(x, u∗n + u)w dx

=

∫Ωn

(g1(x)u− f1(x, u, v))w dx, (2.5)

Thus, undef= u∗n + u ∈ W1,p(Ωn) satisfies both un|∂Ωn = u, in the sense of the

trace, and

∆pun − h1(x, un) = f1(x, u, v)− g1(x)u in D ′(Ωn). (2.6)

Furthermore, since

f1(x, u, v)− g1(x)u+ h1(x, un) ∈ L∞(Ωn),

it follows from [28, Theorem 1] (see also [8, 34]) that un ∈ C 1,β′n(Ωn), forsome 0 < β′n < 1, only depending on p, N , α, ‖g1‖L∞(Ωn), ‖u‖C (Ωn)

and‖f1‖L∞(Ωn×[mn,Mn]), with

mndef= min

x∈Ωn

u(x) and Mndef= max

x∈Ωn

u(x).

Now, it remains to prove that un ∈ [u, u]. Then, combining (2.5) with (1.7), we

get: ∀w ∈W1,p0 (Ωn)

+ def= w ∈W1,p

0 (Ωn), w ≥ 0 a.e in Ωn,∫Ωn

(|∇un|p−2∇un − |∇u|p−2∇u

).∇w dx+

∫Ωn

[h1(x, un)− h1(x, u)]w dx

≥∫

Ωn

[(g1(x)u− f1(x, u, v)

)−(g1(x)u− f1(x, u, v)

)]w dx.

By assumption (H1), applying this inequality with w = (un−u)− ∈W1,p0 (Ωn)

+,we get (by monotonicity of the p-Laplace operator) that un ≥ u in Ωn and, on theway, that un is a weak solution to (Pn1 ). Similarly, combining (2.5) with (1.9), wealso get un ≤ u in Ωn. Then, (2.6) becomes

∆pun − g1(x)un = f1(x, u, v)− g1(x)u in D ′(Ωn). (2.7)

Finally, if we suppose that wn ∈ C 1,β′n(Ωn) is another weak solution to (Pn1 ),

w∗ndef= wn − u ∈ W1,p

0 (Ω) satisfies (2.5). Thus, applying again this comparison


principle, one can show that wn = un in Ωn. And then, (Pn1 ) has a unique weaksolution.

Let us denote

Cndef=(u, v) ∈ C (Ωn)× C (Ωn), u ≤ u ≤ u and v ≤ v ≤ v in Ωn

,

which is a convex set, closed for the usual uniform norm in C (Ωn) × C (Ωn).Then, remarking that Cn is nothing more than the set of the restrictions to Ωn ofthe functions of C, we can compose an order reversing mapping :

Tn : Cn −−−→ Cn

(u, v) 7−→ (Tn1 (u, v), Tn2 (u, v))

def= (un, vn),

where un (resp. vn) is the unique solution to problem (Pn1 ) (resp. (Pn2 )) we juststudy. With this notations, it is now clear that any fixed-point of the mappingTn is a weak solution to (Pn) and vice-versa. Precisely, we aim to apply herethe Schauder fixed-point theorem. Hence, it just remains to prove that Tn is acontinuous mapping and that Tn(Cn) is relatively compact subspace of Cn.

• (Relative compactness of Tn(Cn))It directly follows from the uniform bound we get in the Hölder’s spaceC 1,β′n(Ωn). Precisely, it suffices to remark that since (u, v) ∈ C, in (2.7)the term f1(x, u, v) − g1(x)(u − un) is bounded in L∞(Ωn), uniformly inΩn×Cn. Therefore, from [28, Theorem 1], there exists a constantC > 0 onlydepending on p, N , Ωn, α, ‖g1‖L∞(Ωn), ‖u‖C (Ωn)

and ‖f1‖L∞(Ωn×[mn,Mn])

such that∀(u, v) ∈ Cn, ‖Tn1 (u, v)‖C 1,β′n (Ωn)

≤ C. (2.8)

Similarly, we get the same kind of uniform C 1,β′n estimate for Tn2 on Cn andwe finally get the sufficient equicontinuity in C (Ωn) × C (Ωn) to apply theAscoli-Arzelá theorem.

• (Continuity of Tn)Let (uk, vk)k∈N ⊂ Cn be a sequence converging to an element (u, v) ∈ Cn ask → +∞ for the usual uniform norm in C (Ωn) × C (Ωn). For any k ∈ N,let us denote Tn1 (u

k, vk) = ukn ∈ C 1,β′n(Ωn), the unique solution to (Pn1 )associated to (uk, vk). It follows from (2.8) and the well known compactembedding C 1,β′n(Ωn) W1,p(Ωn)∩C (Ωn) that there exist a subsequence(uk(m)n )m∈N and φn ∈W1,p(Ωn) ∩ C (Ωn) such that

uk(m)n −−−−−→

m→+∞φn in W1,p(Ωn) ∩ C (Ωn). (2.9)


Then, passing to the limit as m → +∞, φn satisfies together φn|∂Ωn = u,u ≤ φn ≤ u in Ωn and

∆φn − g1(x)φn = f1(x, u, v)− g1(x)u in D ′(Ωn).

From the uniqueness of the solution to (Pn1 ), it follows that φn = Tn1 (u, v).Finally, using a standard argument relying on the uniqueness of the solutionto (Pn1 ), we prove that all the sequence (ukn)k∈N converges to Tn1 (u, v). Andwe complete the proof of the continuity of Tn arguing in the same way forproblem (Pn2 ).

Then, from the Schauder fixed-point theorem, we get a fixed-point (un, vn) ∈ Cnof the mapping Tn which is a weak solution to (Pn). Now, for any k ≥ n, wedefine the functions ukn, v

kn on the whole domain Ω as follows:

ukndef= 1Ωn · uk + 1Ω\Ωn · u in Ω,

vkndef= 1Ωn · vk + 1Ω\Ωn · v in Ω.

Then for n = 1, from above estimate (2.8), the sequence (uk1)k≥1 is bounded inC 1,β′(Ω1). Therefore, since C 1,β′1(Ω1) C 1(Ω1), there exist a subsequence(uk1(m)1 )m∈N and u?1 ∈ C 1(Ω1) such that

uk1(m)1 −−−−−→

m→+∞u?1 in C 1(Ω1). (2.10)

In the same way, by iteration we construct for any n ≥ 2, (ukn(m)n )m∈N a subse-

quence of (ukn−1(m)n )m∈N and u?n ∈ C 1(Ωn) such that

ukn(m)n −−−−−→

m→+∞u?n in C 1(Ωn). (2.11)

However, by definition we have for any n′ ≥ n and any m ∈ N,

ukn′ (m)n = u

kn′ (m)n′ in Ωn.

Hence, passing to the limit as m → +∞, it follows that u?n = u?n′ in Ωn. Takinginto account of this remark, we can construct a function u? ∈ C 1

loc(Ω), such thatu ≤ u? ≤ u in Ω, as follows: ∀x ∈ Ω,

u?(x)def= u?i (x), i ≥ n0(x),

with n0(x) ∈ N∗ the unique integer satisfying

D

n0(x) + 1≤ d(x, ∂Ω) <

D

n0(x).


Let Ω′ ⊂⊂ Ω and nΩ′ ∈ N∗ the unique integer such that

ΩnΩ′ ⊂ Ω

′ ⊂ ΩnΩ′+1.

Therefore, by a Cantor’s diagonal argument, the sequence (ukm(m)m )m≥n

Ω′ con-verges to u? in C 1(Ω

′) as m → +∞. Then, we construct in the same way a

function v? ∈ C 1loc(Ω) such that v ≤ v? ≤ v in Ω, and passing to the limit as

m → +∞ in (2.7) and in the similar equation satisfied by vn, (u?, v?) is a largesolution to (P). Finally, the C 1,β

loc (Ω) × C 1,βloc (Ω) regularity of (u?, v?) follows

from the well-known interior regularity result [8, Theorem 2] (see also [34, Theo-rem 2]).

3 Applications

In this section we focus on the following quasilinear elliptic and singular system:

(P)

∆pu = K1(x)ua1vb1 in Ω ; u|∂Ω = +∞, u > 0 in Ω,

∆qv = K2(x)va2ub2 in Ω ; v|∂Ω = +∞, v > 0 in Ω.

In this problem,

(i) Exponents a1 > p − 1, a2 > q − 1 and b1, b2 6= 0 satisfy the super-homogeneity condition:

(a1 − p+ 1)(a2 − q + 1)− |b1b2| > 0; (3.1)

which is equivalent to the existence of a positive constant σ > 0 such that

(a1 − p+ 1)− σ|b1| > 0 and σ(a2 − q + 1)− |b2| > 0. (3.2)

(ii) K1,K2 are two positive functions of C (Ω) satisfying

K1(x) ≈ d(x)−λ1L1(d(x)) and K2(x) ≈ d(x)−λ2L2(d(x)) in Ω,(3.3)

where λ1 < p, λ2 < q and for i = 1, 2, Li is a Karamata function, i.e. alower perturbation in C 2((0, η]), with η > D, which can be written in theform:

∀t ∈ (0, η], Li(t) = exp(∫ η

t

zi(s)

sds), (3.4)

with zi ∈ C ([0, η])∩C 1((0, η]) satisfying zi(0) = 0 and limt→0+

tz′i(t) = 0. We

denote by K the Karamata class (see [24, 25]) of all functions Li satisfyingthe above definition.


Remark 3.1. We just mention here some useful properties of the Karamatafunctions we use next. The interested reader should find the proofs in [1, §2]and more details on the topic in [24, 25].

a. Let L1, L2 ∈ K and p1, p2 ∈ R. Then, L1p1 · L2

p2 also belongs to K.b. Let L ∈ K. Then, it satisfies the following asymptotic behaviour:

∀ε > 0, limt→0+

t−εLi(t) = +∞ and limt→0+

tεLi(t) = 0. (3.5)

c. A function L is in K if and only if L is a positive function of C 2((0, η])satisfying

limt→0+

tL′(t)

L(t)= lim

t→0+

t2L′′(t)

L(t)= 0. (3.6)

d. Let L ∈ K and ε > 0. Then, from (1.5) and the mean value theorem,the asymptotic behaviour of L(εϕ1,r) is given by,

L(εϕ1,r(x)) ≈ L(d(x)) in Ω. (3.7)

Example 3.2. Let m ∈ N∗, C > 0 and ω >> η large enough. Let us define

∀t ∈ (0, η], Li(t) = C

m∏n=1

(logn

(ωt

))µn

,

where, logndef= log · · · log (n times) and µn ∈ R. Then, Li belongs to K.

We discuss here the existence of large solutions to problem (P). For that, re-garding Theorem 2.1, we take

f1(x, u, v) = K1(x)ua1vb1 and f2(x, u, v) = K2(x)v

a2ub2

and construct a suitable sub and supersolutions pair to (P).

Preliminary results

Let 1 < r < +∞. Then, let δ > r − 1 and K ∈ C (Ω), such that

K(x) ≈ d(x)−λL(d(x)) in Ω, (3.8)

with λ ≤ r and L ∈ K a lower perturbation function satisfying (3.4). In view ofconstructing a suitable sub and supersolutions pair to (P), we first introduce thefollowing problem:

(Q)

∆rw = K(x)wδ, w > 0 in Ω

w|∂Ω = +∞.


Proposition 3.3. Under the above hypotheses,

(i) Assume that λ < r. Then, problem (Q) has a positive solutionw ∈ C 1,β′loc (Ω),

for some 0 < β′ < 1, which satisfies

w(x) ≈ d(x)r−λr−1−δL(d(x))

1r−1−δ in Ω. (3.9)

(ii) Assume that λ = r and assume in addition that L satisfies the followingintegrability condition: ∫ η

0t−1L(t)

1r−1 dt < +∞. (3.10)

Then, problem (Q) has a positive solution w ∈ C 1,β′loc (Ω), for some 0 < β′ <

1, which satisfies

w(x) ≈

(∫ d(x)

0t−1L(t)

1r−1 dt

) r−1r−1−δ

in Ω. (3.11)

Proof. See section A in Appendix.

Remark 3.4.

(i) This result extends [1, Theorem 3] to the r-Laplace operator.

(ii) This result may easily be generalized to quasilinear problems with (possi-ble singular) weights K(x) shaped as (3.8), where exponent λ = λ(x) isa Hölder’s continuous function in Ω. As a consequence, it also extends theexistence result of [14, Theorem 1] to quasilinear problems with (possiblesingular) weights containing a Karamata perturbation.

(iii) We recall that the aim of the study of problem (Q) is to guarantee the exis-tence of suitable sub and supersolutions pairs in order to apply Theorem 2.1in the example of this section. For this reason, we deliberately occult hereany question about the non-existence or the uniqueness of solution. Never-theless, may notice that, due to the presence of a Karamata perturbation in theweighted term K(x), the question of the uniqueness, particularly, seems notobvious and, to our knowledge, has not been considered yet in the previouspapers mentioned in [14].


3.1 Main Results

Theorem 3.5. Assume that exponents a1, a2 ∈ R and b1, b2 6= 0 in problem (P)satisfy hypothesis (3.1). Set

α1 =q − 1 − a2

(a1 − p+ 1)(a2 − q + 1)− b1b2, α2 =

p− 1 − a1

(a1 − p+ 1)(a2 − q + 1)− b1b2,

β1 =b1

(a1 − p+ 1)(a2 − q + 1)− b1b2, β2 =

b2

(a1 − p+ 1)(a2 − q + 1)− b1b2

γ1 =(p− λ1)(q − 1 − a2) + (q − λ2)b1

(a1 − p+ 1)(a2 − q + 1)− b1b2, γ2 =

(q − λ2)(p− 1 − a1) + (p− λ1)b2

(a1 − p+ 1)(a2 − q + 1)− b1b2

and assume that

(p− λ1)(q − 1− a2) + (q − λ2)b1 < 0, (3.12)

(q − λ2)(p− 1− a1) + (p− λ1)b2 < 0. (3.13)

Then, problem (P) possesses a large solution (u?, v?) ∈ C 1,β′loc (Ω)× C 1,β

loc (Ω), forsome 0 < β < 1, that satisfies the following estimates:

u?(x) ≈ d(x)γ1L1(d(x))α1L2(d(x))

β1 in Ω, (3.14)

v?(x) ≈ d(x)γ2L2(d(x))α2L1(d(x))

β2 in Ω. (3.15)

Remark 3.6.

(i) Assume that L1(t)α1L2(t)

β1 −−−→t→0+

+∞. Then, conclusion of the above the-

orem still holds if

(p− λ1)(q − 1− a2) + (q − λ2)b1 = 0.

And symmetrically, assume that L2(t)α2L1(t)

β2 −−−→t→0+

+∞. Then, conclu-

sion of the above theorem still holds if

(q − λ2)(p− 1− a1) + (p− λ1)b2 = 0.

(ii) This theorem unifies the different existence results stated in [3, 13, 36].

(iii) For the same reason as in point (iii) of Remark 3.4, we do not investigatehere the questions of non-existence or uniqueness of large solutions to (P).Nevertheless, concerning the non-existence, from the necessary and suffi-cient condition on the existence of a large solution given in [13, Theorem 1],conditions (3.12) and (3.13) seem to be sharp.


Proof. (of Theorem 3.5) We adapt here the method used in [17, 21] to constructa sub and supersolutions pair to (P). Thanks to Proposition 3.3, we apply The-orem 2.1 with a suitable choice of sub and supersolutions pair (u, v), (u, v) ∈C 1,β′

loc (Ω)× C 1,β′loc (Ω), for some 0 < β′ < 1, in the following form:

u = mψ1 and u = m−1ψ1 in Ω, (3.16)

v = mσψ2 and v = m−σψ2 in Ω, (3.17)

where σ > 0 is given in (3.2), 0 < m < 1 is an appropriate constant small enoughand ψ1, ψ2 ∈ C 1,β′

loc (Ω) are given by Proposition 3.3 as respective solutions toproblems

(Q1)

∆pw = d(x)−λ1L1(d(x))wδ1 , w > 0 in Ω,

w|∂Ω = +∞

and

(Q2)

∆qw = d(x)−λ2L2(d(x))wδ2 , w > 0 in Ω,

w|∂Ω = +∞,

satisfying some cone conditions we specify below. We choose suitable perturba-tions L1,L2 ∈ K and suitable values of exponents δ1, δ2 > 1 in order to satisfy

∆pψ1 ≈ K1(x)ψ1a1ψ2

b1 and ∆qψ2 ≈ K2(x)ψ2a2ψ1

b2 in Ω; (3.18)

which provide us inequalities (1.7) to (1.10) in order to apply Theorem 2.1. Here,we look for large solutions (u?, v?) ∈ C 1,β

loc (Ω)× C 1,βloc (Ω), for some 0 < β < 1,

to (P) by making the "Ansatz" that

u?(x) ≈ d(x)γ1L1(d(x))α1L2(d(x))

β1 in Ω,

v?(x) ≈ d(x)γ2L2(d(x))α2L1(d(x))

β2 in Ω,

for some γ1, γ2 < 0 (or γ1, γ2 = 0 under assumption of Remark 3.6) and α1, α2,β1, β2 ∈ R. For that, we take in problems (Q1) and (Q2) :

L1 = L1ν1 · L2

µ1 and L2 = L2ν2 · L1

µ2 in Ω,

where ν1, ν2, µ1, µ2 ∈ R are suitable exponents we fix later. By Proposition 3.3,ψ1, ψ2 ∈ C 1,β′

loc (Ω), for some 0 < β′ < 1 and satisfy

ψ1(x) ≈ d(x)p−λ1p−1−δ1 L1(d(x))

ν1p−1−δ1 L2(d(x))

µ1p−1−δ1 in Ω, (3.19)


ψ2(x) ≈ d(x)q−λ2q−1−δ2 L2(d(x))

ν2q−1−δ2 L1(d(x))

µ2q−1−δ2 in Ω. (3.20)

In view of satisfying estimates given in (3.18), the comparison of the term ∆ψ1with K1(x)ψ1

a1ψ2b1 on one side, and the term ∆ψ2 with K2(x)ψ2

a2ψ1b2 on the

other side, imposes the exponents ν1, ν2, µ1, µ2 and δ1, δ2 to satisfy the followingsystem:

δ1p− λ1

p− 1− δ1= a1

p− λ1

p− 1− δ1+ b1

q − λ2

q − 1− δ2,

δ2q − λ2

q − 1− δ2= a2

q − λ2

q − 1− δ2+ b2

p− λ1

p− 1− δ1,

ν1p− 1

p− 1− δ1= 1 + a1

λ1

p− 1− δ1+ b1

µ2

q − 1− δ2,

ν2q − 1

q − 1− δ2= 1 + a2

λ2

q − 1− δ2+ b2

µ1

p− 1− δ1,

µ1p− 1

p− 1− δ1= a1

µ1

p− 1− δ1+ b1

ν2

q − 1− δ2,

µ2q − 1

q − 1− δ2= a2

µ2

q − 1− δ2+ b2

ν1

p− 1− δ1.

Then, we get

γ1 =p− λ1

p− 1− δ1=

(p− λ1)(q − 1− a2) + (q − λ2)b1

(a1 − p+ 1)(a2 − q + 1)− b1b2, (3.21)

γ2 =q − λ2

q − 1− δ2=

(q − λ2)(p− 1− a1) + (p− λ1)b2

(a1 − p+ 1)(a2 − q + 1)− b1b2, (3.22)

α1 =ν1

p− 1− δ1=

q − 1− a2

(a1 − p+ 1)(a2 − q + 1)− b1b2, (3.23)

α2 =ν2

q − 1− δ2=

p− 1− a1

(a1 − p+ 1)(a2 − q + 1)− b1b2, (3.24)

β1 =µ1

p− 1− δ1=

b1

(a1 − p+ 1)(a2 − q + 1)− b1b2, (3.25)

β2 =µ2

q − 1− δ2=

b1

(a1 − p+ 1)(a2 − q + 1)− b1b2, (3.26)

Then, under assumption (3.1), to get γ1, γ2 < 0, we have to impose conditions(3.12) and (3.13) on exponents a1, a2, b1, b2 and λ1, λ2.


Now, we would like to find a suitable constant 0 < m < 1 in order to satisfyinequalities (1.7) to (1.10) for (u, v), (u, v) ∈ C 1,β′

loc (Ω) × C 1,β′loc (Ω) defined in

(3.16) and (3.17). Then, let (u, v) ∈ [u, u]× [v, v].

(i) Concerning the subsolutions inequalities (1.7) and (1.8). On one hand wehave using estimates (3.19) and (3.20)

∆pu ≥ mC1L1(d(x))ν1+δ1α1L2(d(x))

µ1+δ1β1d(x)δ1γ1−λ1 in Ω,

∆qv ≥ mσC ′1L2(d(x))ν2+δ2α2L1(d(x))

µ2+δ2β2d(x)δ2γ2−λ2 in Ω.

And on the other hand,

K1(x)ua1vb1 ≤ C2m

a1−σ|b1|Λ1(d(x))d(x)a1γ1+b1γ2−λ1 in Ω,

K2(x)va2ub2 ≤ C ′2mσa2−|b2|Λ2(d(x))d(x)

a2γ2+b2γ1−λ2 in Ω,

with Λ1 = L11+a1α1+b1β2 ·L2

a1β1+b1α2 and Λ2 = L21+a2α2+b2β1 ·L1

a2β2+b2α1 .

(ii) Concerning the supersolutions inequalities (1.9) and (1.10), we get similarlyon one hand

∆pu ≤ m−1C3L1(d(x))ν1+δ1α1L2(d(x))

µ1+δ1β1d(x)δ1γ1−λ1 in Ω,

∆qv ≤ m−σC ′3L2(d(x))ν2+δ2α2L1(d(x))

µ2+δ2β2d(x)δ2γ2−λ2 in Ω.

And on the other hand,

K1(x)ua1vb1 ≥ C4m

−a1+σ|b1|Λ1(d(x))d(x)a1γ1+b1γ2−λ1 in Ω,

K2(x)va2ub2 ≥ C ′4m−σa2+|b2|Λ2(d(x))d(x)

a2γ2+b2γ1−λ2 in Ω.

Then from assumption (3.2) and thanks to equalities (3.21) to (3.26), (u, v), (u, v)is a sub and supersolutions pair to problem (P), for m small enough. Therefore,estimates (1.7) to (1.10) hold. Finally, taking into account of the growth propertyof Karamata’s functions (3.5) recalled in Remark 3.1, conditions (2.1) on the right-hand sides hold for this choice of sub and supersolutions pair choosing g1, g2 ∈L∞loc(Ω) as follows:

g1(x)def= k1d(x)

γ1(a1−1)+γ2b1−ε and g2(x)def= k2d(x)

γ2(a2−1)+γ1b2−ε,

with ε > 0 and k1, k2 > 0 large enough. Then, applying Theorem 2.1, the proofof the existence of a large solution (u?, v?) ∈ C 1,β

loc (Ω)× C 1,βloc (Ω), for some 0 <

β < 1, to problem (P) satisfying estimates (3.14) and (3.15) is now complete.

Remark 3.7. Unfortunately, this approach does not allow us to investigate the crit-ical case λ1 = p or λ2 = q. Indeed, in this case, we cannot construct a suitablesub and supersolutions pair shaped has (3.16) and (3.17).


4 Other examples of applications

In this section, we give two extra examples of application of Theorem 2.1 for otherGierer-Meinhardt systems also arising in population dynamics, modelling morecomplex behaviours between two biological species than in the previous section.From now, for sake of clarity we do not consider right-hand sides containing sin-gular weights Ki(x) as in (3.3) anymore. Nevertheless, all the results stated belowcan be easily generalised to systems with singular weights in this form.

4.1 Second example

We consider now the following elliptic system:

(P)

∆pu = ua1vb1 − uα1vβ1 in Ω ; u|∂Ω = +∞, u > 0 in Ω,

∆qv = va2ub2 − vα2uβ2 in Ω ; v|∂Ω = +∞, v > 0 in Ω,

where the above exponents satisfy

(a1 − p+ 1)− σ|b1| > 0 and (a1 − α1)− σ(|b1|+ |β1|) > 0, (4.1)

σ(a2 − q + 1)− |b2| > 0 and σ(a2 − α2)− (|b2|+ |β2|) > 0, (4.2)

for some constant σ > 0. Then, we have the following result:

Theorem 4.1. Set

γ1 =p(q − 1− a2) + qb1

(a1 − p+ 1)(a2 − q + 1)− b1b2, (4.3)

γ2 =q(p− 1− a1) + pb2

(a1 − p+ 1)(a2 − q + 1)− b1b2(4.4)

and assume that

p(q − 1− a2) + qb1 < 0 and (α1 − a1)γ1 + (β1 − b1)γ2 > 0, (4.5)

q(p− 1− a1) + pb2 < 0 and (α2 − a2)γ2 + (β2 − b2)γ1 > 0. (4.6)

Then, problem (P) possesses a large solution (u?, v?) ∈ C 1,βloc (Ω)× C 1,β


u?(x) ∼ d(x)γ1 and v?(x) ∼ d(x)γ2 in Ω. (4.7)


Proof. We apply Theorem 2.1 with

u = mψ1, u = m−1ψ1 and v = mσψ2, v = m−σψ2 in Ω, (4.8)

where σ > 0 is the constant given in (4.1) and (4.2), 0 < m < 1 is a positiveconstant small enough and ψ1, ψ2 ∈ C 1,β′

loc (Ω), for some 0 < β′ < 1, are given byProposition 3.3 as respective solutions to problems

(Q1)

∆pw = wδ1 , w > 0 in Ω,

w|∂Ω = +∞

and

(Q2)

∆qw = wδ2 , w > 0 in Ω,

w|∂Ω = +∞,

with suitable exponents δ1 > p− 1, δ2 > q − 1 such that ψ1 and ψ2 satisfies

∆pψ1 ≈ ψ1a1ψ2

b1 and ∆qψ2 ≈ ψ2a2ψ1

b2 in Ω. (4.9)

From Proposition 3.3,

ψ1(x) ≈ d(x)γ1 and ψ2(x) ≈ d(x)γ2 in Ω, (4.10)

with γ1 = pp−1−δ1

and γ2 = qq−1−δ2

. Therefore, similarly to the previous proof,the above condition (4.9) imposes exponents a1, a2, b2, b2 and δ1, δ2 to satisfy:

δ1p

p− 1− δ1=

a1p

p− 1− δ1+

b1q

q − 1− δ2,

δ2q

q − 1− δ2=

a2q

q − 1− δ2+

b2p

p− 1− δ2.

That is to say, γ1 and γ2 are given by (4.3) and (4.4) and must be negative.

Now, let (u, v) ∈ [u, u]× [v, v].

(i) Concerning the subsolutions inequalities (1.7) and (1.8). On one hand wehave using estimates (4.10),

∆pu ≥ mC1d(x)δ1γ1 and ∆qv ≥ mσC ′1d(x)

δ2γ2 in Ω.

And in the other hand,

ua1vb1 − uα1vβ1 ≤ ma1−σ|b1|C2d(x)a1γ1+b1γ2 in Ω,

va2ub2 − vα2uβ2 ≤ ma2−σ|b2|C ′2d(x)a2γ2+b2γ1 in Ω.


(ii) Concerning the supersolutions inequalities (1.9) and (1.10), we get similarlyon one hand

∆pu ≤ m−1C3d(x)δ1γ1 and ∆qv ≥ m−σC ′3d(x)δ2γ2 in Ω.

And on the other hand, from the second inequalities in assumptions (4.1) and(4.5), we get for m small enough

ua1vb1 − uα1vβ1 ≥ mσ|b1|−a1ψ1a1ψ2

b1[1−m(a1−α1)−σ(|β1|+|b1|)ψ1

α1−a1ψ2β1−b1

]≥ mσ|b1|−a1C4d(x)

a1γ1+b1γ2 in Ω,

and from the second inequalities in assumptions (4.2) and (4.6),

va1ub2 − vα2uβ2 ≥ m|b2|−σa2ψ2a2ψ1

b2[1−mσ(a2−α2)−(|β2|+|b2|)ψ2

α2−a2ψ1β2−b2

]≥ m|b2|−σa2C ′4d(x)

a2γ2+b2γ1 in Ω,

Then, from the first inequalities in (4.1) and (4.2) and thanks to the choice of γ1, γ2,(u, v), (u, v) is a sub and supersolutions pair to problem (P), for m small enough.And we conclude in the same way as in the proof of Theorem 3.5.

4.2 Third example

Finally, we consider for instance the following elliptic system:

(P)

∆pu = ua1vb1 + µ1uα1 + ν1v

β1 in Ω ; u|∂Ω = +∞, u > 0 in Ω,

∆qv = va2ub2 + µ2vα2 + ν2v

β2 in Ω ; v|∂Ω = +∞, v > 0 in Ω,

where µ1, µ2 ∈ R, ν1, ν2 ∈ R and α1, α2, β1, β2, a1, a2, b1, b2 satisfy

(a1 − p+ 1)− σ|b1| > 0 and σ(|b1| − |β1|)− α1 > 0, (4.11)

σ(a2 − q + 1)− |b2| > 0 and (|b2| − |β2|)− σα2 > 0, (4.12)

for some constant σ > 0. Then, we have the following result:

Theorem 4.2. Set

γ1 =p(q − 1− a2) + qb1

(a1 − p+ 1)(a2 − q + 1)− b1b2, γ2 =

q(p− 1− a1) + pb2

(a1 − p+ 1)(a2 − q + 1)− b1b2

and assume that

p(q − 1− a2) + qb1 < 0 and (α1 − a1)γ1 − b1γ2 > 0, (4.13)


q(p− 1− a1) + pb2 < 0 and (α2 − a2)γ2 − b2γ1 > 0. (4.14)

Then, problem (P) possesses a large solution (u?, v?) ∈ C 1,βloc (Ω)× C 1,β


u?(x) ∼ d(x)γ1 and v?(x) ∼ d(x)γ2 in Ω. (4.15)

Proof. It suffices to use the same strategy as in the proof of Theorem (3.5). Thenunder assumptions (4.11) to (4.14), it is also always possible to construct a sub andsupersolutions pair to (P) shaped as (4.8), for m small enough.

A Proof of Proposition 3.3

We adapt the computations which have already been made in [18] to the case ofboundary blow-up solutions. In this section, for sake of simplicity, the eigenvalueλ1,r and the eigenfunction ϕ1,r defined in the introduction are respectively merelydenoted by λ1 and ϕ1.

Case 1 : λ < r. For t ∈ (0, η] we define a Karamata function Θ ∈ K defined asfollows:

Θ(t)def= exp

(∫ η

t

y(s)

sds

), (A.1)

with y ∈ C ([0, η])∩C 1 ((0, η]) such that y(0) = 0 and limt→0+

ty′(t) = 0, carefully

chosen. Let β < 0. Therefore, from (3.6) in Remark 3.1, we get that

limε→0+

(supx∈Ω

εϕ1(x)Θ′(εϕ1(x))

Θ(εϕ1(x))

)= lim

ε→0+

(supx∈Ω

(εϕ1(x))2Θ′′(εϕ1(x))

Θ(εϕ1(x))

)= 0

Hence, there exists ε > 0 (small enough) such that: ∀x ∈ Ω,

−β2≤ −β − ϕ1(x)Θ

′(εϕ1(x))

Θ(εϕ1(x))≤ −3β

2

and

β(β − 1)2

≤ β(β−1)+2βϕ1(x)Θ

′(εϕ1(x))

Θ(εϕ1(x))+ϕ1(x)

2Θ′′(εϕ1(x))

Θ(εϕ1(x))≤ 3

2β(β−1).

Taking into account of this above consideration, let us define

w(x)def= ϕ1(x)

βΘ (εϕ1(x)) in Ω. (A.2)


Then, a direct computation gives us

∆rw =(Θ(εϕ1)

)r−1ϕ1

(β−1)(r−1)−1(−β − εϕ1Θ

′(εϕ1)

Θ(εϕ1)

)r−2

×[λ1ϕ1

r

(−β − εϕ1Θ

′(εϕ1)

Θ(εϕ1)

)+ (r − 1)|∇ϕ1|r

(β(β − 1) + 2β

εϕ1Θ′(εϕ1)

Θ(εϕ1)+

(εϕ1)2Θ′′(εϕ1)

Θ(εϕ1)

)].

Therefore, from the two last inequalities, (1.6) and (3.7), we get

∆rw(x) ≈ Θ(ϕ1(x))r−1ϕ1(x)

(β−1)(r−1)−1 in Ω.

On the other hand, from (3.7), we also have

K(x)w(x)δ ≈ L(ϕ1(x))Θ (ϕ1(x))δ ϕ1(x)

δβ−λ in Ω.

Hence, since δ > r − 1, if we choose β def= r−λ

r−1−δ < 0 and y(t) def= z(t)

r−1−δ fort ∈ [0, η], w both satisfies w|∂Ω = +∞ and

∆rw(x) ≈ K(x)w(x)δ in Ω. (A.3)

Next, we definew = mw and w = m−1w in Ω,

with a suitable 0 < m < 1. Using previous estimate (A.3) and taking into accountof the super-homogeneous nature (with respect to r) of problem (Q), it is easy tosee that form small enough, w,w are respectively a sub and a supersolution to (Q)in Ω. And finally, applying a classic sub and supersolution method in the interiorof Ω (as in the proof of Theorem 2.1), we get the existence of a weak solutionw ∈ C 1,β′

loc (Ω), for some 0 < β′ < 1, to (Q), satisfying

w ≤ w ≤ w in Ω.

Case 2 : λ = r. Similarly to Case 1, for a suitable ε > 0 (small enough), wedefine for x ∈ Ω,

w(x)def= Θ(εϕ1(x)) in Ω;

that is to say with β = 0 in (A.2). Once again, we look for a suitable Θ ∈ Kin order to satisfy (A.3), but this time we furthermore assume that Θ has to bedecreasing in (0, η]. Then, let us remark in particular that, from (1.5) and themean value theorem, we also have

−Θ′(εϕ1(x)) ≈ −Θ′(ϕ1(x)) in Ω. (A.4)


Then, in this case we get

∆rw = ϕ1−1 (−εΘ′(εϕ1)

)r−1[λ1ϕ1

r − (r − 1)|∇ϕ1|rεϕ1Θ

′′(εϕ1)

Θ′(εϕ1)

].

Let us claim, for the moment, that there exists

limε→0+

(supx∈Ω

εϕ1(x)Θ′′(εϕ1(x))

Θ′(εϕ1(x))

)∈ (−∞, 0). (A.5)

Hence, on one hand, together with (1.6) and (A.4), we get that

∆rw(x) ≈ ϕ1(x)−1 (−Θ′(ϕ1(x))

)r−1 in Ω.

And on the other hand, from (3.7), we also have

K(x)w(x)δ ≈ L(ϕ1(x))Θ(ϕ1(x))δϕ1(x)

−r in Ω.

Then, to get (A.3), it suffices that Θ ∈ K satisfies(−Θ′(t)

)r−1 ≈ t1−rL(t)Θ(t)δ in (0, η]. (A.6)

Then, under assumption (3.10), let us define for t ∈ (0, η],

Θ(t)def=

(∫ t

0s−1L(s)

1r−1 ds

) r−1r−1−δ

.

First, by definition of L, we remark that Θ is a positive and decreasing function ofC 2((0, η]) which satisfies (3.6). Hence, from the characterization of Karamata’sfunctions recalled in point c. of Remark 3.1, it follows that Θ ∈ K. Secondly,a direct computation ensures us that Θ satisfies (A.6). Thirdly, an other directcomputation gives us

εϕ1(x)Θ′′(εϕ1(x))

Θ′(εϕ1(x))=

1r − 1

(εϕ1(x)L

′(εϕ1(x))

L(εϕ1(x))

)+

δ

r − 1

(εϕ1(x)Θ

′(εϕ1(x))

Θ(εϕ1(x))

)− 1

Then, taking into account that L,Θ ∈ K satisfy (3.6), claim (A.5) holds. Andfinally, estmate (A.3) also holds for this suitable choice of function Θ and the restof the proof is similar to the first case.


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Author information

Sonia Ben Othman, Département de Mathématiques,Faculté des Sciences de Tunis,Campus universitaire,2092 Tunis, Tunisia.E-mail: [email protected]

Rym Chemmam, Département de Mathématiques,Faculté des Sciences de Tunis,Campus universitaire,2092 Tunis, Tunisia.E-mail: [email protected]

Paul Sauvy, Institut Mathématique de Toulouse - CeReMath,Université Toulouse 1 - Capitole,Manufacture des Tabacs,21, Allée de Brienne,31000 Toulouse, France.E-mail: [email protected]

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