R E S E A R C H
Open Access
Existence of solutions for a class of
degenerate quasilinear elliptic equation in
R
N
with vanishing potentials
Waldemar D Bastos
1, Olimpio H Miyagaki
2*and Rônei S Vieira
3*Correspondence:
[email protected] 2Universidade Federal de Juiz de
Fora, Juiz de Fora, Minas Gerais 36036-330, Brazil
Full list of author information is available at the end of the article
Abstract
We establish the existence of positive solution for the following class of degenerate quasilinear elliptic problem
(P)
–Luap+V(x)|x|–ap ∗
|u|p–2u=f(u) inRN,
u> 0 inRN; u∈D1,p a (RN),
where –Luap= – div(|x|–ap|∇u|p–2∇u), 1 <p<N, –∞<a<N–p
p ,a≤e≤a+ 1,
d= 1 +a–e, andp∗:=p∗(a,e) = Np
N–dpdenote the Hardy-Sobolev’s critical exponent,V is a bounded nonnegative vanishing potential andfhas a subcritical growth at infinity. The technique used here is a truncation argument together with the variational approach.
MSC: 35B09; 35J10; 35J20; 35J70
1 Introduction
Consider the following degenerate quasilinear elliptic problem inRN:
(P)
⎧ ⎨
⎩
–Luap+V(x)|x|–ap∗
|u|p–u=f(u) inRN,
u> inRN; u∈D,p a (RN),
where –Luap= –div(|x|–ap|∇u|p–∇u), <p<N, –∞<a<N–p
p ,a≤e≤a+ ,d= +a–e, andp∗:=p∗(a,e) = Np
N–dp denote the Hardy-Sobolev’s critical exponent,V :R
N →Ris a bounded, nonnegative and vanishing potential andf :R→Ra continuous function with a subcritical growth at infinity. Here,D,ap(RN) is the completion of theC∞
(RN) with the
norm|u|= (
RN|x|–ap|∇u|pdx)
p. We impose the following hypotheses onf andV:
f:R→Ris a continuous function verifying
(f) lim sups→+ sf(s)
|x|–ap∗sp∗<∞, uniformly inx.
(f) There existsα∈(p,p∗)such thatlim sups→∞
sf(s)
|x|–ap∗sα <∞, uniformly inx. (f) There existsθ>psuch thatθF(s)≤sf(s), for alls> .
V:RN→Ris a continuous function verifying
(V) V(x)≥, for allx∈RN.
(V) There are> andr> such thatinf|x|>rV(x)|x|
p[N–p(a+)] (p–)(N–p) ≥.
Remark . The conditions (f) and (f) imply the following:
sf(s)≤C|x|–ap ∗
|s|α withα∈p,p∗ , for alls∈R. ()
Example . An example for the functionf is given by
f(t) =
⎧ ⎨
⎩
|x|–ap∗tλ–, if ≤t≤,
|x|–ap∗tα–, ift≥,
withλ>p∗andαgiven by (f
).
By vanishing potential we mean a potential that vanish on some bounded domain or become very close to zero at infinity. An important example for a such potential is given by
V(x) =
⎧ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
, if|x| ≤r– ,
r–
p[N–p(a+)]
(p–)(N–p) (|x|–r+ ), ifr– <|x| ≤r,
|x|–
p[N–p(a+)]
(p–)(N–p) , if|x| ≥r,
with> .
Consider first the casea= , that is –Luapis thep-Laplacian operator, and the potential is bounded from below by a positive constantV> .
Equations involving thep-Laplacian operator appear in many problems of nonlinear diffusion. Just to mention, in nonlinear optics, plasma physics, condensed matter physics and in modeling problems in non-Newtonian fluids. For more information on the physical background, we refer to [].
For the casep= , we cite [–], and references therein. In [], in addition to the above assumptions, the authors consider a local condition, namely,
min
x∈
V<min
x∈∂V,
where⊂RN is a open bounded set, instead of the global condition imposed by Rabi-nowitz in []. Forp= , see [–].
Now, consider thatV is the zero mass case, that islim|x|→∞V(x) = . Whenp= , we
cite [–] and the recent paper [] by Alves and Souto.
Let us now consider the casea= and the potential bounded from below by a positive constantV> .
In this case, the equations arise in problems of existence of stationary waves for anisotropic Schrödinger equation (see []) and others problems (for example, see [, ]). We cite [] forp= ; and [, ] forp= . For the caseV≡, we cite [], for
p= anda= ; and [], forp= anda= .
provide the convergence of the gradient. In the case studied here, with the absence of this structure, we do not obtain the convergence so directly. To overcome this problem, we use a result found in [, ], whose ideas come from [, ], when the domain is a smooth and bounded. In addition to this difficulty, there are others. For instance, in the present situation, our space is no longer Hilbert, which forces us to obtain new estimates. Since the problem involves singular terms, the estimates are more refined and for which the principal ingredient is the Caffarelli-Kohn-Nirenberg’s inequality (see []). Now we state the main result of this work.
Theorem . Suppose that V and f satisfy,respectively, (V)and(V)and(f)to(f). Then there is a constant∗=∗(V∞,θ,p,c) > such that the problem(P)has a positive solution,for all≥∗,being V∞the maximum of the f in the ball of RN centered in the
origin with radius.
In order to prove this theorem, we first build an auxiliary problem (AP), and then we solve the problem (AP) using variational methods. To finish, we show that the solution of (AP) is also a solution of (P). These steps are the content of the next three sections.
Hereafter,Cis a positive constant which can change value in a sequence of inequalities. We denoteBR=BR() the ball inRN centered in the origin with radiusR. The weak (⇀) and strong (→) convergences are always taken asn→ ∞and
Af means
Af(x)dx. The weightedLpspaces are denoted byLp
α(A) ={u:RN →R:
A|x|–
α|u|p<∞}. Whenα= , we denote · Lp(A)the usual norm inLp(A), with ≤p≤ ∞. ForA=RN, we use · p.
2 The auxiliary problem
As usual, since we are looking for positive solutions of problem (P), we setf(t) = , for all
t≤. The hypothesis (V) allows us to consider the space
E=
u∈D,p
a
RN :
RNV|x|
–ap∗
|u|p<∞
,
with norm
u=
RN
|x|–ap|∇u|p+V|x|–ap∗|u|p
p
.
Associated to the problem (P), we define onE, the Euler-Lagrange functional
I(u) =
RN
|x|–ap|∇u|p+V|x|–ap∗|u|p–
RNF(u),
beingF(s) =s
f(t)dt. From the assumptions on f, it follows thatI isC with Gâteaux
derivative
I′(u)v=
RN
|x|–ap|∇u|p–∇u∇v+V|x|–ap∗|u|p–uv–
To obtain solutions of problem (P), we introduce some truncation of the functionf. Con-siderk=θp–θp >p,r> and define
g(x,t) =
⎧ ⎪ ⎪ ⎨
⎪ ⎪ ⎩
f(t), |x| ≤r,
f(t), |x|>randf(t)≤Vk|x|–ap∗|t|p–t,
V k|x|
–ap∗
|t|p–t, |x|>randf(t) >V k|x|
–ap∗
|t|p–t.
()
Now we define the auxiliary problem:
(AP)
⎧ ⎨
⎩
–Luap+V(x)|x|–ap∗
|u|p–u=g(x,u) inRN,
u> inRN; u∈D,ap(RN). ()
Associated to the problem (AP), we define, onE, the Euler-Lagrange functional
J(u) =
p
RN
|x|–ap|∇u|p+V|x|–ap∗|u|p–
RN
G(x,u) =
pu
p–
RN
G(x,u),
beingG(x,s) =s
g(x,t)dt. From the assumptions onf, it follows thatJisCwith Gâteaux
derivative
J′(u)v=
RN
|x|–ap|∇u|p–∇u∇v+V|x|–ap∗|u|p–uv–
RNg(x,u)v, v ∈E.
3 Solving the problem (AP)
In this section, we show that the problem (AP) has a least energy solution, but first we define some minimax levels. To begin with we set in the space D,ap(B), the norm |u|= (
B|x|
–ap|∇u|p+V
∞|x|–ap
∗
|u|p)p and we define the functionalI
given byI(u) =
p
B|x|
–ap|∇u|p+V
∞|x|–ap
∗
|u|p–
BF(u). Here,D
,p
a (B) is the completion of theC∞(B)
with the norm|u|= (
B|x|
–ap|∇u|p)p.
Lemma . The functional Ihas the mountain pass geometry,namely, . ∃r,ρ> such thatI(u)≥ρfor|u|=r.
. ∃e∈D,ap(B)such that|e| ≥randJ(e)≤.
Proof By using the growth off given in Remark . and the Caffarelli-Kohn-Nirenberg’s inequality (see []), we get
BF(u)≤
Bc|x|
–ap∗|u|p∗≤c
|u|p ∗
, and henceI(u)≥
p|u|p–c|u|p
∗
. Sincep∗ >p, there existsr
such thatρ:= prp–cr
p∗
> . Thus, we
haveI(u)≥ρfor|u|=r. By (f), it follows that there existθ >pandC> such that F(s)≥C|s|θ. Nowu∈Da,p(B) impliesI(tu)≤t
p
p|u| p–tθC
B|u|
θ. Sinceθ>p, there
exists atlarge enough such that, takinge=tu, we haveI(e) < and|e| ≥r.
Lemma . The functional J has the mountain pass geometry,namely,
. ∃r,ρ> such thatJ(u)≥ρforu=r. . ∃e∈Esuch thate ≥randJ(e)≤.
Proof From the definition ofG, we have
RNG(x,u)≤
RNF(u). Thus, like the previous lemma, we haveJ(u)≥ρ:=prp–cr
p∗
proof of the previous lemma. Thus,u∈EandG(x,u) =F(u). With the same argument,
we haveJ(tu)≤t
p
pu
p–tθC
RN|u|θ. Sinceθ >p, there exists atlarge enough such
that, takinge=tu, we haveJ(e) < ande ≥r.
Next, we are going to define two minimax levels, which will play an important role in our arguments. Note that is possible to taketsuch thate=tusatisfies two previous lemmas.
This allows to define the minimax levelscandcby
c=inf γ∈Ŵtmax∈[,]J
γ(t) withŴ=
γ∈C
[, ],E :γ() = andγ() =e
and
c=inf γ∈Ŵtmax∈[,]I
γ(t) withŴ=
γ ∈C[, ],D,p
a (B) :γ() = andγ() =e
respectively. SinceJ(u)≤I(u) inD,ap(B), we havec≤c, by their definitions. Now using
the above lemma together with the mountain pass theorem [, Theorem .], we con-clude that there exists a Palais-Smale sequence ((PS) sequence for short) (un)⊂E forJ,
i.e., (un) satisfiesJ(un)→candJ′(un)→.
Lemma . Suppose(V)and(f)to(f)and let(un)⊂E be a(PS)sequence for the
func-tional J.Then(un)is bounded in E.
Proof Define the setA={x∈RN :|x| ≤Rorf(u(x))≤V(x)
k |x|–ap
∗
|u(x)|p–u(x)}. InA, we
have G(x,u) =F(u). By using (f), we conclude that there is θ >psuch that –G(x,u) +
θug(x,u)≥. So, we have
p
A
|x|–ap|∇u|p+V|x|–ap∗|u|p–
A
G(x,u)
–
θ
A
|x|–ap|∇u|p+V|x|–ap∗|u|p–
A
ug(x,u)
≥
p–
θ A
|x|–ap|∇u|p+V|x|–ap∗|u|p≥(p– )
pk
A
|x|–ap|∇u|p+V|x|–ap∗|u|p.
Now consider the setB=Ac={x∈RN:|x|>Randf(u(x)) >V(x)
k |x|
–ap∗|u(x)|p–u(x)}. In B, we have
Bug(x,u) > andG(x,u) = V pk|x|
–ap∗|u|p. Then
p
B
|x|–ap|∇u|p+V|x|–ap∗|u|p–
B
G(x,u)
–
θ
B
|x|–ap|∇u|p+V|x|–ap∗|u|p–
B
ug(x,u)
≥
k
B
|x|–ap|∇u|p+V|x|–ap∗|u|p–
B
V pk|x|
–ap∗
|u|p
≥(p– )
pk
B
|x|–ap|∇u|p+V|x|–ap∗|u|p.
Therefore, we getJ(u) – θJ
′(u)u≥(p–)
pk u
p. In particular, the above equation holds for
the (PS) sequence (un) forJ, and we haveJ(un)–θJ
′(u
we haveJ(un)→c,θ >p> andJ′(un)unun →, sinceJ ′(u
n)→. Thus, we get J(un) –
θJ
′(u
n)un≤M+θun ≤M+un, for some constantM> . Then we have (p–)
pk un p≤
M+un, which can be rewritten as
un(p– )unp––pk ≤pkM. ()
Assuming un → ∞, equation () implies that (p– )unp––pk→. So un → (ppk–)p–, which is a contradiction. Therefore,u
nis bounded inE.
Lemma . Suppose(V)and(f)to(f).Then the functional J satisfies the Palais-Smale condition,i.e.,every(PS)sequence has a convergent subsequence.
Proof Observe that, given the (PS) sequence (un), by Lemma ., there existsu∈Esuch thatun⇀u, becauseEis reflexive space.aThus, it is enough to show thatun → u. We divide this task in the four claims below.
. Claim
RNung(x,un)→RNug(x,u).
. Claim
RNug(x,un)→RNug(x,u).
. Claim
RNV|x|–ap
∗
|un|p–unu→RNV|x|–ap
∗
|u|p. . Claim
RN|x|–ap|∇un|p–∇un∇u→
RN|x|–ap|∇u|p.
Assuming Claims to for now, we proceed with the proof of lemma. SinceJ′(u
n)un→, we haveRN|x|–ap|∇un|p+V|x|–ap
∗
|un|p–RNung(x,un) =on(), and by Claim , we get
lim sup
n→∞
unp=
RNug
(x,u). ()
As J′(u
n)u→, we get
RN|x|–ap|∇un|p–∇un∇u+V|x|–ap
∗
|un|p–unu–
RNug(x,un) =
on(). Passing the limit in the above equation and using Claims , and , we get
up=
RN
|x|–ap|∇u|p+V|x|–ap∗|u|p=
RN
ug(x,u). ()
Using equations () and (), we haveun → u.
In order to prove the claims, for a givenǫ> , we choosersatisfying, the following two conditions:
. max{
Br\Br|x|
–ap∗|u|p∗,
BcrV|x|
–ap∗|u|p} ≤ǫ.
. η=ηr∈C∞(Bcr)is such thatη≡inBcrand≤η≤,|∇η| ≤rd, for allx∈RN. Observe that condition follows by integrability of|x|–ap∗|u|p∗ andV|x|–ap∗|u|p. Note that, when one of these conditions holds for somer, it also verifies for everyr≥r. Thus,
we can choose anrthat satisfies both conditions.
Remark . From now on, we consider the functiongdefined in () withr=rsatisfying
conditions and above. From the growth ofgand the choice ofr, we conclude:
. g(x,t) =f(t),G(x,t) =F(t)inBr;
. g(x,t)≤Vk|x|–ap∗|t|p–t,G(x,t)≤pkV|x|–ap∗|t|pinBcr; .
Bcr|ug(x,u)| ≤
Bcr|u| V k|x|–ap
∗
|u|p–≤
k
BcrV|x|–ap
∗
Verification of claims
In all proofs, except for Claim , we consider separately integration inBrandBcr. Claim : In Br, by combining the dominated convergence theorem with the com-pact embedding ofE inLr
α(Br), ≤r<p∗ andα< (a+ )r+N( – rp) (see [, Theo-rem .]), we obtain
Brung(x,un)→
Brug(x,u). On the other hand, when integrating in
Bc
rwe do not have the compact embedding ofEinLrα(Bcr). In this case, we first estimate
Bcr|x|–ap|∇un|p+V|x|–ap
∗
|un|p, by using the cut-off functionηdefined before. As (ηun) is also bounded we haveJ′(u
n)(ηun)→, namely,
RN
|x|–ap|∇un|p–∇un∇(ηun) +V|x|–ap
∗
|un|p–unηun=on() +
RN
ηg(x,un)un. ()
Since η ≡ in Br, equation () holds in Brc. Adding to this, the fact that g(x,t)≤ V
k|x|
–ap∗
|t|p–tinBc
rwe have
Bcr
η
|x|–ap|∇un|p+V|x|–ap
∗
|un|p +|x|–ap|∇un|p–∇un(∇η)un
=
Bcr
|x|–ap|∇un|p–∇un∇(ηun) +V|x|–ap
∗
|un|p–unηun
=on() +
Bcr
ηg(x,un)un≤on() +
Bcr
ηV
k|x| –ap∗
|un|p.
From this, we obtain
Bcr
η|x|–ap|∇un|p+V|x|–ap
∗
|un|p
≤on() +
Bcr
ηV
k|x| –ap∗|u
n|p+
Bcr
|x|–ap|un||∇un|p–|∇η|
≤on() +
Bcr
ηV
k|x| –ap∗
|un|p+
Br\Br
|x|–ap|un||∇un|p–|∇η|
+
Bc r
|x|–ap|un||∇un|p–|∇η|
≤on() +
k
Bcr
η|x|–ap|∇un|p+V|x|–ap
∗
|un|p +
rd
Br\Br
|x|–ap|un||∇un|p–.
Thus, we have
Bcr
η|x|–ap|∇un|p+V|x|–ap
∗
|un|p
≤on() +
–
k –
rd
Br\Br
|x|–ap|un||∇un|p–. ()
Using the boundedness of (un) inE,i.e.,un ≤C, strong convergence of (un) inLpap(Br\
Br) and Hölder’s inequality we conclude that
lim sup
n→∞
Br\Br
|x|–ap|un||∇un|p–≤C
Br\Br
|x|–ap|u|p
p
≤Cω
d N N(r)d
Br\Br
|x|–ap∗|u|p∗
p∗
whereωN is the volume of the unit sphere inRN. Therefore, givenǫ> , by the choice of
rand by equations () and (), we infer that
lim sup
n→∞
Bc r
|x|–ap|∇un|p+V|x|–ap
∗
|un|p
≤lim sup
n→∞
Bcr
η|x|–ap|∇un|p+V|x|–ap
∗
|un|p
≤lim sup
n→∞
on() +
–
k –
rd
Br\Br
|x|–ap|un||∇un|p–
≤Cω
d N N
Br\Br
|x|–ap∗|u|p∗
p∗
≤ǫ.
Hence,
lim sup
n→∞
Bcr
|x|–ap|∇un|p+V|x|–ap
∗
|un|p= . ()
Using this fact, we conclude that
lim sup
n→∞
Bcr
ung(x,un)≤lim sup n→∞
k
Bcr
V|x|–ap∗|un|p
≤lim sup
n→∞
k
Bcr
|x|–ap|∇un|p+V|x|–ap
∗
|un|p= .
Now, using Remark ., we havelim supn→∞
Bc
r|ung(x,un) –ug(x,u)|= . Therefore,
Bcrung(x,un)→
Bcrug(x,u) and the proof of Claim is completed. Claim : The proof of this fact is made as in the proof of Claim .
Claim : InBrthe proof proceeds like in Claim . For integration inBcr, we estimate the value of
BrV|x|
–ap∗
|un|p–unuby using the Hölder’s inequality
Bcr
V|x|–ap∗|un|p–unu≤
Bcr
V|x|–ap∗|u|p
p
Bcr
V|x|–ap∗|un|p
p′
≤C
Bcr
V|x|–ap∗|un|p
p′
≤C
Bcr
|x|–ap|∇un|p+V|x|–ap
∗
|un|p
p′
. ()
Using equations () and (), we have
lim sup
n→∞
Bcr
V|x|–ap∗|un|p–unu≤lim sup n→∞
C
Bcr
|x|–ap|∇un|p+V|x|–ap
∗
|un|p
p′
= .
From this fact, we get
BcrV|x|
–ap∗|u
n|p–unu→Bc rV|x|
–ap∗|u|p and the proof of this claim is completed.
Claim : Letp′be dual ofp. Sinceu∈D,p
a (RN) ={u:RN →RN :|x|–au∈Lp
∗
(RN) and |x|–a∇u∈Lp(RN)},bwe see thath=|x|–a|∇u| ∈Lp(RN) andw
Lp′(RN). Asu
n is bounded, we conclude thatwnLp′(RN) is also bounded. Moreover, (un) satisfies the hypothesis of Lemma ., below. So that we have∇un→ ∇ua.e.x∈RN, which give us wn→w=|x|–a(p–)|∇u|p–∇u a.e. x∈RN. Using a theorem [, Theo-rem .], we conclude thatwn⇀winLp
′
(RN). Thus, we have
RNwnh→
RNwh, and hence
RN|x|–ap|∇un|p–∇un∇u→
RN|x|–ap|∇u|p.
Lemma . Let E and J be respectively the space and functional defined in Section.Let
(un)⊂E a bounded sequence such that un⇀u in E and J′(un)→.Then,passing to a
subsequence if necessary,we have∇un→ ∇u,a.e.x∈RN.
The proof of this lemma follows using the same ideas made in [] and [], for a bounded domain. It can be found in [, Claim ] and [, Lemma ].
Using Lemmas ., ., . and the mountain pass theorem, in [, Theorem .], we conclude that there existsu∈Ewhich is a critical point for the functionalJ, in the minimax levelc. Moreover,uis the least energy solution to the problem (AP).
4 The solution of (AP) is solution of (P)
Now, our aim is to show that the solution found in the previous section is also a solution of the problem (P). It is sufficient to verifyf(u)≤Vk|x|–ap∗|u|p–u, for allx∈Bc
r.
Lemma . Any least energy solution u of(AP)satisfies the estimateup≤pkcp–.
Proof Since u is a critical point in the minimax level c≤c, by Lemma ., we have
(p–)
pk up≤J(u) –
θJ
′(u)u=c≤c. So thatup≤pkc
p–.
Remark . The constant pkcp–depends only onV∞,θ andf.
Lemma . Let h be such that|x|–ap∗|h|q is integrable,with pq>N.Consider H:RN×
R→ R, and b:RN → R nonnegative and continuous functions such that |H(x,s)| ≤
h(x)|x|–ap∗|s|p–,for all s> .Let v∈E be a weak solution of(AP):
–Lvap+b|x|–ap∗|v|p–v=H(x,v) in RN.
Then there exists a constant M=M(q,hLq
ap∗(RN)) > such thatv∞≤M|x| –av
p∗.
Proof Givenm∈Nandβ> , setAm={x∈RN:|v|β–≤m},Bm=RN\Amand
vm=
⎧ ⎨
⎩
v|v|p(β–) inAm,
mpv inB m.
Thus
∇vm=
⎧ ⎨
⎩
(pβ–p+ )|v|p(β–)∇v inA
m,
mp∇v inB m.
Thenvm∈Eand using it as the test function in (AP). we have
RN
|x|–ap|∇v|p–∇v∇vm+b|x|–ap
∗
|v|p–vvm=
By definition ofvm, we get
RN|x|
–ap|∇v|p–∇v∇v
m
= (pβ–p+ )
Am
|x|–ap|v|p(β–)|∇v|p+mp
Bm
|x|–ap|∇v|p ()
and
RNb|x|
–ap∗
|v|p–vvm=
Am
b|x|–ap∗|v|pβ+mp
Bm
b|x|–ap∗|v|p> . ()
Using equations () and (), we have
Am
|x|–ap|v|p(β–)|∇v|p
≤(pβ–p+ )–
RN
|x|–ap|∇v|p–∇v∇vm+b|x|–ap
∗
|v|p–vvm. ()
Putting
wm=
⎧ ⎨
⎩
v|v|β– inA
m,
mv inBm,
we have
∇wm=
⎧ ⎨
⎩
β|v|β–∇v inAm,
m∇v inBm.
Thus,
RN|x|
–ap|∇wm|p=βp Am
|x|–ap|v|p(β–)|∇v|p+mp
Bm
|x|–ap|∇v|p. ()
Now, taking into account (), we get
RNb
|x|–ap∗|wm|p=
Am
b|x|–ap∗|v|pβ+mp
Bm
b|x|–ap∗|v|p
=
RNb
|x|–ap∗|v|p–vvm. ()
Using (), () and (), we have
RN
|x|–ap|∇wm|p+b|x|–ap
∗
|wm|p–
RN
|x|–ap|∇v|p–∇v∇vm+b|x|–ap
∗
|v|p–vvm
=
βp–pβ+p–
Am
|x|–ap|v|p(β–)|∇v|p. ()
Combining (), () and (), we obtain
RN
|x|–ap|∇wm|p+b|x|–ap
∗
|wm|p≤βp
LetSbe the best constant for inequality (
RN|x|–ap
∗
|u|p∗)pp∗ ≤ S
RN|x|–ap|∇u|p, for allu∈ D,ap(RN). (See [].) By the inequality (), the boundedness ofHand the definitions of
vmandS, we get (Am|x|–ap
∗
|wm|p
∗
) p p∗ ≤
Sβp
RNh(x)|x|–ap
∗
|v|pβ. Since|w
m|=|v|βinAm, we have
Am
|x|–ap∗|v|p∗β
pp∗
≤Sβp
RNh
(x)|x|–ap∗|v|pβ.
Makingm→ ∞and using the monotone convergence theorem, we get
RN
|x|–ap∗|v|p∗β
pp∗
≤Sβp
RNh(x)
|x|–ap∗|v|pβ.
Sincepq>N,σ=q N
(N–p)> . Thus, we can considerβ=σ
jforj= , , , . . . and, using the Hölder’s inequality, we have
|x|
–a σjv
pσj
p∗σj≤Sσjp
RN
h(x)|x|–ap∗|v|pσj
≤Sσjp
RN
|x|–ap∗h(x)
q
q
RN
|x|
– a σj–|v|
pσjq
q
≤Mσjp
RN
|x|
– a σj–|v|
pσjq
q
≤Mσjp |x|
– a σj–v
pσj
pσjq,
beingM=S[RN|x|–ap
∗
|h(x)|q]q. Note thatM
is independent ofj. Thus, we get
|x|
–a σjv
p∗σj≤M pσj σ
j
σj|x|–
a
σj–v
pσjq for allj= , , , . . . . ()
Forj= andj= , we havepσq=p∗andpσq=p∗σ. Applying this in (), we have
|x|–
a
σv
p∗σ ≤M
pσ σ
σ|x|–av
p∗ and
|x|
–a σv
p∗σ≤M pσ σ
σ|x|–
a
σv
p∗σ.
By iterating, we have
|x|
–a σv
p∗σ≤σ
σ+σM p(σ+σ)
|x|–av
p∗.
Thus, we get
|x|
–a σjv
p∗σj≤σ
σ+σ+···+ j
σjM
p(σ+σ+···+
σj)
|x|–av
p∗ for allj= , , , . . . .
Then we can say that, for allt≥p∗, we have
vt≤σ
σ
(σ–)M p(σ–)
|x|–av
PuttingM:=σ
σ
(σ–)M p(σ–)
, we get
v∞=lim
t→∞vt≤tlim→∞M
|x|–av
p∗=M |x|–av
p∗.
Asσ depends onq, we have, by definition ofM, thatM=M(q,hLq
ap∗(RN)) > .
Remark . In the previous lemma, the constantMdoes not depend on the potentialb
of the problem (AP).
Lemma . There exists a constant M> such thatu∞≤M,for all u positive solution of(AP).
Proof Take r from the definition of g and define A={x∈ RN : |x| ≤rorf(u(x))≤ V(x)
k |x|–ap
∗
|u(x)|p–u(x)} and B=RN \A={x∈RN :|x| >randf(u(x)) > V(x)
k |x|–ap
∗
×
|u(x)|p–u(x)}. Now defineHandbby
H(x,t) =
⎧ ⎨
⎩
f(t) inA,
inB
and b(x) =
⎧ ⎨
⎩
V(x) inA, ( –k)V(x) inB.
We will show that, ifuis solution of (AP), thenuis weak solution of (AP).
In particular, we recall thatH(x,u) =f(u) =g(x,u), inA, andg(x,u) =Vk|x|–ap∗|u|p–u,
inB. Using the fact thatRNis the disjoint union ofAandBand the definitions above, for a givenφ∈E, we write
RN
|x|–ap|∇u|p–∇u∇φ+b|x|–ap∗|u|p–uφ–
RNH (x,u)φ
=
RN
|x|–ap|∇u|p–∇u∇φ+V|x|–ap∗|u|p–uφ–
RN
g(x,u)φ= .
Hence,uverifies
RN|x|–ap|∇u|p–∇u∇φ+b|x|–ap
∗
|u|p–uφ=
RNH(x,u)φ, for allφ∈E. From (f) and (f), we obtain|f(s)| ≤c|x|–ap
∗
|s|α–, for alls> , withα∈(p,p∗). From the
definition ofH, it follows that|H(x,u)| ≤ |f(u)| ≤h(x)|x|–ap∗|u|p–, withh(x) =c|u|α–p.
Takingq= αp–∗p, we have|x|–ap∗|h|q=|x|–ap∗|u|p∗ that is integrable. Thenusatisfies the hypotheses of Lemma ., that is,
u∞≤M
|x|–au
p∗. ()
Using the definition of the constantSand Lemma ., we have
|x|–au
p∗ =
RN
|x|–ap∗|u|p∗
p∗
≤
S
RN
|x|–ap|∇u|p
p
≤
Sup p≤
S pkc
p–
p
. ()
Combining equations () and (), we obtainu∞≤M(Sppkc–)
p:=M
Lemma . Let r be as in the definition of g and consider R≥r.Let u a be any positive solution of(AP).Then we have
u(x)≤ u∞R
N–p p–
|x| –N–pp–(a+)
≤MR
N–p p–
|x| –N–pp–(a+)
for all x∈BcR.
Proof Considerv(x) =MR
N–p p–
|x| –N–pp–(a+)
. By Lemma ., we haveu∞≤M. Sou≤v,
for|x|=R. It follows that (u–v)+= , in|x|=R, and the function given by
w=
⎧ ⎨
⎩
, |x|<R,
(u–v)+, |x| ≥R
is so thatw∈Da,p(RN). Moreover,w∈E, becauseu,v∈E. Let us show now that (u–v)+=
, in|x| ≥R. Takingwas the test function, using the hypotheses ongand V and the fact
thatuis positive solution of (AP), we have
RN
|x|–ap|∇u|p–∇u∇w=
RNg
(x,u)w–
RNV
|x|–ap∗|u|p–uw
=
Bc R
g(x,u)w–
Bc R
V|x|–ap∗|u|p–uw
=
k– BcR
V|x|–ap∗|u|p–uw≤. ()
Here, we considered thatk> . Using the radially symmetric form of the operator –Lvap (see [, ]), we have that –div(|x|–ap|∇v|p–∇v) = inBc
R. In the weak form, it is
BcR
|x|–ap|∇v|p–∇v∇φ= for allφ∈E.
Hence,
RN
|x|–ap|∇v|p–∇v∇w=
BcR
|x|–ap|∇v|p–∇v∇w= . ()
PuttingA={x∈RN:|x| ≥R
andu(x) >v(x)}andB=RN\A, we have
w=
⎧ ⎨
⎩
u–v inA,
inB.
By () and (), we obtain, for all <p<N, that
A |x|–ap
|∇u|p–∇u–|∇v|p–∇v
[∇u–∇v]
=
A
|x|–ap|∇u|p–∇u–|x|–ap|∇v|p–∇v
∇w
=
RN
|x|–ap|∇u|p–∇u∇w–
RN
Consider ≤ p< N. Using the Tolksdorf ’s inequality (see [, Lemma .] or [, Lemma .]) and equation (), we obtain
RN
|x|–ap|∇w|p =
A
|x|–ap|∇u–∇v|p
≤c
A
|x|–ap|∇u|p–∇u–|∇v|p–∇v
[∇u–∇v]≤.
In the case <p< , in addition to the above arguments used, we also use the Hölder’s inequality. Then
RN
|x|–ap|∇w|p
=
A
|x|–ap|∇w|p=
A |x|–ap
|∇u–∇v| p
≤c
A
|x|–ap|∇u|p–∇u–|∇v|p–∇v
[∇u–∇v] p
|∇u|+|∇v| (–p)p
≤c
A
|x|–ap|∇u|p–∇u–|∇v|p–∇v
[∇u–∇v]
p
A
|x|–ap|∇u|+|∇v|p
–p
≤.
Then
RN|x|–ap|∇w|p≤ for all <p<N. Thus, we havew= , inRN, which implies that (u–v)+= , in|x| ≥R
. From this, we conclude thatu≤vinRN, and lemma is proved.
Proof of Theorem. We will show thatf(u)≤Vk|x|–ap∗|u|p–uinBcr, for all solutionuof (AP). By Remark we have|sf(s)| ≤C|x|–ap
∗
|s|p∗, which gives us f(u)
|x|–ap∗|u|p–u≤C|u| P N–p.
Now, note that the hypothesis (V) holds for allR>R. Hence, forR=r>R, we can use
(V) and Lemma .. Thus, for eachxinBcr, and∗=kCM
P N–p
R
p p–
, we have
f(u) |x|–ap∗
|u|p–u≤C|u|
P N–p≤C
MR
N–p p–
|x|
–N–pp–(a+)
P N–p
=CM
P N–p
R
p p–
V
V|x|
p[N–p(a+)] (p–)(N–p)
=
∗
k
V
V|x|
p[N–p(a+)] (p–)(N–p)
≤
∗
V k.
Now, taking∗≤it follows that f(u)
|x|–ap∗|u|p–u≤Vk, for everyxinB c
r, which give usf(u)≤ V
k|x|–ap
∗
|u|p–u, inBc
r.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that all authors collaborated and dedicated the same amount of time in order to perform this article.
Author details
Acknowledgements
The authors would like to thank the referees for valuable comments and suggestions in improving this article. OHM was partially supported by CNPq/Brazil and Fapemig/Brazil (CEX-APQ 00025-11). RSV was partially supported by Capes/Brazil.
Endnotes
a The proof of this fact follows by using the same ideas made in the usual Sobolev spaces; see [40]. b The proof of this fact follows by using the same ideas made in the case
D1,p(RN); see [41].
Received: 28 November 2012 Accepted: 3 April 2013 Published: 17 April 2013 References
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doi:10.1186/1687-2770-2013-92
Cite this article as:Bastos et al.:Existence of solutions for a class of degenerate quasilinear elliptic equation inRN
