**WEAKLY COUPLED SYSTEM VIA BLOW UP**

MARTA GARC´IA-HUIDOBRO, IGNACIO GUERRA AND RA ´UL MAN´ASEVICH Abstract. The existence of positive solutions to certain systems of ordi- nary diﬀerential equations is studied. Particular forms of these systems are satisﬁed by radial solutions of associated partial diﬀerential equations.

1. Introduction

In this paper we will study existence of positive solutions to a system of the form

(D)

*−(r*^{N−1}*φ** _{i}*(u

^{}*(r)))*

_{i}*=*

^{}*r*

^{N−1}*f*

*(u*

_{i}*(r)),*

_{i+1}*i*= 1, . . . , n

*u*

^{}*(0) = 0 =*

_{i}*u*

*i*(R),

where it is understood that *u** _{n+1}*=

*u*

_{1}

*.*Here, for

*i*= 1, . . . , n, the functions

*φ*

*i*are odd increasing homeomorphisms from Ronto R and the

*f*

*i*:R

*→*R are odd continuous functions such that

*sf*

*(s)*

_{i}*>*0 for

*s= 0. Also*

*=*

^{}

_{dr}

^{d}*.*

System (D) is particularly important when the homeomorphisms*φ** _{i}* take
the form

*φ*

*(s) =*

_{i}*sa*

*(|s|), s*

_{i}*∈*Rsince it is satisﬁed by the radial solutions of the system

(P)

div(a*i*(|∇u*i**|)∇u**i*) +*f**i*(u*i+1*(|x|)) = 0, x*∈*Ω,
*i*= 1, . . . , n
*u** _{i}*(|x|) = 0, x

*∈∂Ω,*

where Ω denotes the ball inR* ^{N}* centered at zero and with radius

*R >*0.

1991*Mathematics Subject Classiﬁcation.* Primary 35Jxx; Secondary 34B15.

*Key words and phrases.* Radial solutions, Leray-Schauder degree, blow up, asymptoti-
cally homogeneous functions.

This work was partially supported by EC grant CI 1* ^{∗}* - CT93 - 0323 and Fondecyt
grant 1970332.

Received: October 8, 1997.

c

*1996 Mancorp Publishing, Inc.*

105

Furthermore, concerning the functions*φ**i**, f**i*,*i*= 1, . . . , n, we will assume
that they belong to the class of asymptotically homogeneous functions (AH
for short). We say that *h*:R*→*Ris AH at +∞of exponent*δ >*0 if for any
*σ >*0

*s→+∞*lim
*h(σs)*

*h(s)* =*σ*^{δ}*.*
(1.1)

By replacing +∞ by 0 in (1.1) we obtain a similar equivalent deﬁnition
for a function *h* to be AH of exponent *δ* at zero. AH functions have been
recently used in [GMU] and [GMS] in connection with quasilinear problems.

They form an important class of non homogeneous functions which without being necessarily asymptotic to any power have the suitable homogeneous asymptotic behavior given by (1.1). In a very diﬀerent context they have been used in applied probability and statistics where they are known as regularly varying functions, see for example [R], [S].

By a solution to (D) we understand a vector function **u** = (u_{1}*, . . . , u** _{n}*)
such that

**u**

*∈C*

^{1}([0, T],R

*) and*

^{n}*φ*

*(u*

_{i}

^{}*)*

_{i}*∈C*

^{1}([0, T],R), i = 1, . . . , n, which satisﬁes (D).

In [CMM], the existence of solutions with positive components for a system
of the form (D) with *n* = 2 and with the functions *φ** _{i}* and

*f*

*having the particular form*

_{i}*φ*

*(s) =*

_{i}*|s|*

^{p}

^{i}

^{−2}*s, φ*

*(0) = 0, p*

_{i}

_{i}*>*1, f

*(s) =*

_{i}*|s|*

^{δ}

^{i}

^{−1}*s, f*

*(0) = 0,*

_{i}*δ*

_{i}*>*0, i= 1,2,was done. In [GMU], within the scope of the AH functions, the case of a single equation was considered. In both situations the central idea to obtain a-priori bounds was the blow-up method of Gidas and Spruck, see [GS]. As a consequence of our results in this paper, those in [CMM] and [GMU] are greatly generalized.

Next we develop some preliminaries in order to state our main theorem.

For *i*= 1, . . . , n, let *δ*_{i}*,δ*¯* _{i}* be positive real numbers and

*p*

_{i}*,p*¯

*real numbers greater than one, and assume that the functions*

_{i}*φ*

_{i}*, f*

*,*

_{i}*i*= 1, . . . , nsatisfy

(H1) lim

*s→+∞*

*φ**i*(σs)

*φ** _{i}*(s) =

*σ*

^{p}

^{i}

^{−1}*,*lim

*s→+∞*

*f**i*(σs)
*f** _{i}*(s) =

*σ*

^{δ}

^{i}*,*for all

*σ >*0,

(H2) ^{}^{n}

*i=1*

*δ**i*

(p_{i}*−*1) *>*1.

To the exponents *p**i**, δ**i*, let us associate the system
(AS)

(p_{i}*−*1)E_{i}*−δ*_{i}*E** _{i+1}* =

*−p*

_{i}*, i*= 1, . . . , n,

*E*

*=*

_{n+1}*E*

_{1}

*.*

From (H_{2}), it turns out that (AS) has a unique solution (E_{1}*, . . . , E** _{N}*),such
that

*E*

*i*

*>*0 for each

*i*= 1, . . . , n. An explicit form for these solutions is given in the Appendix at the end of the paper.

Now we can establish our main existence theorem.

**Theorem 1.1.** *For* *i*= 1, . . . , n, *let* *φ**i* *be odd increasing homeomorphisms*
*from*R*onto* R*andf** _{i}*:R

*→*R

*odd continuous functions withxf*

*(x)*

_{i}*>*0

*for*

*x= 0, which satisfy (H*1*), (H*2*), and*

(H_{3}) lim

*s→0*

*φ** _{i}*(σs)

*φ** _{i}*(s) =

*σ*

^{p}^{¯}

^{i}

^{−1}*,*lim

*s→0*

*f** _{i}*(σs)

*f*

*(s) =*

_{i}*σ*

^{δ}^{¯}

^{i}*,*

*for any*

*σ >*0.

*Additionally, for*

*i*= 1, . . . , n,

*let us assume that*

(H_{4}) ^{}^{n}

*i=1*

¯*δ*_{i}

(¯*p*_{i}*−*1) *>*1,
(H_{5}) *p*_{i}*< N, i*= 1, . . . , n, max

*i=1,... ,n**{E*_{i}*−θ*_{i}*} ≥*0,

*where* *θ**i* = ^{N−p}_{p}_{i}_{−1}^{i}*and the* *E**i**s* *are the solutions to (AS). Then problem*
(D) *has a solution* (u1*, . . . , u**n*) *such that* *u**i*(r) *>* 0, r *∈* [0, R), *for each*
*i*= 1, . . . , n.

The plan of this paper is as follows. We begin section 2 by discussing some properties of the AH functions that will be used throughout the paper.

Then we provide an abstract functional analysis setting for problem (D)
so that ﬁnding solutions to that problem is equivalent to solving a ﬁxed
point problem. Section 3 is ﬁrst devoted to the study of a-priori bounds for
positive solutions to problem (D) and then to prove our main theorem by
using Leray Schauder degree arguments. To show the a-priori bounds we
argue by contradiction and thus by using some suitable *rescaling functions*
we ﬁnd that there must exist a vector solution **v** = (v_{1}*, . . . , v** _{n}*) deﬁned on
[0,+∞) (vector ground state) to a system of the form

(D* _{p}*)

*−(r*^{N−1}*|v*_{i}* ^{}*(r)|

^{p}

^{i}

^{−2}*v*

^{}*(r))*

_{i}

^{}=*C**i**r*^{N−1}*|v**i+1*(r)|^{δ}^{i}^{−1}*v**i+1*(r), r*∈*[0,+∞),
*i*= 1, . . . , n,
*v*_{i}* ^{}*(0) = 0, v

*(r)*

_{i}*≥*0, r

*∈*[0,+∞),

where*v**n+1* =*v*1 and*C**i* are positive constants,*i*= 1, . . . , n.We observe here
the interesting fact that in this asymptotic system only properties of *φ*_{i}*, f** _{i}*
at +∞ appear. We reach then a contradiction, and hence the existence of
a-priori bounds, by using hypothesis (H

_{5}) which prevents the existence of such a vector ground state.

In all of our previous argument the existence of suitable rescaling func-
tions is crucial. The lemma for their existence (as well as some of their
key properties) is stated without proof at the beginning of section 3 and its
proof (which is delicate and rather lengthy and technical) is postponed to
section 4. In section 5 we give some applications that illustrate our exis-
tence result. In particular, in example 5.2 we apply our existence results to
a system that contains operators of the form (−∆* _{p}*)

^{n}*,*(−∆

*)*

_{q}

^{m}*,*where for

*t >*1 ∆

_{t}*u*:= div(|∇u|

^{t−2}*∇u).*We end the paper with an Appendix which contains some technical results.

We introduce now some notation. Throughout the paper vectors in R* ^{n}*
will be written in boldface.

*C*# will denote the closed linear subspace of

*C[0, R] deﬁned by*

*C*

_{#}=

*{u*

*∈*

*C[0, R]*

*|u(R) = 0}.*We have that

*C*

_{#}is a

Banach space with respect to the norm* · *:= * · **∞**.* Also we will denote
by *C*_{#}^{n}*,*the Banach space of the *n−tuples of elements of* *C*_{#} endowed with
the norm*||u||** _{n}*:=

^{}

^{n}*i=1**||u*_{i}*||,*where**u**= (u_{1}*, . . . , u** _{n}*)

*∈C*

_{#}

^{n}*.*

Finally we adopt the following conventions. By R+ and R^{+} we mean
[0,+∞) and (0,+∞) respectively. For a function*H*:R*→*R(with lim

*s→0*
*H*(s)

*s* =

0) we deﬁne ˆ*H(s) :=* ^{H(s)}_{s}*, s= 0,* *H(0) = 0,*ˆ and we note that if *H* is AH
of exponent *p* (at +∞ or zero) then ˆ*H* is AH of exponent *p−*1. Also if
*γ*_{i}*, i* = 1, . . . , n, are real numbers or functions, we deﬁne *γ** _{n+i}* =

*γ*

*for all*

_{i}*i*= 1, . . . , n.

2. Preliminaries and Abstract Formulation We begin this section with a proposition.

**Proposition 2.1.** *Let* *h* :R *→* R *be a continuous function with* *h(0) = 0,*
*th(t)>*0 *for* *t= 0, and let* *H(t) :=*^{}_{0}^{t}*h(s)ds* *and* *H*ˆ :R*→* R *as deﬁned in*
*the Introduction.*

(i) *If* *h* *is AHof exponent* *ρ >* 0 *at* +∞, *then there exists* *t*_{0} *>* 0 *and*
*positive constants* *d*_{1} *and* *d*_{2} *with* 1*< d*_{1} *≤d*_{2} *such that*

*d*_{1} *≤* *th(t)*

*H(t)* *≤d*_{2}*,* *for all* *t≥t*_{0}*,*
(2.1)

*h(s)→*+∞ *as* *s→*+∞,*H(t)*ˆ *is increasing for* *t≥t*_{0} *and*
*d*_{1}*h(s)≤d*_{2}*h(t)* *for all* *s, tsuch that* *t*_{0}*≤s≤t.*

(2.2)

(ii) *If* *h* *is AHof exponent* *ρ >*0 *at 0 then there exists* *t*0*>*0 *and positive*
*constants* *d*_{1} *and* *d*_{2}*, with* 1*< d*_{1} *≤d*_{2} *such that*

*d*_{1}*≤* *th(t)*

*H(t)* *≤d*_{2}*,* *for all|t| ≤t*_{0}*,*
*H(t)*ˆ *is increasing in* [−t_{0}*, t*_{0}], and

*d*_{1}*|h(s)| ≤d*_{2}*|h(t)|*

*for alls, t* *with|s| ≤ |t| ≤t*_{0}*.*

*Proof.* We only prove (i), since (ii) is similar. From Karamata’s theorem
(see [R], page 17, Theorem 0.6), it follows that for any *σ >*0

*t→+∞*lim
*h(σt)*

*h(t)* =*σ** ^{ρ}* if and only if lim

*t→+∞*

*H(t)*
*th(t)* = 1

*ρ*+ 1*,*
(2.3)

and thus, if *h*is AH of exponent*ρ >*0, for*ε≥*0 (less than min{ρ,1}) there
is a *t*0*>*0,such that for all*t≥t*0*,*

*ρ*+ 1*−ε*

*t* *≤* *h(t)*

*H(t)* *≤* *ρ*+ 1 +*ε*

*t* *.*

(2.4)

Setting *d*1 := *ρ*+ 1*−ε >* 1 and *d*2 := *ρ*+ 1 +*ε* we have that (2.1) holds.

Now since *h(t) =H** ^{}*(t),from (2.4) we obtain that

*C*_{1}*t*^{d}^{1}^{−1}*≤h(t)≤C*_{2}*t*^{d}^{2}* ^{−1}* for all

*t≥t*

_{0}

*,*(2.5)

for some positive constants *C*_{1}*, C*_{2} and thus *h(t)→*+∞ as*s*+*∞.*

We observe now that the function ˆ*H* is a*C*^{1} function for*t >*0,and that
*H*ˆ* ^{}*(t) =

^{th(t)−H(t)}*2*

_{t}*.*Then from (2.4) and since

*d*1

*>*1,we ﬁnd that ˆ

*H*

*(t)*

^{}*>*0 for

*t≥t*0

*,*i.e., ˆ

*H*is ultimately increasing. Finally, and again from (2.4) for

*t*0

*≤s*

*≤*

*t,*we have that

*d*1

*h(s)*

*≤*

*d*1

*d*2

*H(s)*ˆ

*≤*

*d*1

*d*2

*H(t)*ˆ

*≤*

*d*2

*h(t),*ending the proof of the proposition.

As a consequence of this proposition we have the following result, which will be used to prove our main result.

**Proposition 2.2.** *Let* *h* :R *→* R *be continuous and asymptotically homo-*
*geneous at* +∞ *(at 0) of exponent* *ρ >*0 *satisfying* *th(t)* *>*0 *for* *t= 0. Let*
*{w*_{n}*}and{t*_{n}*} ⊆*R^{+}*be sequences such thatw*_{n}*→wandt*_{n}*→*+∞*(t*_{n}*→*0)
*as* *n→ ∞. Then,*

*n→∞*lim

*h(t*_{n}*w** _{n}*)

*h(t*

*) =*

_{n}*w*

^{ρ}*.*(2.6)

*Proof.* We only prove the case when *h* is AH at +∞, the other case being
similar. Let *H(s) :=* ^{}_{0}^{s}*h(t)dt* and assume ﬁrst *w* *= 0. Then* *t*_{n}*w*_{n}*→* +∞

and by writing

*h(t*_{n}*w** _{n}*)

*h(t**n*) = *t*_{n}*w*_{n}*h(t*_{n}*w** _{n}*)

*H(t*

*n*

*w*

*n*)

*H(t*ˆ _{n}*w** _{n}*)

*H(t*ˆ

*)*

_{n}*H(t** _{n}*)

*t*

*n*

*h(t*

*n*) (2.7)

we see from (2.3) that to obtain (2.6) it suﬃces to prove that

*n→∞*lim

*H(t*ˆ _{n}*w** _{n}*)

*H(t*ˆ

*) =*

_{n}*w*

^{ρ}*.*(2.8)

Since by proposition 2.1, ˆ*H* is ultimately increasing, given*ε >*0 suﬃciently
small, there exists *n*_{0}*>*0 such that for all*n≥n*_{0}

*H(t*ˆ * _{n}*(w

*−ε))*

*H(t*ˆ *n*) *≤* *H(t*ˆ _{n}*w** _{n}*)

*H(t*ˆ *n*) *≤* *H(t*ˆ * _{n}*(w+

*ε))*

*H(t*ˆ

*n*)

and thus (2.8) follows by using the fact that ˆ*H* is AH of exponent *ρ* and
*ε >*0 is arbitrarily small. Assume now that*w*= 0. We claim then that

*n→∞*lim

*h(t*_{n}*w** _{n}*)

*h(t*

*) = 0.*

_{n}If not,

*h(t*_{n}_{k}*w*_{n}* _{k}*)

*h(t*

*n*

*k*)

*≥µ,*

for some subsequences *{t*_{n}_{k}*},* *{w*_{n}_{k}*},* which implies that *t*_{n}_{k}*w*_{n}* _{k}* must tend
to +∞. Let now

*ε >*0 be such that

*ε < µ*

^{1/ρ}. Since

*w*

_{n}

_{k}*→*0,there exists

*k*0 *>* 0 such that *w**n**k* *< ε* and as *t**n**k**w**n**k* *→* +∞, both *t**n**k**w**n**k* and *t**n**k**ε*
belong to the range where ˆ*H* is increasing for*k≥k*_{0}*.. Hence,*

0*≤* *H(t*ˆ *n**k**w**n**k*)

*H(t*ˆ _{n}* _{k}*)

*≤*

*H(t*ˆ

*n*

*k*

*ε)*

*H(t*ˆ

_{n}*)*

_{k}*.*

Using now that ˆ*H* is AH of exponent *ρ, by letting* *k* *→ ∞* we ﬁnd that
lim sup

*k→∞*

*H(t*ˆ _{nk}*w** _{nk}*)

*H(t*ˆ * _{nk}*)

*≤ε*

*and hence, by (2.7),*

^{ρ}*µ≤*lim sup

*k→∞*

*h(t*_{nk}*w** _{nk}*)

*h(t** _{nk}*)

*≤ε*

^{ρ}*< µ,*a contradiction.

Finally, regarding properties of AH (at*∞*or 0) that we will need later on, it
is simple to see that if *χ, ψ*:R*→*Rare AH functions of exponent *p* and *q*
respectively, then*χ◦ψ* is AH of exponent*r* =*pq. Also, ifφ*is an increasing
odd homeomorphism of R onto R which is AH of exponent *p−*1, then its
inverse *φ** ^{−1}* is AH of exponent

*p*

^{∗}*−*1,where

*p*

*=*

^{∗}

_{p−1}

^{p}*.*

We now ﬁnd a functional analysis setting for problem (D).Asimple cal- culation shows that ﬁnding non trivial solutions with positive components to problem (D) is equivalent to ﬁnding non trivial solutions to the problem (A)

*−(r*^{N−1}*φ**i*(u^{}* _{i}*(r)))

*=*

^{}*r*

^{N−1}*f*

*i*(|u

*i+1*(r)|),

*i*= 1, . . . , n,
*u*^{}* _{i}*(0) = 0 =

*u*

*(R).*

_{i}Let (u_{1}(r), . . . , u* _{n}*(r)) be a non trivial solution of (A). Then for each

*i*= 1, . . . , n, we have that

*u*

*i*(r)

*≥*0 and is non increasing on [0, R]. By integrating the equations in (A),it follows that

*u*

*(r) satisﬁes*

_{i}*u**i*=*M**i*(u*i+1*)
where *M** _{i}*:

*C*

_{#}

*→C*

_{#}is given by

*M** _{i}*(v)(r) =

*R*
*r*

*φ*^{−1}* _{i}* [ 1

*s*

^{N−1}*s*
0

*ξ*^{N}^{−1}*f** _{i}*(|v(ξ)|)dξ]ds,
for each

*i*= 1, . . . , n. Let us deﬁne

*T*

_{0}:

*C*

_{#}

^{n}*→C*

_{#}

*by*

^{n}*T*_{0}(u) := (M_{1}(u_{2}), ..., M* _{i}*(u

*), ..., M*

_{i+1}*(u*

_{n}_{1}))

*,*

where* u*= (u1

*, . . . , u*

*n*).Clearly

*T*0 is well deﬁned and ﬁxed points of

*T*0 will provide solutions of (A),and hence componentwise positive solutions of (D).

Deﬁne now the operator*T**h* :*C*_{#}^{n}*×*[0,1]*→C*_{#}* ^{n}* by

*T** _{h}*(u, λ) :=

^{}

*M*˜1(u2

*, λ), ..., M*

*i*(u

*i+1*), ..., M

*n*(u1)

^{}where ˜

*M*1:

*C*

_{#}

*×*[0,1]

*→C*

_{#}is the operator deﬁned by

*M*˜1(v, λ)(r) :

*R*
*r*

*φ*^{−1}_{1} [ 1
*s*^{N}^{−1}

*s*
0

*ξ** ^{N−1}*(f1(|v(ξ)|) +

*λh)dξ]ds*

with*h >*0 a constant to be ﬁxed later. Deﬁne also*S* :*C*_{#}^{n}*×*[0,1]*→C*_{#}* ^{n}* by

*S(u, λ) = (N*1(u2

*, λ), ..., N*

*i*(u

*i+1*

*, λ), ..., N*

*n*(u1

*, λ))*

(2.9)

where *N** _{i}* :

*C*

_{#}

*×*[0,1]

*→C*

_{#}is the operator deﬁned by

*N*

*(v, λ)(r) =*

_{i}*R*
*r*

*φ*^{−1}* _{i}* [

*λ*

*s*

^{N−1}*s*
0

*ξ*^{N−1}*f** _{i}*(|v(ξ)|)dξ]ds, i= 1, . . . , n.

(2.10)

It follows from Proposition 2.2 of [GMU] that all the operators ˜*M*_{1}*, M*_{i}*, N*_{i}*,*
*i* = 1, . . . , n, are completely continuous, hence the operators *T*_{0}*, T** _{h}* and

*S*are also completely continuous. We note that

*T*

*(·,0) =*

_{h}*T*

_{0}=

*S(·,*1).

To prove existence of a ﬁxed point of*T*0 we use suitable a-priori estimates
and degree theory. Indeed, we will show that there exists*R*_{1}*>*0,such that
*deg**LS*(I*−T*0*, B(0, R*1),0) = 0,and also that the index *i(T*0*,*0,0) is deﬁned
and it satisﬁes*i(T*_{0}*,*0,0) = 1,from where the existence of a ﬁxed point of*T*_{0}
follows by the excision property of the degree.

Finally in this section, in our next lemma we will select the constant *h*
that appears in the deﬁnition of the operator*T*_{h}*,*and hence ﬁx this operator
once for all.

**Lemma 2.1.** *Fori*= 1, . . . , n*let the homeomorphismsφ**i**,and the functions*
*f*_{i}*satisfy* (H_{1}) *and* (H_{2}).*Then there exists* *h*_{0} *>*0 *such that the problem*

**u**=*T** _{h}*(u,1)
(2.11)

*has no solutions forh≥h*_{0}*.*

*Proof.* We argue by contradiction and thus we assume that there exists a
sequence*{h*_{k}*}*_{k∈N}*,*with*h*_{k}*→*+∞as *k→ ∞,*such that the problem

**u**=*T*_{h}* _{k}*(u,1)

has a solution **u*** _{k}*= (u

_{1,k}

*, . . . , u*

*),for each*

_{n,k}*k∈*N.Then

**u***satisﬁes*

_{k}*u*

_{1,k}(r) =

*R*
*r*

*φ*^{−1}_{1} [ 1
*s*^{N}^{−1}

*s*
0

*ξ** ^{N−1}*(f

_{1}(|u

_{2,k}(ξ)|) +

*h*

*)dξ]ds (2.12)*

_{k}*u** _{i,k}*(r) =

*R*
*r*

*φ*^{−1}* _{i}* [ 1

*s*

^{N−1}*s*
0

*ξ*^{N−1}*f**i*(|u* _{i+1,k}*(ξ)|)dξ]ds, i= 2, . . . , n,
(2.13)

for each *k* *∈* N. Clearly *u** _{i,k}*(r)

*>*0, r

*∈*[0, R), and is non increasing for

*r*

*∈*[0, R],for all

*k∈*N,and all

*i*= 1, . . . , n. From (2.12)

*u*1,k(r)*≥*(R*−r)φ*^{−1}_{1} (*rh*_{k}

*N* ), for all *r∈*[0, R]

and thus for *r* *∈* [0,^{7R}_{8} ], (we choose this interval for convenience, but any
other interval of the form [0, T]*⊂*[0, R) will work as well) we ﬁnd that

*u*_{1,k}(r)*≥* *R*

8*φ*^{−1}_{1} (*Rh** _{k}*
4N )
(2.14)

where we have used that *u*_{1,k}(r) *≥* *u*_{1,k}(R/4) for all *r* *∈* [0, R/4]. Then,
using that*f** _{i}*(x)

*→*+∞as

*x→*+∞, from (2.13) and (2.14), by iteration, we conclude that for any

*A >*0,there exists

*k*

_{A}*>*0 such that for all

*r∈*[0,

^{3R}

_{4}]

*u**i,k*(r)*≥A* for all *k≥k**A* and all *i*= 1, . . . , n.

(2.15)

Now, from the second of (H_{1}) and (i) of proposition 2.1 there exist *t*_{0} *>*0,
1*< d*_{1} *≤d*_{2} such that

*d*1*f**i*(τ)*≤d*2*f**i*(t)
(2.16)

for all*t≥τ* *≥t*_{0} and all*i*= 1, . . . , n. Hence, by (2.11), and by increasing*A*
if necessary,

*d*1*f**i*(|u* _{i+1,k}*(r)|)

*≤d*2

*f*

*i*(|u

*(ξ)|) (2.17)*

_{i+1,k}for all*ξ* *∈*[0, r] with*r∈*[0,3R/4].Since from (2.12) and (2.13) we also have
that

*u** _{i,k}*(r)

*≥*

3R4

*r*

*φ*^{−1}* _{i}* [ 1

*s*

^{N−1}*r*
0

*ξ*^{N}^{−1}*f** _{i}*(|u

*(ξ)|)dξ]ds, i= 1, . . . , n, (2.18)*

_{i+1,k}then, for *k≥k** _{A}*, from (2.17) and the monotonicity of

*φ*

^{−1}*we have that*

_{i}*u*

*(r)*

_{i,k}*≥*

3R4

*r*

*φ*^{−1}* _{i}* [d f

*(u*

_{i}*(r))dξ]ds, r*

_{i+1,k}*∈*[

*R*4

*,*3R

4 ],
where *d*= _{4Nd}^{d}_{2}^{1}_{3}^{R}*N−1**.*Thus for all*r* *∈*[^{R}_{4}*,*^{R}_{2}],we ﬁnd that

*u** _{i,k}*(r)

*≥*

*R*

4*φ*^{−1}* _{i}* (d f

*(u*

_{i}*(r))), (2.19)*

_{i+1,k}for all *k* large enough and for all*i*= 1, . . . , n. Next, setting
*b** _{i,k}*(r) :=

*R*

4

*φ*^{−1}* _{i}* (d f

*(u*

_{i}*(r)))*

_{i+1,k}*φ*

^{−1}*(f*

_{i}*i*(u

*(r)))*

_{i+1,k}*,*(2.20)

(2.19) becomes

*u** _{i,k}*(r)

*≥b*

*(r)φ*

_{i,k}

^{−1}*(f*

_{i}*(u*

_{i}*(r))), r*

_{i+1,k}*∈*[

*R*4

*,R*

2].

(2.21)

Observing that by (2.15) and (H_{1}), *b** _{i,k}*(r)

*→*

*c*

*as*

_{i}*k*

*→ ∞, uniformly in*[

^{R}_{4}

*,*

^{R}_{2}], where

*c*

*i*is a positive constant, we have that

*b*

*i,k*(r)

*≥*

*C*˜ for all

*r*

*∈*[

^{R}_{4}

*,*

^{R}_{2}], for all

*i*= 1, . . . , n and for all

*k*suﬃciently large and where ˜

*C*is a positive constant. Hence, by (2.5) in the proof of proposition 2.1, for

*ε >*0 small there is a

*k*0

*∈*Nsuch that

*u**i,k*(r)*≥Cu*_{i+1,k}^{pi}^{δi}^{−1}* ^{−ε}*(r), r

*∈*[

*R*4

*,R*

2], (2.22)

for all*k≥k*0 and all*i*= 1, . . . , n, and where*C* is a positive constant. Now,
by iterating in (2.22), we ﬁnd that

*u*_{1,k}(r)*≥C(u*_{1,k}(r))
*n*
*i=1*

*δi*
*pi**−1**−ε*

*,*
(2.23)

where*C* is a new positive constant. Since by (H2) we may choose 0*< ε <*

min{_{p}_{i}^{δ}_{−1}^{i}*, i*= 1, . . . , n} so that ^{}^{n}

*i=1*

*δ**i*

*p**i**−1**−ε*^{}*>*1,from (2.23), we have
(u_{1,k}(r))

*n*
*i=1*

*δi*
*pi**−1**−ε*

*−1* *≤* 1

*C,* for any ﬁxed *r∈*[*R*
4*,R*

2],

which by (2.15) gives a contradiction for large *k. This ends the proof of the*
lemma.

3. A-priori bounds and proof of the main result

In this section we will use the blow up method to ﬁnd a priori bounds
for the positive solutions of problem (D* _{h}*) and then prove Theorem 1.1. Let

*φ*

_{i}*, f*

*,*

_{i}*i*= 1, . . . , n be as in Theorem 1.1 and set

Φ* _{i}*(s) =

^{s}0 *φ** _{i}*(t)dt, F

*(s) =*

_{i}

^{s}0 *f** _{i}*(t)dt, i= 1, . . . , n.

(3.1)

In extending the blow up method to our situation it turns out that a key
step is to ﬁnd a solution (x_{1}*, . . . , x** _{n}*) in terms of

*s*(for

*s*near +∞) to the system

*F**i*(x*i+1*)x*i*=*x**i+1*Φ*i*(x*i**s),* *i*= 1, . . . , n.

(3.2)

In this respect we can prove the following.

**Lemma 3.1.** *Assume that the homeomorphisms* *φ*_{i}*,* *and the functions* *f*_{i}*,*
*i*= 1, . . . , n *satisfy* (H1), (H2), (H3), and (H4). Then

(i) *there exist positive numbers* *s*0*, x*^{0}_{i}*,* *and increasing diﬀeomorphisms* *α**i*

*deﬁned from* [s_{0}*,*+∞) *onto* [x^{0}_{i}*,*+∞), *i*= 1, . . . , n, *which satisfy*
*F** _{i}*(α

*(s))α*

_{i+1}*(s) =*

_{i}*α*

*(s)Φ*

_{i+1}*(α*

_{i}*(s)s),*

_{i}(3.3)

*for alls∈*[s0*,*+∞).

(ii) *The functionsα*_{i}*satisfy*

*s→∞*lim

*f** _{i}*(α

*(s))*

_{i+1}*sφ**i*(α*i*(s)s) = *δ** _{i}*+ 1

*p**i* *, i*= 1, . . . , n.

(3.4)

(iii) *The functionsα*_{i}*satisfy*

*s→+∞*lim
*α** _{i}*(σs)

*α**i*(s) =*σ*^{E}^{i}*for all* *σ∈*(0,+∞) *i*= 1, . . . , n,
*where theE*_{i}*’s are the solutions to* (AS).

We call these *α** _{i}*’s functions

*rescaling*variables for system (D).

The proof of this lemma is rather lengthy and delicate and thus in order not to deviate the attention of the reader we postpone it until section 4.

We next ﬁnd a-priori bounds for positive solutions. To this end let *h*
satisfy the conditions of Lemma 2.1 and consider the family of problems
(D* _{λ}*)

[r^{N−1}*φ*1(u^{}_{1})]* ^{}*+

*r*

*(f1(|u2(r)|) +*

^{N−1}*λh) = 0,*

[r^{N−1}*φ** _{i}*(u

^{}*)]*

_{i}*+*

^{}*r*

^{N−1}*f*

*(|u*

_{i}*(r)|) = 0, λ*

_{i+1}*∈*[0,1], i= 2, . . . , n,

*u*^{}* _{i}*(0) = 0 =

*u*

*i*(R) for

*i*= 1, . . . , n.

Clearly, a solution to (D*λ*) is a ﬁxed point of *T**h*(·, λ).

**Theorem 3.1.** *Under the conditions of Theorem 1.1, solutions to problem*
(D* _{λ}*)

*are a-priori bounded.*

*Proof.* We argue by contradiction and thus we assume that there exists a
sequence *{(u*_{k}*, λ** _{k}*)} ∈

*C*

_{#}

^{n}*×*[0,1], with

**u**

*= (u*

_{k}_{1,k}

*, . . . , u*

*), such that (u*

_{n,k}

_{k}*, λ*

*) satisﬁes (D*

_{k}

_{λ}*) and*

_{k}*u*

*=*

_{k}^{}

^{n}*i=1**u*_{i,k}* → ∞* as *k* *→ ∞.* It is not
diﬃcult to check by using the equations in (D_{λ}* _{k}*) that

^{}

^{n}*i=1**u*_{i,k}* → ∞* as *k→*

*∞* if and only if *u*_{i,k}* → ∞* as *k* *→ ∞* for each *i* = 1, . . . , n. Hence, by
redeﬁning the sequence (u_{k}*, λ** _{k}*) if necessary, we can assume that

*u*

_{i,k}*≥s*

_{0}(s0 as in lemma 3.1) for all

*i*= 1, . . . , n and for all

*k >*0. Let us set

*γ** _{k}* =

^{}

^{n}*i=1*

*α*^{−1}* _{i}* (u

_{i,k}*) and*

*t*

*=*

_{i,k}*α*

*i*(γ

*).*

_{k}(3.5)

Then,*γ*_{k}*→ ∞* as*k* *→ ∞,*and *u*_{i,k}* ≤t*_{i,k}*,*for each *i*= 1, . . . , n. Also by
(3.4)

*k→∞*lim

*f** _{i}*(t

*)*

_{i+1,k}*γ**k**φ**i*(t*i,k**γ**k*) = *δ** _{i}*+ 1

*p*

*i*

*.*(3.6)

Next we deﬁne the change of variables *y* = *γ*_{k}*r, w** _{i,k}*(y) =

^{u}

^{i,k}

_{t}

_{i,k}^{(r)}and set

**w**

*:= (w*

_{k}_{1,k}

*, . . . , w*

*). Clearly we have*

_{n,k}*|w*

*(y)| ≤ 1 for all*

_{i,k}*y*

*∈*[0, γ

_{k}*R].*

In terms of these new variables and since (u_{k}*, λ** _{k}*) satisﬁes (D

_{λ}*), we obtain that (w*

_{k}

_{k}*, λ*

*) satisﬁes*

_{k}*−(y*^{N−1}*φ*_{1}(t_{1,k}*γ*_{k}*w*^{}_{1,k}(y)))* ^{}* =

*y*

^{N}*[*

^{−1}*f*1(t

_{2,k}

*|w*

_{2,k}(y)|)

*γ** _{k}* +

*λ*

_{k}*h*

*γ** _{k}* ],
(3.7)

*−(y*^{N}^{−1}*φ** _{i}*(t

_{i,k}*γ*

_{k}*w*

^{}*(y)))*

_{i,k}*=*

^{}*y*

^{N}

^{−1}*f*

*(t*

_{i}

_{i+1,k}*|w*

*(y)|)*

_{i+1,k}*γ*_{k}*, i*= 2, . . . , n,
(3.8)

*w*^{}* _{i,k}*(0) = 0 =

*w*

*(γ*

_{i,k}

_{k}*R) for*

*i*= 1, . . . , n, (3.9)

where now * ^{}* =

_{dy}*. Let now*

^{d}*T >*0 be ﬁxed and assume, by passing to a subsequence if necessary, that

*γ*

*k*

*R > T*for all

*k*

*∈*N. We observe that by the usual argument,

*w*

^{}*(y)*

_{i,k}*≤*0 and

*w*

*(y)*

_{i,k}*≥*0 for all

*i*= 1, . . . , n, for all

*k∈*N, and for all

*y∈*[0, T].

**Claim.** The sequences *{w*_{i,k}^{}*}*_{k}*, i* = 1, . . . , n, are bounded in *C[0, T*]. In-
deed, assume by contradiction that for some *i*= 1, . . . , n, *{w*^{}_{i,k}*}* contains a
subsequence, renamed the same, with *||w*^{}_{i,k}*||*_{C[0,T}_{]} *→ ∞* as *k* *→ ∞. Then*
there exists a sequence *{y*_{k}*}, y*_{k}*∈*[0, T],such that for any*A >*0 there is*n*_{0}
such that*|w*_{i,k}* ^{}* (y

*)|*

_{k}*> A*for all

*k > n*

_{0}

*.*Integrating (3.7) (resp. (3.8) from 0 to

*y*

_{k}*,*we obtain

(3.10) *φ**i*(t*i,k**γ**k**|w*_{i,k}* ^{}* (y

*k*)|)=

*y*

*k*1−N

*y*

*k*

0

*s*^{N−1}*f** _{i}*(t

_{i+1,k}*w*

*(s))*

_{i+1,k}*γ*_{k}*ds*+*λ*_{k}*hy*_{k}*Nγ*_{k}*.*

Now let*t*0,*d*1,*d*2 be as in Proposition 2.1 and set*M* = max

*i∈{1,... ,n}* sup

*x∈[0,t*0]*f**i*(x).

Since *t*_{i+1,k}*→* +∞ as *k* *→ ∞, by redeﬁning the sequence if necessary, we*
may assume that _{f}_{i}_{(t}^{M}_{i+1,k}_{)} *≤* ^{d}_{d}^{2}_{1} for all*i*= 1, . . . , nand all*k∈*N. Also, since
*w** _{i+1,k}*(s)

*≤*1, if

*t*

_{i+1,k}*w*

*(s)*

_{i+1,k}*≥t*

_{0}, then by Proposition 2.1 we have that

*f** _{i}*(t

_{i+1,k}*w*

*(s))*

_{i+1,k}*f*

*(t*

_{i}*)*

_{i+1,k}*≤*

*d*

_{2}

*d*_{1}*.*
(3.11)

Since if*t*_{i+1,k}*w** _{i+1,k}*(s)

*≤t*

_{0}(3.11) holds by the deﬁnition of

*M, we have that*indeed (3.11) holds for all

*i*= 1, . . . , n, all

*k*

*∈*N and all

*s∈*[0, T]. Hence from (3.10) and the monotonicity of

*φ*

*i*we ﬁnd that

*φ**i*(t*i,k**γ**k**A)*
*φ** _{i}*(t

_{i,k}*γ*

*)*

_{k}*≤*

*d*2

*d*_{1}

*f**i*(t*i+1,k*)T

*φ** _{i}*(t

_{i,k}*γ*

*)γ*

_{k}

_{k}*N*+

*hT*

*Nφ*

*(t*

_{i}

_{i,k}*γ*

*)γ*

_{k}

_{k}*.*

Thus, by (H_{1}) and (3.6), and by letting *k* *→ ∞* in this last inequality we
ﬁnd that

*A*^{p}^{i}^{−1}*≤* *d*2

*d*_{1}

(δ*i*+ 1)T
*p*_{i}*N* *,*

which is a contradiction since*A*can be taken arbitrarily large and hence the
claim follows.

From this claim and Arzela Ascoli Theorem, by passing to a subsequence
if necessary, we have that **w**_{k}*→* **w** := (w1*, . . . , w**n*) in*C** ^{n}*[0, T]. Also, by
(3.5),

1 =^{}^{n}

*i=1*

*α*^{−1}* _{i}* (t

_{i,k}*w*

*(0))*

_{i,k}*γ** _{k}* =

^{}

^{n}*i=1*

*α*^{−1}* _{i}* (t

_{i,k}*w*

*(0))*

_{i,k}*α*

^{−1}*(t*

_{i}*)*

_{i,k}*,*

and hence, by letting*k→ ∞* and using (iii) of lemma 3.1, we obtain
1 =^{}^{n}

*i=1*

*w*_{i}^{Ei}^{1} (0),
which implies that *w* is not identically zero.

Now by integrating (3.7) (respectively (3.8)) from 0 to*y∈*[0, T] and using
(3.9), we obtain

*−φ**i*(t_{i,k}*γ*_{k}*w*^{}* _{i,k}*(y)) = ˜

*f*

*(y)*

_{i,k}*f*

*i*(t

*i+1,k*)

*γ*

_{k}*,*(3.12)

for *i*= 1, . . . , n and all*k∈*N, where
*f*˜_{1,k}(y) =*y*^{1−N} ^{y}

0 *s*^{N}^{−1}*f*1(t2,k*w*2,k(s))

*f*_{1}(t_{2,k}) *ds*+ *λ**k**hy*
*Nf*_{1}(t_{2,k})*,*
(3.13)

and

(3.14) *f*˜* _{i,k}*(y) =

*y*

^{1−N}

^{y}0 *s*^{N−1}*f** _{i}*(t

_{i+1,k}*w*

*(s))*

_{i+1,k}*f**i*(t*i+1,k*) *ds, i*= 2, . . . , n.

Using now Proposition 2.2, we have that ^{f}^{i}^{(t}^{i+1,k}_{f}_{i}_{(t}_{i+1,k}^{w}^{i+1,k}_{)} ^{(s))} *→* (w*i+1*(s))^{δ}* ^{i}*
for each

*s*

*∈*[0, T] and

*i*= 1, . . . , n, and thus by (3.11) we may use the

Lebesgue’s dominated convergence theorem to conclude that

*k→∞*lim *f*˜* _{i,k}*(y) =

*y*

^{1−N}

^{y}0 *s*^{N−1}*w*^{δ}_{i+1}* ^{i}* (s)ds:= ˜

*f*

*(y) (3.15)*

_{i}for each*y* *∈*(0, T].From (3.12),

*−w*_{i,k}* ^{}* (y) =

*φ*

^{−1}*(˜*

_{i}*g*

*(y)µ*

_{i,k}*)*

_{k}*φ*

^{−1}*(µ*

_{i}*)*

_{k}*,*(3.16)

where

˜

*g** _{i,k}*(y) =

*f*˜

*(y)f*

_{i,k}*(α*

_{i}*(γ*

_{i+1}*))*

_{k}*γ**k**φ**i*(γ*k**α**i*(γ*k*)) and *µ** _{k}*=

*φ*

*(γ*

_{i}

_{k}*t*

*).*

_{i,k}Then*µ*_{k}*→*+∞ as*k→ ∞* and by (3.15) and (ii) of lemma 3.1,

˜

*g** _{i,k}*(y)

*→*

*δ*

*i*+ 1

*p*_{i}*f*˜*i*(y) as *k→ ∞* for each *y∈*[0, T].

(3.17)

Integrating (3.16) over [0, y],we obtain
*w** _{i,k}*(0)

*−w*

*(y) =*

_{i,k}

^{y}0

*φ*^{−1}* _{i}* (˜

*g*

*(s)µ*

_{i,k}*)*

_{k}*φ*

^{−1}*(µ*

_{i}*k*)

*ds.*

(3.18)

Then, since by (3.15) there exists *A >* 0 such that *|f*˜* _{i}*(y)| ≤

*A*for all

*i*= 1, . . . , n and all

*y*

*∈*[0, T], using (3.17) and the monotonicity of

*φ*

*, by another application of the Lebesgue’s dominated convergence theorem, we ﬁnd that*

^{−1}*w** _{i}*(0)

*−w*

*(y) = (*

_{i}*δ*

*+ 1*

_{i}*p** _{i}* )

^{pi}^{1}

^{−1}

^{y}0

*s*^{1−N} ^{s}

0 *t*^{N−1}*w*^{δ}_{i+1}* ^{i}* (t)dt

^{}

^{pi}^{1}

^{−1}*ds,*and hence that

*w*

*satisﬁes*

_{i}(D* _{p}*)

_{T}*−(y*^{N−1}*|w*_{i}* ^{}*(y)|

^{p}

^{i}

^{−2}*w*

_{i}*(y))*

^{}*= (*

^{}

^{δ}

^{i}

_{p}^{+1}

*)y*

_{i}

^{N−1}*w*

^{δ}

_{i+1}*(y), y*

^{i}*∈*(0, T],

*w*

_{i}*(0) = 0, w*

^{}*i*(y)

*≥*0 for all

*y∈*[0, T].

We observe next that each component *w** _{i}*(y) is decreasing on [0, T].Thus
if for some

*i, w*

*i*(0) = 0,then necessarily

*w*

*i*(y) = 0 for all

*y∈*[0, T].But from (D

*)*

_{p}*it follows that*

_{T}*w*

*(y) = 0 for all*

_{i+1}*y*

*∈*[0, T] and hence by iterating, that

**w**

*≡*

**0**on [0, T], which cannot be. Now, for the purpose of our next argument let us call

*{w*

^{T}

_{k}*}*the ﬁnal subsequence, solution to (3.7), (3.8) and (3.9), which by the limiting process provided us with the non trivial solution

**w**to (D

*p*)

*T*deﬁned in [0, T]. We also set

**w**

^{T}*≡*

**w.**Let us choose next

*T*

_{1}

*> T.*By repeating the limiting process following (3.9), this time starting from the sequence

*{w*

^{T}

_{k}*}, we will ﬁnd a subsequence*

*{w*

^{T}

_{k}^{1}

*}, which*as

*k→ ∞*will provide us with a non trivial solution

**w**

^{T}^{1}to (D

*p*)

*T*1

*.*Clearly

**w**

^{T}^{1}is an extension of

**w**

*to the interval [0, T*

^{T}_{1}],which satisﬁes

*w*

^{T}

_{i}^{1}(y)

*≥*0,

*i*= 1, . . . , n. It is then clear that by this argument we can obtain a non trivial solution (called again

**w) to (D**

*),i.e.*

_{p}**w**satisﬁes

(D* _{p}*)

*−(y*^{N−1}*|w*^{}* _{i}*(y)|

^{p}

^{i}

^{−2}*w*

_{i}*(y))*

^{}*=(*

^{}

^{δ}

^{i}

_{p}^{+1}

*)y*

_{i}

^{N}

^{−1}*w*

_{i+1}

^{δ}*(y), y*

^{i}*∈*(0,+∞),

*w*

_{i}*(0) = 0, w*

^{}*(y)*

_{i}*≥*0 for all

*y*

*∈*[0,+∞).

We claim now that under the hypotheses of Theorem 1.1 such a non trivial
solution cannot exist. The proof of this claim is entirely similar to lemma
2.1 in [CMM] so we just sketch it. An integration of the equations of (D*p*)
over [0, r], r*∈*(0,+∞), shows that*w*_{i}* ^{}*(r)

*≤*0,for all

*r >*0,and that (3.19)

*−r*

^{N−1}*|w*

_{i}*(r)|*

^{}

^{p}

^{i}

^{−2}*w*

_{i}*(r)*

^{}*≥(δ*

*+ 1*

_{i}*p**i* )^{pi}^{1}^{−1}*r*^{N}

*N* *w*_{i+1}^{δ}* ^{i}* (r), for all

*r >*0, Also it must be that

*w*

*(r)*

_{i}*>*0 for all

*r >*0 and all

*i*= 1, . . . , n. Now by Proposition 2.1 and Lemma 2.1 in [CMM], see also [MI] for related results, we have that for all

*i*= 1, . . . , n,

*w*

_{i}*∈C*

^{2}(0,+∞) and that

*rw*_{i}* ^{}*(r) +

*θ*

_{i}*w*

*(r)*

_{i}*≥*0, for all

*r >*0.

(3.20)

Hence, from (3.19)
*θ*_{i}*w** _{i}*(r)

*r* *≥ −w*_{i}* ^{}*(r)

*≥Cr*

^{pi}^{1}

^{−1}*w*

_{i+1}

^{pi}

^{δi}*(r) for all*

^{−1}*r >*0,

where*C* is a positive constant. (In the rest of this argument*C* will denote a
positive constant that may change from one step to the other). Multiplying
this inequality by*r*^{E}^{i}^{+1}, using (3.20) and system (AS), we obtain

(3.21) *r*^{E}^{i}*w** _{i}*(r)

*≥C(r*

^{E}

^{i+1}*w*

*(r))*

_{i+1}

^{p}

^{i}

^{δ}

^{−1}

^{i}*,*for all

*r >*0 and

*i*= 1, . . . , n.

Iterating this expression *n−*1 times, we ﬁnd ﬁrst that
(3.22) *r*^{E}^{i}*w** _{i}*(r)

*≥C(r*

^{E}

^{i}*w*

*(r))*

_{i}*n*
*j=1*

*δ**j*

*p**j**−1*

for each *i*= 1, . . . , n,
and thus by hypothesis (H_{2}),

(3.23) *w** _{i}*(r)

*≤Cr*

^{−E}*for each*

^{i}*i*= 1, . . . , n.

By (3.20), *r*^{θ}^{i}*w** _{i}*(r) is non decreasing, and thus combining with (3.22),
(3.24)

*C*

_{i}*≡w*

*(r*

_{i}_{0})r

^{θ}_{0}

^{i}*≤w*

*(r)r*

_{i}

^{θ}

^{i}*≤Cr*

^{−E}

^{i}*r*

^{θ}*=*

^{i}*Cr*

^{−(E}

^{i}

^{−θ}

^{i}^{)}

*,*

for all *r > r*_{0} *>* 0, for all *i* = 1, . . . , n and where the *C** _{i}*’s are positive
constants. If the strict inequality holds in hypothesis (H

_{5}), we obtain a contradiction by letting

*r→*+∞in (3.24) and the claim follows in this case.

Next, let us assume that for some *j* *∈ {1, . . . , n}* we have that *E** _{j}* =

*θ*

_{j}*.*Integrating the

*j-th equation of (D*

*p*) on (r0

*, r), r*0

*>*0, using (3.21) and iterating

*n−*2 times, we obtain

*r*^{N−1}*|w*^{}* _{j}*(r)|

^{p}

^{j}

^{−1}*≥C*

*r*
*r*0

*s*^{N}^{−1−E}^{j+1}^{δ}* ^{j}*(s

^{E}

^{j}*w*

*j*(s))

^{P}

^{j}

^{}*ds*where

*P*

_{j}*:=*

^{}^{}

^{n}*i=1,i =j*
*δ**i*

(p*i**−1)**.*Hence, since*w** _{j}*(r)r

^{θ}*is non decreasing, and using that*

^{j}*E*

*=*

_{j}*θ*

_{j}*,*

*r*^{N−1}*|w*_{j}* ^{}*(r)|

^{p}

^{j}

^{−1}*≥C*

*r*
*r*0

*s*^{−1}*ds* for all *r > r*_{0}*,*