Quantitative Local Bounds for Subcritical Semilinear Elliptic Equations

0.78 MB·English

by  Matteo Bonforte

Quantitative Local Bounds for Subcritical
Semilinear Elliptic Equations
Matteo Bonfortea Gabriele Grillob and Juan Luis Vazquezc
January 30, 2012
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1
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2 Abstract

n The purpose of this paper is to prove local upper and lower bounds for weak solutions of
a semilinearellipticequationsoftheform−∆u=cup, with0<p<p =(d+2)/(d−2), definedon
s
J bounded domains of Rd, d≥3, without reference to the boundary behaviour. We give an explicit

7 expression for all the involved constants. As a consequence, we obtain local Harnack inequalities
2 with explicit constant, as well as gradient bounds.

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P
A
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t
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m Keywords. Local bounds, semilinear elliptic equations, regularity, Harnack inequality.
[ Mathematics Subject Classification. 35B45, 35B65, 35K55, 35K65.

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v
1 (a) Departamento de Matema´ticas, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain. E-mail
0 address: [email protected].
8 Web-page: http://www.uam.es/matteo.bonforte
5
. (b) Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20183 Milano,
1
Italy. E-mail address: [email protected]
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2 (c) Departamento de Matem´aticas, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain. E-mail
1 address: [email protected].
:
v Web-page: http://www.uam.es/juanluis.vazquez
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X
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a
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Contents
1 Introduction 3
2 Preliminaries. Local energy estimate 4
2.1 More general nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Local Upper Bounds 6
3.1 Local upper bounds I. The upper Moser iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Local upper bounds II. The linear case with unbounded coefficients . . . . . . . . . . . . . . . . . 14
3.2.1 Energy Estimates and Reverse Poincar´e inequalities . . . . . . . . . . . . . . . . . . . . . 14
3.2.2 Extending local upper bounds. A lemma by E. De Giorgi . . . . . . . . . . . . . . . . . . 20
4 Lower bounds 24
4.1 A short reminder about the spaces Mp(Ω). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 The John-Nirenberg Lemma and reverse H¨older inequalities. . . . . . . . . . . . . . . . . . . . . 25
4.3 Lower Moser iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 Reverse Ho¨lder inequalities and additional local lower bounds when 1<p<p . . . . . . . . . . 30
c
5 Harnack inequalities 33
6 Local Absolute bounds 36
7 Regularity. Local bounds for the gradients 37
8 Table of results 41
9 Appendix. Numerical Identities and Inequalities 42
2
1 Introduction
In this paper we obtain local upper and lower estimates for the weak solutions of semilinear elliptic
equations of the form
(1.1) −∆u=f(u)
posed in a bounded domain Ω ⊂ Rd. The choice of right-hand side we have in mind is f(u) = λup
with λ,p > 0. The range of exponents of interest will be 1 < p < p := (d+2)/(d−2) if d ≥ 3, or
s
p > 1 if d = 1,2. This problem is one of the most popular problems in nonlinear elliptic theory and
enjoys a large bibliography [2, 8, 9, 12, 16, 17, 18, 19, 20, 21, 23, 24, 29, 30, 31, 32] for 0 ≤ p < p
s
and [7, 11] for p = p . We refrain from attempting to give a complete bibliography for this nowadays
s
classical problem.
We focus our attention on obtaining local estimates for solutions that are defined inside the domain
without reference to their boundary behaviour. This is the notion of solution we use.
Definition 1.1 Alocalweaksolutiontoequation−∆u=f(u)inΩisdefinedasafunctionu∈W1,2(Ω)
loc
with f(u)∈L1 (Ω) which satisfies
loc
ˆ
(1.2) [∇u·∇ϕ−f(u)ϕ] dx=0
K
for any subdomain with compact closure K ⊂Ω and all bounded ϕ∈C1(K).
0
Our aim is to contribute quantitative estimates in the form of upper bounds for solutions of any
sign, lower bounds for positive solutions, and also local Harnack inequalities and gradient bounds. By
quantitative estimates we mean keeping track of all the constants during the proofs. As far as we
know, there does not exist in literature a systematic set of quantitative estimates of local upper and
lower bounds, and neither of the Harnack constant, in the form we explicitly provide here. We recall
that the quantitative control of the constants of such inequalities may have an important role in the
applications; it is needed for instance in the results of [3] on the asymptotic properties of solutions of
the fast diffusion equation in bounded domains.
Contents and main results. We start with a section devoted to basic energy estimates. We then
consider in Section 3 the upper estimates for nonnegative solutions of the equation −∆u = λup. The
exponent range is 0 ≤ p < p , a main restriction of the theory, as it is already well known. See also
s
[10]forL∞-boundsofdifferenttypeforEquation(1.1)withmoregeneralnonlinearities. Ourfirstmain
result, Theorem 3.1, can be considered as a smoothing effect with very precise constants; it is much
simpler for p ≤ 1, but we also obtain the more complicated and novel estimates for 1 < p < p . Next
s
we obtain local upper estimates for −∆u = b(x)u with unbounded coefficients in Theorem 3.8 and we
apply them to the case b(x)=up−1 in Theorem 3.9.
In Section 4 we prove quantitative lower estimates, Theorems 4.6, 4.8. We prove Harnack inequalities
in Theorems 5.1, 5.2 and 5.3. All of these results appear to be well known from a qualitative point
of view. Let us mention that, as far as we know, the Harnack inequality for solutions to (1.2) when
p>1 is not stated explicitly in the literature. The fact that the “constant” involved has to depend on
u when p ≤ p < p is confirmed by the results of [6], [14] applied to separation of variable solutions
c s
of parabolic problems, see also the very recent monograph [15]. This is also related to the fact that, in
the range p ≤p<p , there exist (very weak) singular solutions. Notice also that in such a range the
c s
notion of weak and very weak solution is really different, cf. [13, 22, 25, 26, 27, 28].
In Section 6 we derive quantitative absolute upper (for 1<p<p ) and lower bounds (for 0≤p<1)
c
which are new as far as we know, cf. Theorem 6.1. The last section is devoted to quantitative gradient
estimates, cf. Theorem 7.1, and absolute upper bounds for the gradient when 1<p<p , cf. Theorem
c
7.2.
3
As a consequence of the above theory, we conclude that functions in the so-called De Giorgi class
(satisfying Sobolev and local reverse Sobolev inequalities, at least at the level of truncates) are indeed
locally bounded functions.
Much of the known theory takes into account boundary conditions of different types: Dirichlet, Neu-
mann, Robin, orother. Ourresultsapplytoallthose cases. Wewill studythepreciseestimatesfor the
Dirichlet problem in an upcoming paper [4].
2 Preliminaries. Local energy estimate
We shall pursue in the sequel the well-known idea that local weak solutions satisfy reverse Sobolev
or Poincar´e inequalities. Such local reverse inequalities are the key to prove local upper and lower
estimates of next sections, and indeed imply that such functions are H¨older continuous. We comment
thatasimilarlineofreasoningcouldbeadaptedtodealwithfunctionbelongingtosuitablydefinedDe
Giorgi classes.
Lemma 2.1 (Energy Estimates) Let Ω⊂Rd be a bounded domain, and let p ≥0 and λ>0. Let u
be a local nonnegative weak solution in Ω to −∆u=λup. Then the following energy equality holds true
for any δ >0, α(cid:54)=−1 and any positive test function ϕ∈C2(Ω) and compactly supported in Ω:
ˆ ˆ ˆ
(2.1) 4α (cid:12)(cid:12)∇(cid:16)(u+δ)α+21(cid:17)(cid:12)(cid:12)2ϕdx=λ(α+1)2 up(u+δ)αϕdx+(α+1) (u+δ)α+1∆ϕdx.
Ω Ω Ω
Moreover, for any δ ≥0 we have the Caccioppoli estimates
ˆ ˆ ˆ
up (cid:12) (cid:12)2 |∇ϕ|2
(2.2) λ ϕdx+ (cid:12)∇log(u+δ)(cid:12) ϕ dx≤ dx.
u+δ ϕ
Ω Ω Ω
Local subsolutions u of −∆u≤λup satisfy, for α(cid:54)=−1 and δ >0:
ˆ ˆ ˆ
(2.3) 4α (cid:12)(cid:12)∇(cid:16)(u+δ)α+21(cid:17)(cid:12)(cid:12)2ϕdx≤λ(α+1)2 up(u+δ)αϕdx+(α+1) (u+δ)α+1∆ϕdx
Ω Ω Ω
while local supersolution −∆u≥λup satisfy, for any α(cid:54)=−1 and δ >0:
ˆ ˆ ˆ
(2.4) 4α (cid:12)(cid:12)∇(cid:16)(u+δ)α+21(cid:17)(cid:12)(cid:12)2ϕdx≥λ up(u+δ)αϕdx+ 1 (u+δ)α+1∆ϕdx,
(α+1)2 α+1
Ω Ω Ω
and the Caccioppoli estimates also work.
Remark. Notice that when α>−1, we can let δ =0 in the energy identity (2.1) to get
ˆ ˆ ˆ
(2.5) 4α (cid:12)(cid:12)∇(cid:16)uα+21(cid:17)(cid:12)(cid:12)2ϕdx=λ(α+1)2 up+αϕdx+(α+1) uα+1∆ϕdx.
Ω Ω Ω
The same remark applies to subsolutions:
ˆ ˆ ˆ
(2.6) 4α (cid:12)(cid:12)∇(cid:16)uα+21(cid:17)(cid:12)(cid:12)2ϕdx≤λ(α+1)2 up+αϕdx+(α+1) uα+1∆ϕdx
Ω Ω Ω
Proof. Let ϕ∈C2(Ω)∩C1(Ω) and δ ≥0. Multiply −∆u by (u+δ)αϕ, with α(cid:54)=−1 and integrate by
0
parts to get
ˆ ˆ ˆ
− ϕ(u+δ)α∆udx= ∇ϕ·(cid:0)∇u(cid:1)(u+δ)αdx+α ϕ(u+δ)α−1(cid:12)(cid:12)∇u(cid:12)(cid:12)2dx
(2.7) Ω Ω ˆ Ω ˆ
=− 1 (u+δ)α+1∆ϕdx+ 4α (cid:12)(cid:12)∇(u+δ)α+21(cid:12)(cid:12)2ϕdx.
α+1 (α+1)2
Ω Ω
4
For local weak solutions of −∆u=λup, the above equality immediately gives the energy identity (2.1)
forα(cid:54)=−1. Similarconsiderationshold, inthestatedrangeofα, forsubandsupersolutions. Toderive
the Cacciopoli estimate we use the test function ϕ/(u+δ) to get
ˆ ˆ ˆ ˆ √
up ϕ ϕ (cid:12) (cid:12)2 ∇ϕ·∇u ϕ
0≤λ ϕdx=− ∆udx=− (cid:12)∇u(cid:12) dx+ √ dx
u+δ u+δ (u+δ)2 u+δ ϕ
ˆΩ Ω ˆ Ω ˆ Ω
(cid:12) (cid:12)2 1 |∇ϕ|2 1 (cid:12) (cid:12)2
≤− ϕ(cid:12)∇log(u+δ)(cid:12) dx+ dx+ (cid:12)∇log(u+δ)(cid:12) ϕdx
2 ϕ 2
Ωˆ Ωˆ Ω
1 (cid:12) (cid:12)2 1 |∇ϕ|2
≤− ϕ(cid:12)∇log(u+δ)(cid:12) dx+ dx,
2 2 ϕ
Ω Ω
where we have used the inequality a·b≤(|a|2+|b|2)/2.
We shall also need the following particular computation.
Lemma 2.2 Fix two balls B ⊂ B ⊂⊂ Ω. Then there exists a test function ϕ ∈ C1(B ), with
R1 R0 0 R0
∇ϕ≡0on∂Ω,whichisradiallysymmetricandpiecewiseC2 asafunctionofr,satisfiessupp(ϕ)=B
R0
and ϕ=1 on B , and moreover satisfies the bounds
R1
4 4d
(2.8) (cid:107)∇ϕ(cid:107) ≤ and (cid:107)∆ϕ(cid:107) ≤ .
∞ R −R ∞ (R −R )2
0 1 0 1
Proof. Consider the radial test function defined on B
R0
 1 if 0≤|x|≤R
 1− 2((R|x0|−−RR11))22 if R1 <|x|≤ 1R0+2R1
(2.9) ϕ(|x|)=
 02((RR00−−R|x1|))22 iiff |Rx0|+2>R1R<0 |x|≤R0
for any 0<R <R . We have
1 0

0 if 0≤|x|≤R or if |x|>R
∇ϕ(|x|)= −(4R(|0x−|−RR11)2)|xx| if R1 <|x|≤ 1R0+2R1 0
 −(4R(R0−0−R|1x)|2)|xx| if R0+2R1 <|x|≤R0
and, recalling that ∆ϕ(|x|)=ϕ(cid:48)(cid:48)(|x|)+(d−1)ϕ(cid:48)(|x|)/|x|,

0 if 0≤|x|≤R or if |x|>R
∆ϕ(|x|)= −(R0−4R1)2 − d|−x|1(4R(|0x−|−RR11)2) if R1 <|x|≤ 1R0+2R1 0
 −(R0−4R1)2 − d|−x|1(4R(R0−0−R|1x)|2) if R0+2R1 <|x|≤R0
As a consequence we easily obtain the bounds (2.8).
Corollary 2.3 (Quantitative Caccioppoli Estimates) Let δ ≥ 0. Let Ω ⊂ Rd be a bounded do-
main, and let p ≥ 0 and λ > 0. Let u be a local positive weak solution in Ω to −∆u = λup. For any
B ⊂B ⊂⊂Ω we have
R R0 ˆ ˆ
(2.10) λ up dx+ (cid:12)(cid:12)∇log(u+δ)(cid:12)(cid:12)2dx≤ 8ωdR0d
u+δ (R −R)2
BR BR 0
where ω denotes the volume of the unit ball in Rd.
d
5
Proof. We use (2.2), using the test function ϕ of Lemma 2.2 with R replacing R :
1
ˆ ˆ ˆ ˆ
up (cid:12) (cid:12)2 up (cid:12) (cid:12)2
λ dx+ (cid:12)∇log(u+δ)(cid:12) dx≤λ ϕdx+ ϕ(cid:12)∇log(u+δ)(cid:12) dx
u+δ u+δ
BR BR ˆ Ω Ω
|∇ϕ|2 8|supp(ϕ)| 8ω Rd
≤ dx≤ = d 0 .
ϕ (R −R)2 (R −R)2
Ω 0 0
Note that the case δ >0 follows immediately from the case δ =0 since u≥0.
Remark. Letting δ =0 in the Caccioppoli estimates (2.10) shows that
ˆ
8ω Rd
(2.11) λ up−1dx≤ d 0
(R −R)2
BR 0
When p > 1 this yields a local absolute upper bound for the local Lp−1-norm, a fact that will allow to
conclude an absolute local L∞-bound in the range 1<p<p :=d/(d−2), as we shall see in Section 6.
c
This absolute upper bound represents a novelty both because it is quantitative and because it is local:
to our knowledge this is the first absolute local bound for elliptic equations. When p=1 such absolute
bound is easily seen to be impossible, while in the case 0 < p < 1 we get an absolute lower bound for
the local Lp−1-integral, which is new, at least as far as we know. It will be used below.
2.1 More general nonlinearities
As long as we deal with local estimates, we can apply the method to a larger class of operators and
nonlinearities. (i) First of all, namely we can treat local solutions of:
(2.12) −∇·A(x,u,∇u)=λup,
where A is a Carath´eodory function such that
ν |ξ|2 ≤A(x,u,ξ)·ξ ≤ν |ξ|2 and |A(x,u,ξ)|2 ≤ν |ξ|2
1 2 2
for suitable constants 0<ν <ν . The proofs of the inequalities are the same, and the results contain
1 2
ν (resp. ν ) depending on whether you consider subsolutions (resp. supersolutions).
1 2
(ii) Second we can consider supersolutions of the problem
(2.13) −∇·A(x,u,∇u)=f(x,u),
as long as f(u)≥a up with a >0, since they are supersolutions of −∇·A(x,u,∇u)=a up.
0 0 0
(iii) We can consider subsolutions of (2.13) with f(u) ≤ a (u+b )p, and a ,b ≥ 0. Then we can
1 1 1 1
obtain an estimate for v =u+b .
1
The only thing that changes a bit are the energy estimates, and it is not so difficult to keep track of
the new constants throughout the proof. We have decided here to consider the model case, to simplify
the presentation and to focus on the main ideas.
(iv) Other semilinear problems of this type are treated in the literature. Thus, Ambrosetti and Prodi’s
book [2] discusses right-hand sides of the form f(x,u)=λu+c(u)+h(x), with a∈R, c(·)∈C0(R)∩
L∞(R) and h ∈ C0,α(Ω), for some α ∈ (0,1). Such nonlinearities can be treated with the methods
presented here as well. We refrain from dealing with it in this work.
3 Local Upper Bounds
Thissectionisdevotedtotheproofoftheupperboundsandwewillprovidetwokindsofestimates. We
prove local upper bounds for nonnegative subsolutions, then by Kato’s inequality it is easy to extend
such results to solutions with any sign.
6
3.1 Local upper bounds I. The upper Moser iteration
The local upper bounds follow from the local Sobolev imbedding theorem on balls B ⊂Rd
R
(cid:18) (cid:19)
1
(3.1) (cid:107)f(cid:107)2 ≤S2 (cid:107)∇f(cid:107)2 + (cid:107)f(cid:107)2
L2∗(BR) 2 L2(BR) R2 L2(BR)
where S = S (B ) is the best constant and 2∗ = 2d/(d−2). We are requiring hereafter without any
2 2 1
further comment that d ≥ 3. The Sobolev inequality combines with the energy inequalities of Lemma
2.1 which can be considered as local reverse Sobolev (or Poincar´e) inequalities. The proof of the local
upper bounds goes though the celebrated Moser iteration. We adopt the notation (cid:107)f(cid:107) =(cid:107)f(cid:107) ,
ffl ´ Lq(BR) q,R
we recall that |B | = ω Rd and that f(x)dx = f(x)dx/|X|. Throughout this section we are
R d X X
considering nonnegative subsolutions u to −∆u=λup, unless otherwise explicitly stated.
Theorem 3.1 (Local Upper Estimates) Let Ω ⊆ Rd and let λ > 0. (i) Let u ≥ 0 be a local weak
subsolution to −∆u = λup in Ω, with 1 < p < p = 2∗ − 1 = (d + 2)/(d − 2). Then, for any
s
q >q :=d(p−1) /2 and for any B ⊂B ⊆Ω, the following bound holds true
+ R∞ R0
(cid:32) (cid:33)1+(p−1)µ (cid:32) (cid:33)−µ
q
(3.2) (cid:107)u(cid:107) ≤I uqdx up−1dx
L∞(BR∞) ∞,q
BR0 BR∞
where µ = d/(2q−d(p−1)) = d/2(q−q), and the constant I > 0 depends on d,p,q,R ,R , but
∞,q 0 ∞
not on λ.
(ii) For 0≤p≤1 the estimate simplifies into
(cid:32) (cid:33)1/q
(3.3) (cid:107)u(cid:107) ≤I uqdx .
L∞(BR∞) ∞,q
BR0
valid for all q >0. I >0 has the same dependence as before, and it also depends on λ when p=1,
∞,q
but not otherwise.
Exponents of the
local upper estimates.
Remarks on the result. (i) Inequality (3.2) is a kind of reverse H¨older inequality, indeed we can
rewrite it as:
(3.4) (cid:107)u(cid:107)µ(p−1) (cid:107)u(cid:107) ≤C (cid:107)u(cid:107)1+µ(p−1).
Lp−1(BR∞) L∞(BR∞) Lq(BR0)
7
Written in this form, it is clear from H¨older’s inequality that a constant which makes (3.4) true for a
q >q, make the same inequality true also for all q(cid:48) >q. The same applies to (3.3).
(ii) The linear case p=1 is well known, cf. [16, 19, 20].
Remarks on the constant. (i) The proof below allows to find the following expression for the
constant:
  d
c S2ω2d(p(p−−11)+) 2q−d(p−1)+ (cid:40)(cid:18) d (cid:19)d 2(d−2)
I∞,q = 1(12 −dρ)2  d−2 (cid:0)√d−√d−2(cid:1)2 ×
(3.5)
(cid:20) (cid:26) (cid:27)(cid:21)(cid:27) d
× Λ + d−2 +(1−ρ)2max d−2|dq−(d−2)|, 1 2q−d(p−1)+
p q (dq)2 4
where ρ=R /R <1 and we have used the convention x /x=0 when x=0 and, moreover, we have
∞ 0 +
set Λ =2 if p(cid:54)=1, Λ =λ/4 if p=1, with
p p

(d−2)q if q > d
 (d−2)q−d d−2
(3.6) c1 := im=a0,x1(cid:12)(cid:12)(cid:12)(d(−dd−2d)2k)0k−0−1+1+i(cid:104)iq(cid:104)−q−d(dp(−p2−12)1+)+(cid:105)+(cid:105)+(p(−p−1)1+)+d−2d−22−21(cid:12)(cid:12)(cid:12) if 0<q < d−d2.
(iii) When q also satisfies 0<q <d/(d−2), we will require in the proof the additional condition
log2∗−d(p−1)+ log2∗−d(p−1)+
(3.7) l2oqg−d(dp−1)+ is not an integer, and we let k0 =i.p. l2oqg−d(dp−1)+  ,
d−2 d−2
(i.p. is the integer part of a real number). Notice that taking q =p+1>d(p−1)/2 is possible if and
only if p<p =(d+2)/(d−2).
s
(iv) Of course, condition (3.7) is not essential, in view of the remark after formula (3.4). In fact, let
q > d(p−1)+ be such that that A(q) := log2∗−d(p−1)+/log d is an integer. Take qˆ∈ (d(p−1) /2,q)
2 2q−d(p−1)+ d−2 +
such that A(qˆ) is not an integer. Then (3.2) is valid with qˆinstead of q.
Proof. We are going to use the energy identity (2.1) for any α>−1, α(cid:54)=0, in the form (2.3) valid for
subsolution, to prove Lq −L∞ local estimates via Moser iteration, keeping track of all the constants.
We divide the proof in several steps.
•Step 1. LetuasinLemma2.1andϕthetestfunctionofLemma2.2,whichissupportedinB and
R0
such that ϕ ≡ 1 on B . The local Sobolev inequality (3.1) on the ball B applied to f = u(α+1)/2,
R1 R1
together with the energy inequality (2.3) (we can take δ =0 as in (2.6)), gives
(3.8)
(cid:34)ˆ (cid:35) 2 (cid:32)ˆ ˆ (cid:33)
u22∗(α+1)dx 2∗ ≤S22 (cid:12)(cid:12)∇uα+21(cid:12)(cid:12)2dx+ R12 uα+1dx
BR1 (cid:32)ˆBR1 1 BˆR1 (cid:33)
≤S22 (cid:12)(cid:12)∇uα+21(cid:12)(cid:12)2ϕdx+ R12 uα+1dx
(cid:32) BR0 ˆ 1 BR1 ˆ ˆ (cid:33)
λ(α+1)2 α+1 1
=S2 up+αϕdx+ uα+1∆ϕdx+ uα+1dx
2 4|α| 4|α| R2
(cid:32)λ(α+1)2 ˆBR0 (cid:20)(α+1)(cid:107)B∆R0ϕ(cid:107) 1 (cid:21)ˆ 1 BR1 (cid:33)
≤S2 up+αdx+ ∞ + uα+1dx
2 4|α| 4|α| R2
(cid:32)λ(α+1)2 ˆBR0 (cid:20) d(α+1) 1 1(cid:21)ˆ BR0 (cid:33)
≤S2 up+αdx+ + uα+1dx
2 4|α| |α|(R −R )2 R2
BR0 0 1 1 BR0
8
in the last step we have used the inequality (cid:107)∆ϕ(cid:107) ≤4d/(R −R )2 of Lemma 2.2.
∞ 0 1
• Step 2. Caccioppoli estimates and the first iteration step. Now we need to split two cases, namely
0 ≤ p ≤ 1 and 1 < p < p , and in both cases we will use the Caccioppoli estimate (2.10) with δ = 0
s
which holds for any p>0 and reads
(cid:107)u(cid:107)p−1 8
(3.9) λ p−1,R∞ ≤ .
|B | (R −R )2
R0 0 ∞
Superlinear case: 1<p<p . We continue estimate (3.8) as follows:
s
(cid:34)ˆ (cid:35) 2
2∗
u22∗(α+1)dx
BR1 ´
≤S2(cid:32)λ(α+1)2 +(cid:20) d(α+1) + 1 (cid:21)´BR0 uα+1dx(cid:33)ˆ up+αdx
2 4|α| |α|(R −R )2 R2 up+αdx
(cid:32)λ(α+1)2 (cid:20) d0(α+11) 1 1 (cid:21)BR0 |B | (cid:33)BˆR0
≤ S2 + + ´ R0 up+αdx
(a) 2 4|α| |α|(R −R )2 R2 up−1dx
(3.10) S2|B | (cid:32)λ(α+1)2(cid:107)u(cid:107)p0−1 1 (cid:20) d1(α+BR10) 1 (cid:21)(cid:33)BˆR0
= 2 R0 p−1,R0 + + up+αdx
(cid:107)u(cid:107)pp−−11,R0 4|α| |BR0| |α|(R0−R1)2 R12ˆ BR0
S2|B | (cid:18) 2(α+1)2 (cid:20) d(α+1) 1 (cid:21)(cid:19)
≤ 2 R0 + + up+αdx
(b) (cid:107)u(cid:107)pp−−11,R0 |α|(R0−R1)2 |α|(R0−R1)2 R12 BR0 ˆ
= S22|BR0| (cid:20) 1 (cid:0)2(α+1)2+d(α+1)(cid:1)+ (R0−R1)2(cid:21) up+αdx
(R −R )2(cid:107)u(cid:107)p−1 |α| R2
0 1 p−1,R0 1 BR0
where in (a) we have used the convexity in the variable r > 0 of the function N(r) = log(cid:107)u(cid:107)r, the
r
incremental quotient is increasing, hence choosing α+1≥α>0, we obtain
N(p−1+α)−N(α) N(α+p)−N(α+1) (cid:107)u(cid:107)p−1+α (cid:107)u(cid:107)α+p
≤ that is p−1+α ≤ α+p
p−1 p−1 (cid:107)u(cid:107)α (cid:107)u(cid:107)α+1
α α+1
Then we have
(cid:107)u(cid:107)α+p (cid:107)u(cid:107)p−1+α (cid:107)u(cid:107)α (cid:107)u(cid:107)p−1
(cid:107)u(cid:107)αα++p1 ≥ (cid:107)up−(cid:107)α1+α = (cid:107)up−(cid:107)α1+α(cid:107)u(cid:107)pp−−11+α ≥|BR0|−α+(pp−−11) |BR0|α+p−p−11−1(cid:107)u(cid:107)pp−−11 = |Bp−|1
α+1 α α R0
since by H¨older inequality:
(cid:107)u(cid:107)(cid:107)up−(cid:107)1+α ≥|BR|−α+(pp−−11) and (cid:107)u(cid:107)p−1+α ≥|BR|α+1p−1−p−11(cid:107)u(cid:107)p−1.
α
In (b) we have used the Caccioppoli estimate (3.9).
Sublinear case: 0 ≤ p ≤ 1. We first assume 0 ≤ p < 1, we discuss the case p = 1 separately. We
continue estimate (3.8) as follows:
(3.11)
´
(cid:34)ˆ u22∗(α+1)dx(cid:35)22∗ ≤S22(cid:32)λ(α4|+α|1)2´BR0 uupα++α1ddxx + |α|(dR(α+−1R) )2 + R12(cid:33)ˆ uα+1dx
BR1 BR0 0 1 ˆ 1 BR0
(cid:18) 2(α+1)2 d(α+1) 1 (cid:19)
≤S2 + + uα+1dx
2 |α|(R −R )2 |α|(R −R )2 R2
0 ∞ 0 1 1 BR0 ˆ
= S22 (cid:20) 1 (cid:0)2(α+1)2+d(α+1)(cid:1)+ (R0−R1)2(cid:21) uα+1dx
(R −R )2 |α| R2
0 ∞ 1 BR0
9
whichfollowsbytheconvexityinthevariabler >0ofthefunctionN(r)=log(cid:107)u(cid:107)r,whichimpliesthat
r
the incremental quotient is increasing, hence choosing α+1≥α:=β >0, we obtain
0
N(p−1+α)−N(α) N(α+p)−N(α+1) (cid:107)u(cid:107)p−1+α (cid:107)u(cid:107)α+p
≤ that is p−1+α ≤ α+p
p−1 p−1 (cid:107)u(cid:107)α (cid:107)u(cid:107)α+1
α α+1
hence
´
´BR0 up+αdx = (cid:107)u(cid:107)αα++pp ≤ (cid:107)u(cid:107)αα−−((11−−pp)) ≤ |BR0|1−αp ≤ (cid:107)u(cid:107)pp−−11,R0 ≤ 8
BR0 uα+1dx (cid:107)u(cid:107)αα++11 (cid:107)u(cid:107)αα (cid:107)u(cid:107)1α−p |BR0| λ(R0−R∞)2
again by H¨older inequalities, we just stress on the last step in which we have used that
(cid:107)u(cid:107)p−1,R0 ≤ (cid:107)u(cid:107)α hence |BR0|1−αp ≤ (cid:107)u(cid:107)pp−−11,R0 ≤ 8
|BR0|p−11 |BR0|α1 (cid:107)u(cid:107)1α−p |BR0| λ(R0−R∞)2
which is true since p−1<0<α, and in the last step we have used the Caccioppoli estimate (3.9).
Notice that when p=1, we obtain directly that
(cid:34)ˆ u22∗(α+1)dx(cid:35)22∗ ≤S22(cid:18)λ(α4|+α|1)2 + |α|(dR(α+−1R) )2 + R12(cid:19)ˆ uα+1dx
(3.12) BR1 0 1 ˆ1 BR0
S2 (cid:20) 1 (cid:18)λ (cid:19) (R −R )2(cid:21)
= 2 (α+1)2+d(α+1) + 0 1 uα+1dx
(R −R )2 |α| 4 R2
0 ∞ 1 BR0
The first iteration step. We can write the first iteration step for all p ≥ 0 in the following way: let
β = α+1 ≥ β > 0 and recall that we are requiring β (cid:54)= 1 as well, then inequalities (3.10) and (3.11)
0
can be written as
(cid:34)ˆ (cid:35) 2 ˆ
2∗
(3.13) u22∗βdx ≤I(p,β,R1,R0) uβ+(p−1)+dx
BR1 BR0
where
S2 |B | (cid:20)Λ β2+dβ (R −R )2(cid:21)
(3.14) I(p,β,R ,R )= 2 ´ R0 p + 0 1
1 0 (R0−R1)2 BR0 u(p−1)+dx |β−1| R12
where Λ =2 if p(cid:54)=1 and Λ =λ/4 if p=1.
p p
• Step 3. The Moser iteration. Let us define the sequence of exponents β >0 so that
n
2∗ 2∗
β +(p−1) = β that is β = β −(p−1)
n + 2 n−1 n 2 n−1 +
it turns out that, for any given β and all n≥1:
0
(3.15)
(cid:20)2∗(cid:21)n (cid:34) n(cid:88)−1(cid:18)2∗(cid:19)k−n(cid:35) (cid:20)2∗(cid:21)n  (cid:88)n (cid:18) 2 (cid:19)j
βn = 2 β0−(p−1)+ 2 = 2 β0−(p−1)+ 2∗ 
k=0 j=1
(cid:20)2∗(cid:21)n (cid:20) d−2(cid:18) (cid:18) 2 (cid:19)n(cid:19)(cid:21) (cid:20)2∗(cid:21)n (cid:20) d−2(cid:21) d−2
= β −(p−1) 1− = β −(p−1) +(p−1)
2 0 + 2 2∗ 2 0 + 2 + 2
since (cid:80)k sj =(1−sk)s/(1−s). Moreover we have that for all p≥1,
j=1
(cid:18)2∗(cid:19)−n d−2
β −−−−→β − (p−1) .
2 n n→∞ 0 2 +
10