On qualitative properties of solutions for elliptic problems with the $p$-Laplacian through domain perturbations

0.34 MB·

by  Vladimir Bobkov

On qualitative properties of solutions for elliptic problems
with the p-Laplacian through domain perturbations
Vladimir Bobkov∗
Department of Mathematics and NTIS, Faculty of Applied Sciences,
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1 University of West Bohemia, Univerzitn´ı 8, 306 14 Plzenˇ, Czech Republic
0
2
Sergei Kolonitskii†

n
a
J St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034 Russia

5
2

]
P Abstract
A We study the dependence of least nontrivial critical levels of the energy functional corre-
. sponding to the zero Dirichlet problem −∆pu = f(u) in a bounded domain Ω ⊂ RN upon
h
domain perturbations. The nonlinearity f is assumed to be superlinear and subcritical. We
t
a show that among all (generally eccentric) spherical annuli Ω least nontrivial critical levels
m attain maximums if and only if Ω is concentric. As a consequence of this fact we prove the
[ nonradiality of least energy nodal solutions whenever Ω is a ball or concentric annulus.

Keywords: p-Laplacian; superlinear nonlinearity; domain derivative; shape optimization;
1
Neharimanifold; least energy solution; nodal solution; radiality.
v
MSC2010: 35J92, 35B06, 49Q10, 35B30, 49K30, 35B51.
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0
4
1 Introduction and main results
7
0
1. Let Ω be a bounded domain in RN, N > 2, with the boundary ∂Ω of class C2,ς, ς ∈ (0,1).
0 Consider the boundary value problem
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1 −∆ u=f(u) in Ω,
p
: (D)
v ( u=0 on ∂Ω,
i
X
where∆ u:=div(|∇u|p−2∇u)isthep-Laplacian,p>1. Denotep∗ = Np ifp<N andp∗ =+∞
ar if p>Np. We will always impose the following assumptions on the nonNli−npearity f :R→R:
(A ) f ∈C1(R\{0})∩C0,γ(R) for some γ ∈(0,1).
1 loc
(A ) There exist q ∈ (p,p∗) and C > 0 such that |sf′(s)|,|f(s)| 6 C(|s|q−1 +1) for all s ∈
2
R\{0}.1
f(s) f(s)
(A ) f′(s)>(p−1) >0 for all s∈R\{0}, and limsup <λ (Ω), where
3 s |s|p−2s p
s→0
|∇u|pdx
λ (Ω):= min Ω . (1.1)
p ◦ |u|pdx
u∈Wp1(Ω)\{0}R Ω
∗E-mail: [email protected] R
†E-mail: [email protected]
1Ifp>N,thisassumptioncanberelaxed,see[31],condition(F4)andLemma5.6.
1
(A ) There exist s >0 and θ >p such that 0<θF(s)6sf(s) for all |s|>s , where
4 0 0
s
F(s):= f(t)dt.
Z0
◦
Problem (D) corresponds to the energy functional E :W1(Ω)→R defined as
p
1
E[u]= |∇u|pdx− F(u)dx.
p
ZΩ ZΩ
◦
ThefunctionalE isweaklylowersemicontinuousandbelongstoC1(W1(Ω)). Bydefinition,aweak
p
solution of (D) is a critical point of E. Moreover, any weak solution of (D) is C1,β(Ω)-smooth,
β ∈(0,1), and any constant-sign weak solution satisfies the Hopf maximum principle2.
If for some c ∈ R there exists a nontrivial critical point u of E such that E[u] = c, then c is
called a nontrivial critical level of E. We are interested in least nontrivial critical levels µ (Ω)
+
and µ (Ω) among positive and negative solutions of (D), respectively. In Appendix A below we
−
discuss that, under assumptions (A )−(A ), µ (Ω) and µ (Ω) can be defined as
1 4 + −
µ (Ω)= min E[v] and µ (Ω)= min E[v], (1.2)
+ −
v∈N(Ω),v>0 v∈N(Ω),v60
where
◦
N(Ω):={u∈W1(Ω)\{0}: E′[u]u≡ |∇u|pdx− uf(u)dx=0}
p
ZΩ ZΩ
isthe Neharimanifold. Minimizersof (1.2)existandtheyareleastenergyconstant-signsolutions
of (D). Moreover, µ (Ω)>0.
±
The first goal of the present article is to study the behavior of µ (Ω) under smooth domain
±
perturbations Ω :=Φ (Ω) driven by a family of diffeomorphisms
t t
Φ (x)=x+tR(x), R∈C1(RN,RN), |t|<δ, (1.3)
t
where δ > 0 is small enough. Let us take an arbitrary minimizer v of µ (Ω) and consider a
0 +
◦
function v (y) := v (Φ−1(y)), y ∈ Ω . It is not hard to see that v ∈ W1(Ω ) and v > 0 on Ω .
t 0 t t t p t t t
By Lemma A.1 and Remark A.9 from Appendix A, for each|t|<δ we canfind a unique constant
α(v )>0 such that α(v )v ∈N(Ω ). Consequently, µ (Ω )6E[α(v )v ]. (We alwaysassume by
t t t t + t t t
default that domains of integrationin E[α(v )v ] are Ω .) Analogous facts remainvalid if we take
t t t
any minimizer w of µ (Ω) and consider w (y):=w (Φ−1(y)), y ∈Ω .
0 − t 0 t t
We prove the following result.
Theorem 1.1. Assume that (A )−(A ) are satisfied. Then µ (Ω ) and µ (Ω ) are continuous
1 4 + t − t
at t = 0. Moreover, E[α(v )v ] and E[α(w )w ] are differentiable with respect to t ∈ (−δ,δ) and
t t t t
the following Hadamard-type formulas hold:
∂E[α(v )v ] p−1 ∂v p
t t =− 0 hR,ni dσ,
∂t p ∂n
(cid:12)t=0 Z∂Ω(cid:12) (cid:12)
∂E[α(wt)wt](cid:12)(cid:12)(cid:12) =−p−1 (cid:12)(cid:12)(cid:12)∂w0(cid:12)(cid:12)(cid:12)phR,ni dσ, (1.4)
∂t p ∂n
(cid:12)t=0 Z∂Ω(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
where n is the outward unit normal v(cid:12)(cid:12)ector to ∂Ω and h·,(cid:12)(cid:12)·i sta(cid:12)(cid:12)nds for the scalar product in RN.
Remark 1.2. It is rather counterintuitive that the domain derivative does not explicitly depend
on the weak term f.
2seeRemarksA.6andA.7inAppendixA.
2
Origins of this problematic go back to the work of Hadamard [21], where he proved that the
first eigenvalue λ (Ω ) of the zero Dirichlet Laplace operator in Ω is differentiable at t = 0 and
2 t t
deduced its expression (see (1.5) below with p = 2) which nowadays is known as the Hadamard
formula. We refer the reader, for instance, to [34, 23, 15] for the general theory of the shape
optimization and related historical remarks. The first eigenvalue (1.1) in the general case p > 1
was treated in [18] (see also [28]), and it was proved that
∂λ (Ω ) ∂ϕ p
p t =−(p−1) p hR,ni dσ, (1.5)
∂t ∂n
(cid:12)t=0 Z∂Ω(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
where ϕ is the eigenfunction assoc(cid:12)iatedwith λ (Ω) an(cid:12)d nor(cid:12)malizedsuch that kϕ k =1. At
p (cid:12) p (cid:12) (cid:12) p Lp(Ω)
the same time, in Remark 3.5 below we discuss that µ (Ω ) and µ (Ω ) are not differentiable at
+ t − t
t=0, in general.
Note that the main prototypical nonlinearity for (D) is given by f(u) = |u|q−2u, where q ∈
(p,p∗). Itcaneasilybecheckedthatassumptions(A )−(A )aresatisfied. Duetothehomogeneity
1 4
and oddness of f, the problem of finding the least critical levels µ (Ω) can be rewritten in the
±
form of the nonlinear Rayleigh quotient
|∇u|pdx
µ (Ω)= min J(u):= min Ω . (1.6)
q u∈W◦1(Ω)\{0} u∈W◦1(Ω)\{0} R |u|qdx pq
p p Ω
The minimum is achieved, and, after an appropriate normal(cid:0)iRzation, co(cid:1)rresponding minimizers
satisfy (D). These facts remain valid for all q ∈[1,p∗).
As a corollary of the proof of Theorem 1.1 we obtain the following fact.
Theorem 1.3. Let q ∈ [1,p∗). Then µ (Ω ) is continuous at t = 0. Moreover, if u is a
q t 0
minimizer of µ (Ω) normalized such that ku k = 1, and u (y) := u (Φ−1(y)), y ∈ Ω , then
q 0 Lq(Ω) t 0 t t
J(u ) is differentiable with respect to t and
t
∂J(u ) ∂u p
t =−(p−1) 0 hR,ni dσ.
∂t ∂n
(cid:12)t=0 Z∂Ω(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
ThesecondaimofourworkistouseTheorem1.1forstudyingashapeoptimizationproblemfor
µ (Ω)overaspecialclassofdomains. Namely,letΩbeanopensphericalannulusB (x)\B (y),
± R1 R0
where |x − y| < R − R . Due to the invariance of (D) upon orthogonal transformations of
1 0
coordinates, we can take x=0 and y =se , where s∈[0,R −R ) and e is the first coordinate
1 1 0 1
vector. For simplicity and to avoid ambiguity, we denote
µ˜ (s):=µ (B (0)\B (se )).
± ± R1 R0 1
Inordertoguaranteetheexistenceofminimizersofµ˜ (s)foralls∈[0,R −R )(seeAppendixA)
± 1 0
the second part of assumption (A ) must be satisfied for any annulus B (0)\B (se ), s ∈
3 R1 R0 1
[0,R −R ). Forthisendweimposethe followingadditionalassumption(seeadiscussionbelow):
1 0
f(s)
(A∗) limsup <λ (B (0)\B ((R −R )e )).
3 |s|p−2s p R1 R0 1 0 1
s→0
We consider the following question:
What value of the displacement s∈[0,R −R ) maximizes/minimizes µ˜ (s)?
1 0 ±
In the case of the eigenvalue problem (1.1) this question was addressed in several articles,
see [24, 32, 22, 26] for the linear case p = 2, and [11, 3] for general p > 1. It was proved in
[3, Theorem 1.1] that λ (s) := λ (B (0)\B (se )) is continuous and strictly decreasing on
p p R1 R0 1
[0,R −R ), which implies that λ (s) attains its maximum if and only if s = 0 and attains its
1 0 p
minimumifandonlyifs=R −R . Thesefactsjustifythechoices=R −R inassumption(A∗).
1 0 1 0 3
3
The common approach to prove the monotonicity of λ (s) is to consider a perturbation Φ
p t
which “shifts” the inner boundary ∂B (se ) along the direction e while the outer boundary
R0 1 1
∂B (0)remainsfixed. ThentheHadamardformula(1.5)allowstofindaderivativeofλ (s)with
R1 p
respect to the displacement s in terms of an integral over the inner boundary. Hence, to show
that λ′(s) < 0, one can try to compare values of the normal derivatives of the eigenfunction of
p
λ (s) on hemispheres {x ∈ ∂B (se ) : x < s} and {x ∈ ∂B (se ) : x > s}. In the linear
p R0 1 1 R0 1 1
case p=2, reflection arguments together with the strong comparison principle can be applied to
show that such values are strictly ordered, which leads to λ′(s) < 0 for any s ∈ (0,R −R ).
p 1 0
(Note that λ′(0)=0 due to symmetry reasons.) At the same time, the lack of strong comparison
p
principles for the general nonlinear case p > 1 entails the use of additional arguments. In [11],
applying an appropriate version of the weak comparison principle, it was shown that λ′(s) 6 0
p
for all s ∈(0,R −R ). The strict negativity of λ′(s) was obtained recently in [3] bypassing the
1 0 p
usage of (global) strong comparison results.
Considering the least nontrivial critical levels µ˜ (s), we follow the strategy described above.
±
For this end, we employ two symmetrization methods: polarization, cf. [8], and spherical sym-
metrization (i.e., foliated Schwarz symmetrization), cf. [25, 7]. With the help of these methods,
we use the ideas from [3], to derive the following result.
Theorem 1.4. Assume that (A ) − (A ) and (A∗) are satisfied. Then µ˜ (s) and µ˜ (s) are
1 4 3 + −
continuous and strictly decreasing on [0,R −R ).
1 0
As a corollaryof the proofof Theorem 1.4 we have the following fact which will be used later.
Proposition 1.5. Assume that (A )−(A ) are satisfied. Then µ˜ (s) and µ˜ (s) are continuous
1 4 + −
and strictly decreasing for sufficiently small s>0.
The last (but not least) aim of the present article is the investigation of symmetry properties
of least energy nodal solutions to problem (D) via the results stated above. By nodal (or sign-
changing) solution of (D) we mean a weak solution u such that u± :=max{±u,0}6≡0 in Ω. By
definition,anodalsetofuisasetZ ={x∈Ω:u(x)=0},andanyconnectedcomponentofΩ\Z
is a nodal domain of u.
Consider the nodal Nehari set
◦
M:={u∈W1(Ω): u+ ∈N(Ω), −u− ∈N(Ω)}. (1.7)
p
Evidently, M contains all nodal solutions of (D). Moreover, in Appendix A we discuss that a
least energy nodal solution of (D) can be found as a minimizer of the problem
ν = min E[u]. (1.8)
u∈M
LetΩ be aboundedradialdomaininRN, thatis,Ωis aballorconcentricannulus. The study
ofsymmetricpropertiesofleastenergynodalsolutionsto(D)insuchΩwasinitiatedin[7],where
it was shown that in the linear case p = 2 any minimizer of ν is a foliated Schwartz symmetric
function with precisely two nodal domains. Here we consider the following question:
Is it true that any least energy nodal solution of (D) in a bounded radial Ω is nonradial?
This question was first stated and answered affirmatively in [2] for the linear case p = 2. The
authorsobtainedthelowerestimateN+1ontheMorseindexofradialnodalsolutionsto(D)and
usedthe fact that the Morseindex ofany leastenergynodalsolutionof (D) in exactly 2 (see [6]).
◦
NotethattheassumptionE ∈C2(W1(Ω))isessentialfortheargumentsof[2]and[6]. Later,under
2
◦
weakerassumptionsonf whichallowEtobeonlyinC1(W1(Ω)),nonradialitywasprovedin[5]by
2
performingtheideaof[2]intermsofa“generalized”Morseindex. Nevertheless,necessitytowork
withthe linearizedproblemassociatedwith(D)atsign-changingsolutionsmakesageneralization
of the methods of [2] and [5] to the case p > 1 nonobvious. (See [9] about linearization of the
p-Laplacian). Here we give the affirmative answer on nonradiality in the general case p>1 using
different arguments based on shape optimization techniques.
4
Theorem 1.6. Let Ω be a ball or annulus and let (A )−(A ) be satisfied. Then any minimizer
1 4
of ν is nonradial and has precisely two nodal domains.
The fact that any minimizer of ν has exactly two nodal domains can be easily obtained by
generalizing arguments from [10, p. 1051] or, equivalently, [6, p. 6]. To prove nonradiality, we
argue by contradiction and apply Proposition 1.5 to the least critical levels on eccentric annuli
generated by small shifts of the nodal set of a radial nodal solution. Then, the union of least
energy constant-sign solutions on modified in such a way nodal domains defines a function from
M which energy is strictly smaller than ν.
To the best of our knowledge, the idea to use shape optimization techniques for studying
properties of nodal solutions was firstly performed in [4], where it was proved that any second
eigenfunction of the p-Laplacian on a ball cannot be radial. See also [3] about a development of
this result.
Itisworthmentioningthatourapproachhasanintrinsicsimilaritywiththemethodsof[2]and
[5]. Consideraradialnodalsolutionof (D)withknodaldomains. Itsnodalsetistheunionofk−1
concentricspheresS ,...,S insideΩ. Foranyfixedi∈{1,...,k−1}andj ∈{1,...,N},small
1 k−1
shifts of S along coordinate axis e generate a family of functions along which energy functional
i j
E strictlydecreases. Thus,intotal,wehave(k−1)N suchfamiliesgeneratedbyshifts. Moreover,
scaling each of k nodal components, we produce k additional families of functions with strictly
decreasingenergy. Therefore,wehave(k−1)N+k suchfamilies(comparewith[7,Theorem2.2]).
Without rigorousjustification, we mention that this number can be seen as a weak variant of the
Morse index of a radial nodal solution of (D) with k nodal domains.
Therestofthearticleisorganizedasfollows. InSection2,westudythedependenceofµ (Ω )
± t
on t and proveTheorems 1.1 and1.3. Section 3 is devotedto the study of the shape optimization
problem for annular domains and contains the proof of Theorem 1.4. In Section 4, we prove the
nonradiality of least energy nodal solutions to (D) stated in Theorem 1.6. Appendix A contains
auxiliary results.
2 Domain perturbations for least nontrivial critical levels
For the proof of Theorems 1.1 and 1.3 we need to prepare several auxiliary facts. Recall that
Ω =Φ (Ω) is the deformation of Ω, where the diffeomorphism Φ is given by (1.3):
t t t
Φ (x)=x+tR(x), R∈C1(RN,RN), |t|<δ,
t
and δ >0 is sufficiently small.
Noting that any weak solution of (D) in Ω belongs to C1,β(Ω) (see Remark A.6), we state
the following partial case of the generalized Pohozaev identity (see [14, Lemma 2, p. 323] with
L(x,s,ξ)= 1|ξ|p−F(s)).
p
Proposition 2.1. Let u be any weak solution of (D) in Ω. Then u satisfies
1
|∇u|pdiv(R)dx− |∇u|p−2h∇u,∇u·R′idx− F(u)div(R)dx
p
ZΩ ZΩ ZΩ
p−1 ∂u p
=− hR,nidσ, (2.1)
p ∂n
Z∂Ω(cid:12) (cid:12)
(cid:12) (cid:12)
where n is the outward unit normal vector to ∂Ω, and R′ is the Jacobi m(cid:12)(cid:12) atr(cid:12)(cid:12)ix of R.
◦ ◦
Fix now a nontrivial u ∈ W1(Ω) and let u ∈ W1(Ω ) be a function defined as u (y) :=
0 p t p t t
u (Φ−1(y)), y ∈ Ω . Although assertions of the following two lemmas can be deduced from
0 t t
general results [34, 23, 15], we give sketches of their proofs for the sake of completeness.
5
Lemma 2.2. Let φ ∈ C1(−δ,δ). Then F(φ(t)u (y))dy is differentiable with respect to t ∈
Ωt t
(−δ,δ) and
R
∂
F(φ(t)u (y))dy =φ′(0) u f(φ(0)u )dx+ F(φ(0)u )div(R)dx. (2.2)
∂t t 0 0 0
ZΩt (cid:12)t=0 ZΩ ZΩ
(cid:12)
(cid:12)
Proof. Changing variables by(cid:12)the rule y = Φ (x) and noting that dy = det dΦt dx for |t| < δ,
t dx
we obtain that
(cid:0) (cid:1)
dΦ
F(φ(t)u (y))dy = F(φ(t)u (Φ (x)))det t dx= F(φ(t)u )det(I+tR′)dx,
t t t dx 0
ZΩt ZΩ (cid:18) (cid:19) ZΩ
where R′ is the Jacobi matrix of R. This implies the differentiability of F(φ(t)u (y))dy on
Ωt t
(−δ,δ). On the other hand, from Jacobi’s formula we know that
R
∂
det(I+tR′) =Tr(R′)=div(R). (2.3)
∂t
(cid:12)t=0
(cid:12)
(cid:12)
Thus, differentiating ΩtF(φ(t)ut(y))dy by t(cid:12)at zero, we derive (2.2).
R
By the same arguments as in the proof of Lemma 2.2 we get the following fact.
Corollary 2.3. u (y)f(αu (y))dy is differentiable with respect to t∈(−δ,δ) for any α∈R.
Ωt t t
Lemma 2.4. LeRt φ ∈ C1(−δ,δ). Then |∇(φ(t)u (y))|pdy is differentiable with respect to
Ωt t
t∈(−δ,δ) and
R
∂
|∇(φ(t)u (y))|pdy =p|φ(0)|p−2φ(0)φ′(0) |∇u |pdx
∂t t 0
ZΩt (cid:12)t=0 ZΩ
(cid:12)
+|φ(0)|p |∇u0|pdiv(R(cid:12)(cid:12))dx−p|φ(0)|p |∇u0|p−2h∇u0,∇u0·R′idx. (2.4)
ZΩ ZΩ
Proof. First, after the same change of variables as in the proof of Lemma 2.2, we obtain
|∇u (y)|pdy = |∇u (Φ−1(y))·(Φ−1(y))′|pdy = |∇u ·(Φ′)−1|pdet(I +tR′)dx, (2.5)
t 0 t t 0 t
ZΩt ZΩt ZΩ
where by (Φ−1)′ and Φ′ we denoted the corresponding Jacobi matrices and used the inversion
t t
property (Φ−1(y))′ = (Φ′(x))−1. Hence, (2.5) implies the differentiability of |∇u (y)|pdy on
t t Ωt t
(−δ,δ). Note that the derivative of the inverse Jacobi matrix (Φ′)−1 is given by
t R
∂ ∂Φ′
(Φ′)−1 =−(Φ′)−1· t ·(Φ′)−1 =−R′
∂t t t ∂t t
(cid:12)t=0 (cid:12)t=0
(cid:12) (cid:12)
since Φ′t =I for t=0. Hence, di(cid:12)(cid:12)fferentiating (2.5) by t at zero(cid:12)(cid:12)and taking into account (2.3), we
obtain
∂
|∇u (y)|pdy = |∇u |pdiv(R)dx−p |∇u |p−2h∇u ,∇u ·R′i dx.
∂t t 0 0 0 0
ZΩt (cid:12)t=0 ZΩ ZΩ
(cid:12)
(cid:12)
Finally, noting that ∇(φ(t)(cid:12)ut)=φ(t)∇ut, we arrive at (2.4).
Recall the definition (1.2) of the least nontrivial critical levels of E in perturbed domains Ω :
t
µ (Ω )= min E[v] and µ (Ω )= min E[v].
+ t − t
v∈N(Ωt),v>0 v∈N(Ωt),v60
6
FromAppendixA(seeLemmaA.3andRemarkA.9)weknowthatδ >0canbechosensufficiently
small such that µ (Ω ) and µ (Ω ) possess minimizers for any |t| < δ which are constant-sign
+ t − t
C1,β(Ω )-solutions of (D) in Ω .
t t
Below in this section, we always denote by v an arbitrary minimizer of µ (Ω), that is, v ∈
0 + 0
N(Ω), v > 0 in Ω, and E[v ] = µ (Ω). As above, consider the family of nonnegative functions
0 0 +
v (y) := v (Φ−1(y)), where y ∈ Ω and |t| < δ. We do not know that v ∈ N(Ω ). However,
t 0 t t t t
for each |t| < δ Lemma A.1 yields the existence of a unique α = α(v ) such that α > 0 and
t t t
α v ∈N(Ω ).
t t t
Lemma 2.5. α ∈C1(−δ,δ) and α =1.
t 0
Proof. Define Ψ:(0,+∞)×(−δ,δ)→R by
Ψ(α,t)=αp−1 |∇v |pdy− v f(αv )dy.
t t t
ZΩt ZΩt
From Lemma 2.4 and Corollary 2.3 we see that Ψ(α,·) is differentiable on (−δ,δ) for any α > 0.
On the other hand, we know that v > 0 in Ω (see Remark A.7) and hence v > 0 in Ω . Thus,
0 t t
from(A )itfollowsthatv f(αv )isdifferentiablewithrespecttoα>0foreachx∈Ωand|t|<δ.
1 t t
Therefore, using (A ), we see that Ψ(·,t)∈C1(0,+∞) for any |t|<δ.
2
Since v ∈N(Ω), we have Ψ(1,0)=0. Moreover,in view of the first part of (A ) we have
0 3
f(v )
Ψ′ (1,0)=(p−1) |∇v |pdx− v2f′(v )dx= v2 (p−1) 0 −f′(v ) dx<0.
α 0 0 0 0 v 0
ZΩ ZΩ ZΩ (cid:18) 0 (cid:19)
Hence,takingδ >0smaller(ifnecessary),the implicitfunctiontheoremassurestheexistenceofa
differentiablefunctionα :(−δ,δ)→(0,+∞)suchthatα =1andΨ(α ,t)=0forallt∈(−δ,δ),
t 0 t
that is, α v ∈N(Ω ).
t t t
Remark 2.6. Consideranyminimizerw ofµ (Ω)anditsdeformationw =w (Φ−1(y)),y ∈Ω .
0 − t 0 t t
Then the result of Lemma 2.5 remains valid for α =α(w ) such that α w ∈N(Ω ).
t t t t t
Now we are ready to prove Theorem 1.1. We give the proof of each statement separately.
Proposition 2.7. E[α v ] is differentiable with respect to t∈(−δ,δ) and
t t
∂E[α v ] p−1 ∂v p
t t =− 0 hR,ni dσ. (2.6)
∂t p ∂n
(cid:12)t=0 Z∂Ω(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
Proof. First, E[αtvt] is differentiable(cid:12) due to Lemmas 2(cid:12).5, 2.(cid:12)2 and 2.4. Moreover,applying equali-
ties (2.2) and (2.4) with φ(t)=α and recalling that α =1, we compute
t 0
∂E[α v ] 1
t t =α′ |∇v |pdx+ |∇v |pdiv(R)dx− |∇v |p−2h∇v ,∇v ·R′idx
∂t 0 0 p 0 0 0 0
(cid:12)t=0 ZΩ ZΩ ZΩ
(cid:12)
(cid:12)(cid:12) −α′0 v0f(v0)dx− F(v0)div(R)dx.
ZΩ ZΩ
Since v ∈N(Ω), the terms containing α′ cancel out and we arrive at
0 0
∂E[α v ] 1
t t = |∇v |pdiv(R)dx− |∇v |p−2h∇v ,∇v ·R′i dx− F(v )div(R)dx.
∂t p 0 0 0 0 0
(cid:12)t=0 ZΩ ZΩ ZΩ
(cid:12)
(cid:12)
Finally, app(cid:12)lying the Pohozaev identity (2.1), we derive (2.6).
Remark 2.8. Consider any minimizer w of µ (Ω) and its deformation w = w (Φ−1(y)), y ∈
0 − t 0 t
Ω . Arguing as in Proposition 2.7, we see that E[α(w )w ] is also differentiable with respect to
t t t
t∈(−δ,δ) and satisfies the Hadamard-type formula (1.4).
7
Proposition 2.9. µ (Ω ) and µ (Ω ) are continuous at t=0.
+ t − t
Proof. We give the proof for µ (Ω ) only. The case of µ (Ω ) can be handled in much the same
+ t − t
way. Let us show that
limsupµ (Ω )6µ (Ω)6liminfµ (Ω ).
+ t + + t
t→0 t→0
Suppose, by contradiction, that the first inequality does not hold. Consider a minimizer v of
0
µ (Ω) and its deformation v (y) := v (Φ−1(y)), y ∈ Ω . We know that E[α(v )v ] > µ (Ω ).
+ t 0 t t t t + t
Moreover, E[α(v )v ] is continuous with respect to t∈(−δ,δ), see Proposition 2.7. Therefore, we
t t
get
limsupE[α(v )v ]>limsupµ (Ω )>µ (Ω)=limsupE[α(v )v ],
t t + t + t t
t→0 t→0 t→0
which is impossible.
Suppose now, contrary to our claim, that µ+(Ω)>liminfµ+(Ωt). Let {tk}k∈N be a sequence
t→0
◦
such that µ (Ω)> lim µ (Ω ), and let u ∈W1(Ω ) be a minimizer of µ (Ω ), k ∈N. We
+ k→+∞ + tk k p tk + tk
wanttoshowthat{uk}k∈N converges,uptoasubsequence,toaminimizerofµ+(Ω). Thiswillget
a contradiction. Consider a smooth (nonempty) domain Ωˆ ⊂ Ω . Extending each element
k∈N tk
ofW◦ 1(Ωˆ)outsideofΩˆ byzero,weseethatN(Ωˆ)⊂N(Ω )foranyk ∈N. Takinganyξ ∈C∞(Ωˆ),
p tk T 0
we apply Lemma A.1 to find an appropriate multiplier α > 0 such that αξ ∈ N(Ωˆ), and hence
µ (Ω )6 E[αξ] for any k ∈N. This implies that all k∇u k are uniformly bounded from
+ tk k Lp(Ωtk)
above. Indeed, using (A ), (A ), and the first part of (A ), we get
2 4 3
1 1
0< F(u )dx6C + u f(u )dx6C + u f(u )dx,
k 1 θ k k 1 θ k k
ZΩtk Z{x∈Ωtk:uk(x)>s0} ZΩtk
where C > 0 is chosen sufficiently large to be independent of k. Therefore, supposing that
1
k∇u k →+∞ as k →+∞ and recalling that u ∈N(Ω ), we obtain a contradiction:
k Lp(Ωtk) k tn
1
µ (Ω )= |∇u |pdx− F(u )dx
+ tk p k k
ZΩtk ZΩtk
1 1 1 1
> |∇u |pdx− u f(u )dx−C = − |∇u |pdx−C →+∞,
p k θ k k 1 p θ k 1
ZΩtk ZΩtk (cid:18) (cid:19)ZΩtk
since θ >p.
Consider now a bounded domain Ω ⊃ Ω . Extending each u by zero outside of Ω ,
k∈N tk k tk
◦
we get uk ∈ Wp◦1(Ω) and k∇ukkLp(Ωe) =e k∇SukkLp(Ω◦tk) for all k ∈ N. Therefore, the boundedness
of {uk}k∈N in Wp1(Ω) implies the existence of u∈Wp1(Ω) such that, up to a subsequence, uk →u
◦ e
weakly in W1(Ω) and strongly in Lq(Ω), q ∈(p,p∗). Moreover, since u >0 in Ω for all k ∈N,
p k tk
e e
we get u>0 a.e. in Ω.
e e f(s)
In Remark A.9 below we show that the second part of (A ) yields limsup < C <
e 3 |s|p−2s
s→0
λ (Ω ) for some C > 0 and all k large enough. Thus, due to the previous inequality and (A ),
p tk e2
we can find µ ∈ (0,C) and C > 0 such that |f(s)| 6 µ|s|p−1 +C |s|q−1 for all s ∈ R, where
2 2
q ∈(p,p∗). Therefoere, we get
e
q
µ p
|∇u |pdx= u f(u )dx6 |∇u |pdx+C |∇u |pdx
k k k λ (Ω ) k 3 k
ZΩtk ZΩtk p tk ZΩtk ZΩtk !
for some C > 0. If we suppose that k∇u k →0 as k → +∞, then for sufficiently large k
3 k Lp(Ωtk)
we obtain a contradiction since µ<C <λ (Ω ) and q >p. Thus, there exists C >0 such that
p tk 4
b
8
k∇u k > C for any k large enough. This implies that u f(u )dx > C and hence
k Lp(Ωtk) 4 Ωtk k k 4
u6≡0 a.e. in Ω. R
Applying the fundamental lemma of calculus ofvariations,it is not hardto see that u≡0 a.e.
in RN \Ω. Sience ∂Ω ∈ C2,γ, we conclude that u ∈ W◦ 1(Ω) (cf. [1, Theorem 5.29]), and hence by
p
the weak convergence we have
k∇ukLp(Ω) =k∇ukLp(Ωe) 6lki→m+in∞fk∇ukkLp(Ωe) =lki→m+in∞fk∇ukkLp(Ωtk).
Further, Lemma A.1 implies the existence of α(u) > 0 such that α(u)u ∈ N(Ω). Moreover,
E[αu] achieves its unique maximum with respect to α > 0 at α(u). On the other hand, since
each u ∈ N(Ω ), we deduce from Lemma A.1 that a unique point of maximum of E[αu ] with
k tk k
respect to α>0 is achieved at α=1. Therefore,
E[α(u)u]6liminfE[α(u)u ]6liminfE[u ]= lim µ (Ω )<µ (Ω).
k→+∞ k k→+∞ k k→+∞ + tk +
Thus, recalling that u>0 a.e. in Ω, we get a contradiction to the definition of µ (Ω), and hence
+
µ (Ω)6liminfµ (Ω ). This completes the proof.
+ + t
t→0
Remark 2.10. Proposition 2.9 implies that from any sequence of minimizers u of µ (Ω )
k + tk
(or µ (Ω )), k ∈ N, one can extract a subsequence which converges strongly in W1(RN) to
− tk p
a minimizer of µ (Ω) (or µ (Ω)). In view of possible nonuniqueness, the limit minimizer may
+ −
depend on a sequence {tk}k∈N.
Theorem 1.3 can be proved using the same arguments as for Theorem 1.1 (even without
normalization by α in view of homogeneity of the functional J in (1.6)).
t
3 Optimization problem in annuli
In this section we prove Theorem 1.4. Let us fix R > R > 0 and s ∈ [0,R −R ). Consider
1 0 1 0
problem (D) in the open spherical annulus B (0)\B (se ):
R1 R0 1
−∆ u=f(u) in B (0)\B (se ),
p R1 R0 1 (3.1)
( u=0 on ∂BR1(0) and ∂BR0(se1).
Recallthenotationµ˜ (s)=µ (B (0)\B (se ))fortheleastnontrivialcriticallevelsdefinedby
± ± R1 R0 1
(1.2)andconsideradiffeomorphismΦ (x)=x+tR(x),|t|<δ,withthevectorfieldR(x)=̺(x)e ,
t 1
where ̺ is a smooth function equal to zero in a neighborhood of ∂B (0) and equal to one in a
R1
neighborhood of ∂B (se ). It is not hard to see that
R0 1
Φ (B (0)\B (se ))=B (0)\B ((s+t)e ).
t R1 R0 1 R1 R0 1
Thereforeµ (Φ (B (0)\B (se )))=µ˜ (s+t). Thisfact,togetherwithProposition2.9,implies
± t R1 R0 1 ±
the first part of Theorem 1.4.
Lemma 3.1. Let (A )−(A ) be satisfied. Then µ˜ (s) and µ˜ (s) are continuous for sufficiently
1 4 + −
small s>0. If moreover (A∗) holds, then µ˜ (s) and µ˜ (s) are continuous on [0,R −R ).
3 + − 1 0
Recall that imposing (A∗) we can find minimizers of µ˜ (s) for each s ∈ [0,R − R ), see
3 ± 1 0
the discussion in Section 1. Without (A∗) we can guarantee the existence of minimizers only for
3
sufficiently small s>0.
For simplicity of expositionwe will give the proofof the second partof Theorem1.4 for µ˜ (s)
+
only. Thecaseofµ˜ (s)canbeprovedalongthesamelines. Wewillalwaysassumethat(A )−(A )
− 1 4
and (A∗) are satisfied.
3
9
Let v be anarbitrary minimizer of µ˜ (s), that is, v is a least energy positive solution of (3.1).
+
Recall that v ∈ C1,β(Ω) and satisfies the Hopf maximum principle (see Remarks A.6 and A.7
below). Definingv (y):=v(Φ−1(y)), y ∈B (0)\B ((s+t)e ),wehaveµ˜ (s+t)6E[α(v )v ],
t t R1 R0 1 + t t
where α(v ) is given by Lemma A.1. Hence, noting that α(v)=1, from Theorem 1.1 we obtain
t
µ˜ (s+t)−µ˜ (s)
Dµ˜ (s):=limsup + +
+ t
t→0+
E[α(v )v ]−E[α(v)v] ∂E[α(v )v ] p−1 ∂v p
6limsup t t = t t =− n dσ, (3.2)
t ∂t p ∂n 1
t→0+ (cid:12)t=0 Z∂BR0(se1)(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
where n1 =n1(x) is the first component of the outwa(cid:12)rd unit normal vector n(cid:12)to ∂(cid:12)BR0(se1).
Our main aim is to prove that Dµ˜ (s) < 0 for all s ∈ (0,R −R ). In combination with the
+ 1 0
continuity of µ˜ (s) (see Lemma 3.1), this will immediately imply the desired strict monotonicity
+
of µ˜ (s) on [0,R −R ).
+ 1 0
For the fixed s ∈[0,R −R ) we write Ω:= B (0)\B (se ), for simplicity. Denote by H
1 0 R1 R0 1 a
a hyperplane passing through the point se (center of the inner ball) perpendicularly to a vector
1
a6=0 which satisfies ha,e i>0. Let ρ :RN →RN be a map which reflects a point x∈RN with
1 a
respect to H , and Σ := {x∈ RN : ha,x−se i > 0}. Note that under the assumption on a we
a a 1
have ρ (Ω∩Σ )⊆{x∈Ω: ha,x−se i<0}.
a a 1
First we prove the following fact.
Lemma 3.2. Dµ˜ (s) 6 0 for all s ∈ [0,R −R ). Moreover, if Dµ˜ (s) = 0 for some s ∈
+ 1 0 +
[0,R −R ), then for any minimizer v of µ˜ (s) there exists ε >0 such that v(x)=v(ρ (x)) for
1 0 + 0 e1
all x∈∂B (se ) and ε∈(0,ε ).
R0+ε 1 0
Proof. Let v be a minimizer of µ˜ (s) for some s ∈ [0,R −R ). Extend v by zero outside of Ω
+ 1 0
and consider the following function:
min(v(x),v(ρ (x))), x∈Σ ,
V(x)= e1 e1 (3.3)
(max(v(x),v(ρe1(x))), x∈RN \Σe1.
◦
ThefunctionV isthepolarization ofv withrespecttoH ,cf.[8,7]. ItisknownthatV ∈W1(Ω),
e1 p
V >0 in Ω, and
|∇V|pdx= |∇v|pdx, Vf(V)dx= vf(v)dx, F(V)dx= F(v)dx,
ZΩ ZΩ ZΩ ZΩ ZΩ ZΩ
see [8, Corollary5.1]and [7, Lemma 2.2]. In particular, V ∈N(Ω) and E[V]=E[v], that is, V is
also a minimizer of µ˜ (s). Since (3.2) holds for an arbitrary minimizer of µ˜ (s), we arrive at
+ +
p−1 ∂V p
Dµ˜ (s)6− n dσ.
+ p ∂n 1
Z∂BR0(se1)(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Now, since V(x) = V(ρe1(x)) = 0 for all x ∈ ∂BR0(se1),(cid:12) and(cid:12) V(x) 6 V(ρe1(x)) for all x ∈ Σe1,
we get
∂V ∂V
(ρ (x))6 (x)<0 for all x∈∂B (se )∩Σ . (3.4)
∂n e1 ∂n R0 1 e1
Moreover,noting that n (x)=−n (ρ (x)) and n (x)<0 for all x∈∂B (se )∩Σ , we get
1 1 e1 1 R0 1 e1
p−1 ∂V p ∂V p
Dµ˜ (s)6− (x) − (ρ (x)) n (x)dσ 60. (3.5)
+ p ∂n ∂n e1 1
Z∂BR0(se1)∩Σe1 (cid:18)(cid:12) (cid:12) (cid:12) (cid:12) (cid:19)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
This is the desired conclusion. (cid:12) (cid:12) (cid:12) (cid:12)
10