Compactness results and applications to some "zero mass" elliptic problems

0.22 MB·English

by  Antonio Azzollini

6
Compactness results and applications to
0
0
2 some “zero mass” elliptic problems

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J

7 A. Azzollini & A. Pomponio
∗ †
1

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A
. In ricordodiGiulioMinervini
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1 1 Introduction and statement of the main results
v
0
1
Inthis paperwestudy the elliptic problem,
4
1
0 ∆v = f′(v) in Ω, (1)
6 −
0
/ inthesocalled“zeromasscase”thatis,roughlyspeaking,whenf′′(0) = 0.
h
A particular exampleis
t
a
m
: ∆v = vNN−+22 in RN,
v −
i
X with N > 3. This problem has been studied very intensely (see [4,16,24])
r and weknow the explicit expression ofthe positive solutions
a
[N(N 2)λ2](N−2)/4
v(x) = − , with λ > 0, x RN.
[λ2 + x x 2](N−2)/2 0 ∈
0
| − |
If f is not the critical power, we are led to require particular growth
conditions onthenonlinearity f. Infact,whileinthe “positive masscase”
(namely when f′′(0) < 0) the natural functional setting is H1(Ω) and we
havesuitablecompactembeddingsjustassumingasubcriticalbehaviorof
∗DipartimentodiMatematica,Universita`degliStudidiBari,ViaE.Orabona4,I-70125
Bari,Italy,e-mail: [email protected]
†DipartimentodiMatematica,PolitecnicodiBari,ViaAmendola126/B,I-70126Bari,
Italy,e-mail: [email protected]
1
2 A. Azzollini and A. Pomponio
f,inthe“zeromasscase”theproblemisstudiedin 1,2(Ω)thatisdefined
D
asthe completion ofC∞(Ω) with respectto the norm
0
1
2
u = u 2dx .
k k |∇ |
(cid:18)ZΩ (cid:19)
Inordertorecoveranalogouscompactnessresults,weneedtoassumethat
f issupercritical nearthe origin andsubcritical atinfinity.
With these assumptions on f, the problem (1) has been dealt with by
Berestycki & Lions [13–15], when Ω = RN, N > 3, and existence and
multiplicity results have beenproved.
Recently, Benci & Fortunato [8] have introduced a new functional set-
ting, namely the Orlicz space Lp + Lq, which arises very simply from the
growth conditions on f and seemsto be the natural framework forstudy-
ing“zeromass” problems asshown also byPisani in [23].
Usingthisnewfunctional setting, Benci &Micheletti in[9]studied the
problem (1), with Dirichlet boundary conditions, in the case of exterior
domain, namely when RN Ω is contained into a ball B . Under suitable
ε
\
assumptions,iftheballradiusεissufficientlysmall,theyareabletoprove
the existence of apositive solution.
The functional setting introduced in [8] seems to be the natural one
alsofor studying the nonlinearSchro¨dinger equationswith vanishing po-
tentials, namely
∆v +V(x)v = f′(v), in RN, (2)
−
with
lim V(x) = 0.
x→∞
SomeexistenceresultsforsuchaproblemhavebeenfoundbyBenci,Grisanti
&Micheletti [10,11]and byGhimenti & Micheletti [18].
Even if in a different context, we need also to mention the paper of
Ambrosetti, Felli & Malchiodi [2], where problem (2) is studied when the
nonlinearityf(v)isreplacedbyafunction f(x,v)ofthetype K(x)vp,with
K vanishingat infinity.
In this paper, we study problem (1) in two different situations. In Sec-
tion 4, we look for complexvalued solutions ofthe following problem
∆v = f′(v) in R3, (3)
−
assumingthat f C1(C,R)satisfies the following assumptions:
∈
(f1) f(0) = 0;
Compactness resultsand “zeromass”ellipticproblems 3
(f2) M > 0 such that f(M) > 0;
∃
(f3) ξ C : f′(ξ) 6 cmin( ξ p−1, ξ q−1);
∀ ∈ | | | | | |
(f4) f(eiαρ) = f(ρ),for all ξ = eiαρ C;
∈
where 1 < p < 6 < q andc > 0.
Observe that an example of function satisfying the previous hypothe-
ses can be obtained as follows. Let us consider the function f˜ : R+ R
→
definedas
atp +b if t > 1
˜
f(t) :=
tq if t 6 1,
(cid:26)
with a,b R chosen in order to have f˜ C1 and let us define f : C R
∈˜ ∈ →
asf(ξ) = f( ξ ).
| |
Introducing the cylindrical coordinates (r,z,θ), for all n Z, we look
∈
for solutions ofthe type
vn(x,y,z) = un(r,z)einθ with un R. (4)
∈
Weobtain the following existence result for problem (3):
Theorem 1.1. Let f satisfy the hypotheses (f1-f4). Then there exists a sequence
(vn) of complex-valued solutions of problem (3), such that, for every n Z,
n
∈
vn(x,y,z) = un(r,z)einθ, with un R.
∈
Actually,anexistenceresultinthesamespiritofoursispresentin[22].
However, in [22] the problem is studied using different tools and the de-
tails are omitted. Moreover in [12] an interesting physical interpretation
has been given to the complex valued solutions of the equation (3) in the
positivemasscase. Infacttherehasbeenshownthestrictrelationbetween
such solutions and the standing waves of the Schro¨dinger equation with
nonvanishing angularmomentum.
In Section 5,we study
∆v = f′(v) in R2 I,
− × (5)
v = 0 in R2 ∂I,
(cid:26) ×
whereI isaboundedintervalofRandf C1(R,R)satisfiesthefollowing
∈
assumptions:
(f1’) f(0) = 0;
(f2’) ξ R :f(ξ) > c min( ξ p, ξ q);
1
∀ ∈ | | | |
4 A. Azzollini and A. Pomponio
(f3’) ξ R : f′(ξ) 6 c min( ξ p−1, ξ q−1);
2
∀ ∈ | | | | | |
(f4’) there exists α > 2 such that ξ R : αf(ξ) 6 f′(ξ)ξ;
∀ ∈
with 2 < p < 6 < q and c ,c > 0.
1 2
Wewill prove the following multiplicity result:
Theorem 1.2. Let f satisfy the hypotheses (f1’-f4’). Then there exist infinitely
many solutions withcylindricalsymmetryof problem(5).
In order to approach to our problems, we use a functional framework
related to the Orlicz space Lp + Lq. The main difficulty in dealing with
suchspacesconsistsinthelackofsuitablecompactnessresults. Inviewof
this,thekeypointsofthispaperaretwocompactnesstheoremspresented
in Section 3. They are obtained adapting a well known lemma of Esteban
&Lions [17]to our situation.
The paperisorganized asfollows: Section 2 isdevoted to a briefrecall
on the space Lp +Lq; in Section 3, we present our compactness results; in
Sections 4 and 5 we solve problems (3) and (5); finally, in the Appendix
we prove a compact embedding theorem using similar arguments as in
Section 3.
2 Some properties of the Lp + Lq spaces
Inthissection,wepresentsomebasicfactsontheOrliczspaceLp+Lq. For
more details, see [8,19,23].
Let Ω R3. For 1 < p < 6 < q, denote by (Lp(Ω), ) and by
Lp
⊂ k · k
(Lq(Ω), ) the usual Lebesgue spaceswith their norms, andset
Lq
k·k
Lp +Lq(Ω) := v : Ω R (v ,v ) Lp(Ω) Lq(Ω) s.t. v = v +v .
1 2 1 2
{ → |∃ ∈ × }
Thespace Lp +Lq(Ω) isa Banachspace with the norm
v (Ω) : inf v + v (v ,v ) Lp(Ω) Lq(Ω),v +v = v
Lp+Lq 1 Lp 2 Lq 1 2 1 2
k k {k k k k | ∈ × }
anditsdualistheBanachspace Lp′(Ω)∩Lq′(Ω),k·kLp′∩Lq′ ,wherep′ = p−p1,
q′ = q and
q−1 (cid:0) (cid:1)
kϕkLp′∩Lq′ : kϕkLp′ +kϕkLq′.
In the sequel,for allv Lp +Lq(Ω), weset
∈
Ω> := x Ω v(x) > 1 ,
∈ | | |
Ω6 := x Ω v(x) 6 1 .
(cid:8) (cid:9)
∈ | | |
The following theorem su(cid:8)mmarizes some p(cid:9)roperties about Lp + Lq
spaces
Compactness resultsand “zeromass”ellipticproblems 5
Theorem 2.1. 1. Let v Lp +Lq(Ω). Then
∈
1
max v 1, v
k kLq(Ω6) − 1+meas(Ω>)1/rk kLp(Ω>)
(cid:18) (cid:19)
6 v 6 max v , v (6)
Lp+Lq Lq(Ω6) Lp(Ω>)
k k k k k k
wherer = pq/(q p). (cid:0) (cid:1)
−
2. ThespaceLp +Lq iscontinuously embeddedinLp .
loc
3. For everyr [p,q] :Lr(Ω) ֒ Lp +Lq(Ω) continuously.
∈ →
4. Theembedding
1,2(Ω) ֒ Lp +Lq(Ω) (7)
D →
iscontinuous.
Proof
1. See Lemma1 in[8].
2. See Proposition 6 of[23].
3. See Corollary 9 in [23].
4. Itfollows from the point 3and the Sobolev continuous embedding
1,2(Ω) ֒ L6(Ω).
D →
(cid:3)
The following theorem hasbeenproved in [23]:
Theorem2.2. Letf beaC1(C,R)function(resp. C1(R,R))satisfyingassump-
tion(f3)(resp. (f3’)). Thenthefunctional
v Lp +Lq(Ω) f(v)dx
∈ 7−→
ZΩ
isof classC1. Moreover theNemytskioperator
f′ : v Lp +Lq(Ω) f′(v) (Lp +Lq(Ω))′
∈ 7−→ ∈
isbounded.
UsingTheorem2.2wegetaveryusefulinequalityfortheLp+Lq-norm.
6 A. Azzollini and A. Pomponio
Theorem2.3. For allR > 0,thereexistsapositiveconstantc = c(R)suchthat,
forall v Lp +Lq(Ω) with v 6 R,
Lp+Lq
∈ k k
max v pdx, v qdx 6 c(R) v . (8)
Lp+Lq
| | | | k k
(cid:18)ZΩ> ZΩ6 (cid:19)
Proof Let us introduce g C1(R,R) such that g(0) = 0 and with the
∈
following growth conditions:
(g1) ξ R :g(ξ) > c min( ξ p, ξ q);
1
∀ ∈ | | | |
(g2) ξ R : g′(ξ) 6 c min( ξ p−1, ξ q−1).
2
∀ ∈ | | | | | |
Integrating in (g1)we get
g(v)dx > c v pdx+ v qdx .
1
| | | |
ZΩ (cid:18)ZΩ> ZΩ6 (cid:19)
ByLagrange theorem, there exists t [0,1]such that
∈
g′(tv)vdx = g(v)dx > c v pdx+ v qdx .
1
| | | |
ZΩ ZΩ (cid:18)ZΩ> ZΩ6 (cid:19)
Then, by the boundness of g′ (see Theorem 2.2), there exists M > 0 such
that
M v > g′(tv)v dx > c v pdx+ v qdx
Lp+Lq 1
k k | | | | | |
ZΩ (cid:18)ZΩ> ZΩ6 (cid:19)
andhence the conclusion. (cid:3)
Remark 2.4. Combining the inequality (6) with the estimate (8) we deduce that
thefollowing statementsare equivalent:
a) v v inLp +Lq(Ω),
n
→
b) v v 0 and v v 0,
k n − kLp(Ω>n) → k n − kLq(Ω6n) →
whereΩ> = x Ω v (x) v(x) > 1 and Ω6 isanalogously defined.
n { ∈ | | n − | } n
Compactness resultsand “zeromass”ellipticproblems 7
3 Compactness results
In this section we present the main tools of this paper, namely a compact-
ness theorem for sequences with “a particular symmetry” and a compact
embeddingofasuitable subspace of 1,2 into Lp+Lq. The proofs ofthese
D
results are both modelled on that of Theorem 1 of [17], which states that a
suitable subspace ofH1 iscompactly embeddedinto Lp, forp subcritical.
Firstofall,foreveryintervalI ofR,possiblyunbounded,weintroduce
the following subspace of 1,2(R2 I):
D ×
1,2(R2 I) = u 1,2(R2 I) u( , ,z)isradial, for a.e. z I .
Dcyl × ∈ D × | · · ∈
(cid:8) (cid:9)
Moreover weassume the following
Definition3.1. Ifu : R2 I Risameasurablefunction,wecallz-symmetrical
× →
rearrangementofuin(x,y)theSchwarzsymmetricalrearrangementofthefunc-
tion
u(x,y, ) : z I u(x,y,z) R.
· ∈ 7→ ∈
Moreover we call z-symmetrical rearrangement of u the function v defined as
follows
u˜ : (x,y,z) R2 I u˜ (z)
x,y
∈ × 7→
whereu˜ is thez-symmetricalrearrangementof uin(x,y).
x,y
In our first compactness result, weconsider I = R.
Theorem 3.2. Let (u ) be a bounded sequence in 1,2(R3) such that u is the
j j Dcyl j
z-symmetrical rearrangement of itself. Then (u ) possesses a converging subse-
j j
quence inLp +Lq(R3), for all1 < p < 6 < q.
Proof With an abuse of notations, in the sequel for every v 1,2(R3),
∈ Dcyl
we denote byv alsothe function defined in R+ R as
×
v( x2 +y2,z) = v(x,y,z).
p
Beingtheproofquitelongandinvolved,wedivideitintoseveralsteps,for
reader’s convenience. Since (u ) is bounded in the 1,2(R3) norm, there
j j
D
exists u 1,2(R3) such that
∈ Dcyl
u ⇀ u weaklyin 1,2 R3 andin Lp +Lq(R3), 1 < p 6 6 6 q, (9)
j Dcyl
u u a.e. in R3, (10)
j (cid:0) (cid:1)
→
u u inLp(K), for all K R3, 1 6 p < 6. (11)
j
→ ⊂⊂
8 A. Azzollini and A. Pomponio
ByLions[20],
C
j > 1, r > 0,z = 0 : u (x,y,z) 6 , (12)
∀ ∀ 6 | j | r41 z 41
| |
where r = x2 +y2. By (12), for R > 0 large enough, j > 1 and for all
(r, z ) (R,+ ) (R,+ ),we have
| | ∈ p∞ × ∞
u (x,y,z) < 1,
j
| |
u(x,y,z) < 1, (13)
| |
(u u)(x,y,z) < 1.
j
| − |
Let
D := (r,z) R+ R r > R, z > R ,
1
{ ∈ × | | | }
D := (r,z) R+ R 0 6 r 6 R, z 6 R ,
2
{ ∈ × | | | }
D := (r,z) R+ R 0 6 r 6 R, z > R ,
3
{ ∈ × | | | }
D := (r,z) R+ R r > R, z 6 R .
4
{ ∈ × | | | }
4
Obviously D = R+ R. Moreover denote by χ the characteristic
i × Di
i=1
[
function ofD and observe that, since
i
4
u u = (u u)χ
k j − kLp+Lq j − Di Lp+Lq
(cid:13)Xi=1 (cid:13)
(cid:13)4 (cid:13) 4
(cid:13) (cid:13)
6 (u u)χ = u u ,
k j − DikLp+Lq k j − kLp+Lq(Di)
i=1 i=1
X X
then weget the conclusion ifweprove that, forall i = 1,...,4,
u u in Lp +Lq(D ).
j i
→
CLAIM 1: uj u inLp +Lq(D1).
→
Suppose for a moment that q > 8. By (13), for every (x,y,z) D , we
1
∈
have (u u)(x,y,z) < 1,then the inequality(6)implies
j
| − |
u u 6 u u . (14)
k j − kLp+Lq(D1) k j − kLq(D1)
On the other hand,since
C
u u a.e. and (u u)(r,z) q 6 L1(D ),
j j q q 1
→ | − | r 4 z 4 ∈
| | | |
Compactness resultsand “zeromass”ellipticproblems 9
byLebesgue theorem u uin Lq(D ).
j 1
→
If6 < q 6 8,thentake r q 6,4(q 6) andsetα = 6 and β = 6 .
∈ − − 6−q+r q−r
Observe that 1 + 1 = 1 so,(cid:16)byHolder, (cid:17)
α β
u u qdxdydz u u r u u q−rdxdydz
j j j
| − | | − | | − |
ZD1 ZD1
1 1
6 u u αr α u u (βq−r) β
j j
| − | | − |
(cid:16)ZD1 (cid:17) (cid:16)ZD1 (cid:17)
6−q+r
6 u u 6−6qr+r 6 u u q−r. (15)
| j − | k j − kL6
(cid:16)ZD1 (cid:17)
Since (u ) is bounded in 1,2(R3),it isbounded in L6(R3).
j j
D
Moreover, since q 6 < r < 4(q 6),certainly 6r > 8,and thenthe last
− − 6−q+r
integral in inequality(15)goesto zero.
Hencethe Claim1 isproved.
CLAIM 2: uj u in Lp +Lq(D2).
→
Itisenoughtoobserve that,sinceD hasfinitemeasure,Lp+Lq(D ) =
2 2
Lp(D ) (see [23, Remark5])and then we getthe conclusion by(11).
2
CLAIM 3: uj u in Lp +Lq(D3).
→
First suppose p < 4 and consider g C1(R,R), g(0) = 0, such that the
∈
following growth and strong convexity conditions hold
(G) k > 0s.t. t R : g′(t) 6 k min( t p−1, t q−1),
1 1
∃ ∀ ∈ | | | | | |
(SC) k > 0s.t. s,t R :g(s) g(t) g′(t)(s t)
2
∃ ∀ ∈ − − −
> k min( s t p, s t q).
2
| − | | − |
Since g(0) = 0, from (G)and (SC)wededuce that
k ,k > 0s.t. s R : k min( s p, s q) 6 g(s) 6 k min( s p, s q). (16)
3 4 3 4
∃ ∀ ∈ | | | | | | | |
The condition (SC) has been introduced in [5], where an explicit example
offunction satisfying (SC)is alsogiven.
For almost every (x,y) R2, we set ux,y : R R defined as ux,y(z) :=
∈ →
u(x,y,z). Wegive an analogous definition forux,y, forall j > 1.
j
For almost every(x,y) R2 with (x2 +y2)1/2 6 R, we set
∈
w (x,y) := g(ux,y(z))dz.
j j
Z(−R,R)c
10 A. Azzollini and A. Pomponio
Weshowthat
w (x,y) g(ux,y(z))dz for a.e. (x,y) B . (17)
j R
→ ∈
Z(−R,R)c
Consider
w (x,y) g(ux,y(z))dz 6 g(ux,y) g(ux,y) dz
j − | j − |
(cid:12) Z(−R,R)c (cid:12) Z(−R,R)c
(cid:12) (cid:12)
(cid:12) (cid:12) = g′(θx,y) ux,y ux,y dz (18)
(cid:12) (cid:12) | j || j − |
Z(−R,R)c
where, for almost every (x,y) B , θx,y is a suitable convex combination
∈ R j
of ux,y and ux,y. Since (u ) is bounded in Lp + Lq, g′(θx,y) is bounded in
j j j j
′
Lp + Lq (see Theorem 2.2) so, by (18), to prove (17) we are reduced to
s(cid:16)howtha(cid:17)t
ux,y ux,y in Lp +Lq ( R,R)c for a.e. (x,y) B .
j → − ∈ R
For, define (cid:0) (cid:1)
Ωx,y = z R z > R, ux,y(z) ux,y(z) > 1
j ∈ | | | | j − |
sothat, by(6), (cid:8) (cid:9)
ux,y ux,y 6
k j − kLp+Lq((−R,R)c)
max kuxj,y −ux,ykLp(Ωxj,y),kuxj,y −ux,ykLq((−R,R)c\Ωxj,y) . (19)
(cid:16) (cid:17)
ByLebesgue theorem and by(12),
kuxj,y −ux,ykLq((−R,R)c\Ωxj,y) → 0 for a.e. (x,y) ∈ BR. (20)
Moreover there exists R′ = R′(x,y) Rsuch thatfor all z > R′
∈ | |
2C
ux,y(z) ux,y(z) 6 6 1.
| j − | r1/4 z 1/4
| |
LetR˜ = max(R,R′). Wehave
ux,y ux,y p 6 ux,y ux,y pdz
k j − kLp(Ωxj,y) Z(−R˜,−R)∪(R,R˜)| j − |
andthen
kuxj,y −ux,ykLp(Ωxj,y) → 0 for a.e. (x,y) ∈ BR, (21)