Ground states of critical and supercritical problems of Brezis-Nirenberg type

0.23 MB·English

by  Mónica Clapp

GROUND STATES OF CRITICAL AND SUPERCRITICAL
PROBLEMS OF BREZIS-NIRENBERG TYPE
5
1 MO´NICACLAPP,ANGELAPISTOIA,ANDANDRZEJSZULKIN
0
2

n
a
Abstract. We studytheexistence of symmetricgroundstates to thesuper-
J
criticalproblem

7 −∆v=λv+|v|p−2v inΩ, v=0on∂Ω,

] inadomainoftheform
P Ω={(y,z)∈Rk+1×RN−k−1:(|y|,z)∈Θ},
A
where Θ is abounded smooth domain such that Θ⊂(0,∞)×RN−k−1, 1≤
h. k ≤ N −3, λ ∈ R, and p = 2(N−k) is the (k+1)-st critical exponent. We
N−k−2
t show that symmetric ground states exist for λ in some interval to the left of
a
eachsymmetriceigenvalue,andthatnosymmetricgroundstatesexistinsome
m
interval(−∞,λ∗)withλ∗>0ifk≥2.
[ Relatedtothisquestionistheexistenceofgroundstatestotheanisotropic

criticalproblem
1
v −div(a(x)∇u)=λb(x)u+c(x)|u|2∗−2u inΘ, u=0 on∂Θ,
9 wherea,b,carepositivecontinuousfunctionsonΘ.Wegiveaminimaxcharac-
1 terizationforthegroundstatesofthisproblem,studythegroundstateenergy
5 levelasafunctionofλ,andobtainabifurcationresultforgroundstates.
1
Key words: Supercritical elliptic problem, anisotropic critical problem,
0
groundstates,bifurcation.
.
1 2010MSC:35J61(35J20, 35J25).
0
5 1. Introduction
1
: We consider the supercritical Brezis-Nirenberg type problem
v
i −∆v =λv+|v|2∗N,k−2v in Ω,
X (℘ )
λ v =0 on ∂Ω,
r (cid:26)
a where Ω is given by
(1.1) Ω:={(y,z)∈Rk+1×RN−k−1 :(|y|,z)∈Θ}
for some bounded smooth domain Θ in RN−k such that Θ ⊂ (0,∞)×RN−k−1,
1 ≤ k ≤ N −3, λ ∈ R, and 2∗ := 2(N−k) is the so-called (k +1)-st critical
N,k N−k−2
exponent.
If k = 0 then 2∗ = 2∗ is the critical Sobolev exponent and problem (℘ )
N,0 λ
becomes
−∆v =λv+|v|2∗−2v in Θ,
(1.2)
v =0 on ∂Θ.
(cid:26)
Date:August20,2016.
M. Clapp is partially supported by CONACYT grant 237661 and PAPIIT-DGAPA-UNAM
grantIN104315(Mexico),A.PistoiaispartiallysupportedbyPRIN2009-WRJ3W7grant(Italy).
1
2 MO´NICACLAPP,ANGELAPISTOIA,ANDANDRZEJSZULKIN
AcelebratedresultbyBrezisandNirenberg[1]statesthat(1.2)hasagroundstate
v > 0 if and only if λ∈(0,λ ) and N ≥4, or if λ∈(λ ,λ ) and N =3, where λ
1 ∗ 1 ∗
is some number in (0,λ1). Moreover,they show that λ∗ = λ41 >0 if Θ is a ball. As
usual, λ denotes the m-th Dirichlet eigenvalue of −∆ in Θ.
m
Problem (1.2) has been widely investigated. Capozzi, Fortunato and Palmieri
[2] established the existence of solutions for all λ > 0 if N ≥5 and for all λ 6= λ
m
if N = 4 (see also [11, 24]). Several multiplicity results are also available, see e.g.
[3, 7, 8, 9, 25] and the references therein.
Recently, Szulkin, Weth and Willem [22] gave a minimax characterization for
the ground states of problem (1.2) when λ≥λ . They established the existence of
1
ground states for λ6=λ if N =4 and for all λ≥λ if N ≥5.
m 1
Concerning the supercritical problem (℘ ) with k ≥1, Passaseo[16, 17] showed
λ
that a nontrivial solution does not exist if λ = 0 and Θ is a ball. This statement
was extended in [5] to more general domains Θ, and to some unbounded domains
in [6]. On the other hand, existence of multiple solutions has been established in
[4, 14, 23].
This work is concerned with the existence of symmetric ground states for the
supercritical problem (℘ ) with k ≥ 1. Note that the domain Ω is invariant under
λ
the action of the group O(k +1) of linear isometries of Rk+1 on the first k +1
coordinates. A function v :Ω→R is called O(k+1)-invariant if v(gy,z)=v(y,z)
for every g ∈O(k+1), (y,z)∈Rk+1×RN−k−1. The subspace
H1(Ω)O(k+1) :={v ∈H1(Ω):v is O(k+1)-invariant}
0 0
of H01(Ω) is continuously embedded in L2∗N,k(Ω), so the energy functional Jλ :
H1(Ω)O(k+1) →R given by
0
Jλ(v):= 21 |∇v|2− λ2 v2− 21∗ |v|2∗N,k
ZΩ ZΩ ZΩ
is well defined. Its critical points are the O(k+1)-invariant solutions to problem
(℘ ). An O(k+1)-invariant (PS) -sequence for J is a sequence (v ) such that
λ τ λ k
v ∈H1(Ω)O(k+1), J (v )→τ and J′(v )→0 in H−1(Ω).
k 0 λ k λ k
We set
ℓO(k+1) :=inf{τ >0: there exists an O(k+1)-invariant (PS) -sequence for J }.
λ τ λ
This is the lowest possible energy level for a nontrivial O(k+1)-invariant solution
to problem (℘ ). An O(k+1)-invariant ground state of problem (℘ ) is a critical
λ λ
point v ∈H1(Ω)O(k+1) of J such that J (v)=ℓO(k+1). Since J does not satisfy
0 λ λ λ λ
thePalais-Smalecondition,anO(k+1)-invariantgroundstatedoesnotnecessarily
exist.
Let 0 < λ[k] < λ[k] ≤ λ[k] ≤ ··· be the O(k +1)-invariant eigenvalues of the
1 2 3
problem
−∆v =λv in Ω, v ∈H1(Ω)O(k+1),
0
counted with their multiplicity. Set λ[k] := 0. We shall prove the following result
0
for O(k+1)-invariant ground states.
Theorem 1.1. For every 1≤k ≤N −3, the following statements hold true:
(a) Problem (℘ ) does not have an O(k+1)-invariant ground state if λ≤0.
λ
SUPERCRITICAL PROBLEMS OF BREZIS-NIRENBERG TYPE 3
(b) For each m ∈ N ∪ {0}, there is a number λ[k] ∈ [λ[k],λ[k] ) with the
m,∗ m m+1
property thatproblem (℘ )has an O(k+1)-invariant groundstatefor every
λ
λ∈(λ[k] ,λ[k] ) and does not have an O(k+1)-invariant ground state for
m,∗ m+1
any λ∈[λ[k],λ[k] ).
m m,∗
(c) Let β :=max{dist(x,{0}×RN−k−1):x∈Θ}. Then,
(k−1)2 if 3k≥N,
λ[k] ≥ 4β2
0,∗  k (2∗ −1)k−2∗ if 3k≤N.
 (2∗N,kβ)2 N,k N,k
(cid:16) (cid:17)
In particular,λ[k] >0 if k ≥2.
0,∗
This last statement stands in contrastwith the case k =0 where a groundstate
to problem (1.2) exists for every λ ∈ [0,λ ) if N ≥ 4. We also show that λ[1] > 0
1 0,∗
if Θ is thin enough, see Proposition 4.4.
Asweshallsee,theO(k+1)-invariantgroundstatesofproblem(℘ )correspond
λ
to the ground states of the critical problem
(1.3) −div(a(x)∇u)=λb(x)u+c(x)|u|2∗−2u in Θ, u=0 on ∂Θ,
with 2∗ = 2n , n:=dimΘ, a(x ,...,x )=xk and a=b=c.
n−2 1 n 1
The critical problem (1.3) with general coefficients a ∈ C1(Θ), b,c ∈ C0(Θ) has
aninterestinitsown. Westudyitinsection2andgiveaminimaxcharacterization
for its ground states, similar to that in [22]. We study the properties of its ground
state energy level as a function of λ, and obtain a bifurcation result for ground
states, see Theorem 2.1.
Anisotropiccriticalproblemsoftheform(1.3)havebeenstudied,forexample,by
Egnell[10]and,morerecently,byHadijietal. [12,13]. Theyobtainedexistenceand
multiplicityresultsundersomeassumptionswhichinvolveflatnessofthecoefficient
functionsatsomelocalmaximumorminimumpointinthe interiorofΘ.Notethat
the function a(x ,...,x ) = xk attains its minimum on the boundary of Θ. This
1 n 1
produces a quite different behavior regarding the existence of ground states, as we
shall see in the following sections.
Section 2 is devoted to the study of the general anisotropic critical problem. In
section 3 we prove a nonexistence result for supercritical problems. It will be used
in Section 4 where we prove Theorem 1.1. In the last section we include some
questions and remarks.
2. Ground states of the anisotropic critical problem
In this section we consider the anisotropic Brezis-Nirenberg type problem
−div(a(x)∇u)=λb(x)u+c(x)|u|2∗−2u in Θ,
(2.1)
u=0 on ∂Θ,
(cid:26)
where Θ is a bounded smooth domain in Rn, n ≥ 3, λ ∈ R, a ∈ C1(Θ), b,c ∈
C0(Θ) are strictly positive on Θ, and 2∗ := 2n is the critical Sobolev exponent in
n−2
dimension n.
We take
1/2
(2.2) hu,vi := a(x)∇u·∇v, kuk := a(x)|∇u|2 ,
a a
ZΘ (cid:18)ZΘ (cid:19)
4 MO´NICACLAPP,ANGELAPISTOIA,ANDANDRZEJSZULKIN
to be the scalar product and the norm in H1(Θ), and
0
1/2 1/2∗
|u| := b(x)u2 , |u| := c(x)|u|2∗ ,
b,2 c,2∗
(cid:18)ZΘ (cid:19) (cid:18)ZΘ (cid:19)
to be the norms in L2(Θ) and L2∗(Θ) respectively. They are,clearly, equivalent to
the standard ones.
Let 0<λa,b <λa,b ≤λa,b ≤··· be the eigenvalues of the problem
1 2 3
−div(a(x)∇u)=λb(x)u in Θ, u=0 on ∂Θ,
counted with their multiplicity, and e ,e ,e ,... be the corresponding normalized
1 2 3
eigenfunctions, i.e. |e | =1. Set
j b,2
Z :={0}, Z :=span{e ,...,e },
0 m 1 m
Y :={w∈H1(Θ):hw,zi =0 for all z ∈Z },
m 0 a m
T :=(−∞,λa,b), and T :=[λa,b,λa,b ) if m∈N.
0 1 m m m+1
The solutions to problem (2.1) are the critical points of the functional J :
λ
H1(Θ)→R given by
0
J (u):= 1kuk2 − λ|u|2 − 1 |u|2∗ .
λ 2 a 2 b,2 2∗ c,2∗
If λ∈T we define
m
N ≡N (Θ):={u∈H1(Θ)rZ :J′(u)u=0 and J′(u)z =0 for all z ∈Z }.
λ λ 0 m λ λ m
This is a C1-submanifold of codimension m+1 in H1(Θ), cf. [22]. If λ < λa,b it
0 1
is the usual Nehari manifold, and if λ≥λa,b it is the generalized Nehari manifold,
1
introduced by Pankov in [15] and studied by Szulkin and Weth in [20, 21]. Note
that J′(z)z < 0 for all z ∈ Z r {0}. Clearly, the nontrivial critical points of
λ m
J belong to N . Moreover, they coincide with the critical points of its restriction
λ λ
J | :N →R. The proof of these facts is completely analogousto the one given
λ Nλ λ
in [22] for the autonomous case. Set
ℓ ≡ℓa,b,c :=infJ .
λ λ λ
Nλ
Following [20] one shows that, for every w ∈ Y r{0}, there exist unique t ∈
m λ,w
(0,∞) and z ∈Z such that
λ,w m
t w+z ∈N ,
λ,w λ,w λ
and that
J (t w+z )= max J (tw+z).
λ λ,w λ,w λ
t>0,z∈Zm
Let Σ :={w∈Y :kwk =1} be the unit sphere in Y . Then,
m m a m
(2.3) ℓ = inf max J (tw+z).
λ λ
w∈Σm t>0,
z∈Zm
As usual, we denote the best Sobolev constant for the embedding H1(Rn) ֒→
L2∗(Rn) by S. We set
κa,c := mx∈iΘnca(x(x))n−n222!n1Sn2,
and define
λa,b,c :=inf{λ∈T :ℓ <κa,c}.
m,∗ m λ
SUPERCRITICAL PROBLEMS OF BREZIS-NIRENBERG TYPE 5
Theorem 2.1. For every m∈N∪{0} the following statements hold true:
(a) The function λ7−→ℓ is nonincreasing in T and
λ m
0<ℓ ≤κa,c for all λ∈T .
λ m
(b) ℓ is attained on N if ℓ <κa,c.
λ λ λ
(c) The function λ7−→ℓ is continuous in T and
λ m
lim ℓ =0.
λ
λրλa,b
m+1
Hence, λa,b,c <λa,b .
m,∗ m+1
(d) ℓ is not attained if λ ∈ (−∞,λa,b,c) or λ ∈ [λa,b,λa,b,c), m ≥ 1, and is
λ 0,∗ m m,∗
attained if λ∈(λa,b,c,λa,b ).
m,∗ m+1
Remark 2.2. It follows from part (c) above that bifurcation (to the left) occurs
at each λa,b. This fact is essentially known and can be obtained by other meth-
m
ods. However, we would like to emphasize that here we show that our bifurcating
solutions are ground states.
Proof of Theorem 2.1. (a): Let λ,µ ∈T . If λ≤µ then J (u)≥J (u) for every
m λ µ
u∈H1(Θ). So ℓ ≥ℓ accordingto (2.3). This provesthatλ7−→ℓ is nonincreas-
0 λ µ λ
ing in T .
m
If λ∈T and w ∈Σ we have that
m m
1 kwk2−λ|w|2 n/2
(2.4) max J (tw+z)≥maxJ (tw)= a b,2
t>0,z∈Zm λ t>0 λ n |w|2c,2∗ !
1 1− λ n/2
≥ λm+1 .
n |w|2c,2∗ !
Using Sobolev’s inequality we conclude that there is a positive constant C such
that
max J (tw+z)≥C for all w ∈Σ .
λ m
t>0,z∈Zm
Therefore, ℓ >0.
λ
Let ϕ ∈ C∞(Rn) be a positive function such that supp(ϕ ) ⊂ B (0) and
k c k 1/k
|∇ϕ |2 → Sn/2, |ϕ |2∗ → Sn/2, where B (ξ) = {x ∈ Rn : |x−ξ| < r}. Let
k k r
ξ ∈Θ be such that
R R
a(ξ)n2 a(x)n2
=min
c(ξ)n−22 x∈Θ c(x)n−22
andchooseν ∈Rn with|ν|=1suchthatν is the inwardpointing unitnormalatξ
if ξ ∈∂Θ. Set ξ :=ξ+ 1ν and u (x):=ϕ (x−ξ ). Then u ∈H1(Θ) for k large
k k k k k k 0
6 MO´NICACLAPP,ANGELAPISTOIA,ANDANDRZEJSZULKIN
enough, and we have that
n
(2.5) maxJ (tu )= 1 kukk2a−λ|uk|2b,2 2
t>0 λ k n |uk|2c,2∗ !
n
a(x)|∇u |2−λ b(x)u2 2
= 1 B1/k(ξk) k B1/k(ξk) k
nR c(x)|u R|2∗ 2/2∗ 
 B1/k(ξk) k 
−→ n1 ca(ξ()ξ)n−n22(cid:16)2R!Sn2 =κa,c as(cid:17)k →∞. 
Hence, ℓ ≤κa,c for λ<λa,b.
λ 1
Next, we assume that λ ∈ T with m ∈ N. We fix an open subset θ of Θ such
m
that θ∩B (ξ ) = ∅ for k large enough. If z ∈ Z and z = 0 in θ then z = 0
1/k k m
in Θ, see [22, Lemma 3.3]. Hence, ( c(x)|z|2∗)1/2∗ is a norm in Z and, since
θ m
Z is finite-dimensional, this norm is equivalent to kzk . In particular, there is a
pomsitive constantA suchthat c(x)|Rz|2∗ ≥2∗Akzk2∗ foar all z ∈Z . It follows by
θ a m
convexity that, for every t>0 and every z ∈Z , we have
m
R
|tu +z|2∗ = c(x)|tu +z|2∗ + c(x)|z|2∗
k c,2∗ k
ZΘrθ Zθ
≥t2∗ c(x)u2∗ +2∗t2∗−1 c(x)u2∗−1z+2∗Akzk2∗.
k k a
ZΘ ZΘ
Therefore,
λ
(2.6) J (tu +z)≤J (tu )− |tu |2 +t (a(x)∇u ∇z−λb(x)u z)
λ k 0 k 2 k b,2 k k
ZΘ
+ 1 kzk2−λ|z|2 −t2∗−1 c(x)u2∗−1z−Akzk2∗
2 a b,2 k a
(cid:16) (cid:17) ZΘ
≤J (tu )+t (a(x)∇u ∇z−λb(x)u z)
0 k k k
ZΘ
−t2∗−1 c(x)u2∗−1z−Akzk2∗.
k a
ZΘ
Consequently,
J (tu +z)≤B(t2+tkzk +t2∗−1kzk )−C(t2∗ +kzk2∗)
λ k a a a
for some positive constants B and C. This implies that there exists R > 0 such
thatJ (tu +z)≤0 for all t≥R, z ∈Z andk largeenough. On the other hand,
λ k m
for t≤R, z ∈Z and k large enough, since ϕ ⇀0 weakly in H1(Θ), inequalities
m k 0
(2.6) and (2.5) imply that
J (tu +z)≤J (tu )+o(1)=κa,c+o(1).
λ k 0 k
This proves that ℓ ≤κa,c for λ≥λa,b and concludes the proof of statement (a).
λ 1
(b): Let I :Σ →R be the function given by
λ m
I (w):=J (t w+z ).
λ λ λ,w λ,w
SUPERCRITICAL PROBLEMS OF BREZIS-NIRENBERG TYPE 7
Then ℓ := inf I (w). It is shown in [20, 21] that I ∈ C1(Σ ,R). Since Σ
λ w∈Σm λ λ m m
is a smooth submanifold of H1(Θ), Ekeland’s variational principle yields a Palais-
0
Smale sequence (w ) for I such that I (w ) → ℓ , cf. [24, Theorem 8.5]. Set
k λ λ k λ
u := t w +z . By Corollary 2.10 in [20] or Corollary 33 in [21], (u ) is a
k λ,wk k λ,wk k
Palais-Smale sequence for J . Now, Corollary 3.2 in [4] asserts that every Palais-
λ
Smale sequence (u ) for J such that J (u ) → τ < κa,c, contains a convergent
k λ λ k
subsequence. It follows that ℓ is attained on N if ℓ <κa,c.
λ λ λ
(c): Let w ∈Σ . First, we will show that the function λ7−→I (w) is continu-
m λ
ous in T . Let µ ,µ ∈ T be such that µ →µ. A standard argument shows that
m j m j
J (tw+z)≤0foreveryj ∈Nift2+kzk2 islargeenough. Therefore,thesequences
µj a
(t ) and (z ) are bounded and, after passing to a subsequence, t →t in
µj,w µj,w µj,w 0
[0,∞) and z →z in Z . Hence,
µj,w 0 m
I (w)=J (t w+z )→J (t w+z )≤I (w).
µj µj µj,w µj,w µ 0 0 µ
If J (t w+z )<I (w) then, since
µ 0 0 µ
J (t w+z )→J (t w+z )=I (w),
µj µ,w µ,w µ µ,w µ,w µ
we would have that, for j large enough,
max J (tw+z)=J (t w+z )<J (t w+z ),
t>0,z∈Zm µj µj µj,w µj,w µj µ,w µ,w
which is a contradiction. Consequently, I (w) → I (w). This proves that λ 7−→
µj µ
I (w) is continuous in T for each w∈Σ .
λ m m
Next, we prove that the function λ 7−→ ℓ is continuous from the left in T .
λ m
Let µ ,µ ∈ T be such that µ ≤ µ and µ →µ. Since the infimum of any family
j m j j
of continuous functions is upper semicontinuous and λ7−→ℓ is nonincreasing, we
λ
have that
limsupℓ ≤ℓ ≤liminfℓ .
j→∞ µj µ j→∞ µj
This proves that λ7−→ℓ is continuous from the left in T .
λ m
Toprovethatλ7−→ℓ iscontinuousfromtherightinT wearguebycontradic-
λ m
tion. Assume there are µ ,µ∈T such that µ ≥µ, µ →µ and sup ℓ <ℓ .
j m j j j∈N µj µ
Then ℓ < κa,c and, by statement (b), there exists w ∈ Σ such that ℓ =
µj j m µj
J (t w +z ). Inequality (2.4) asserts that
µj µj,w j µj,w
1 1− µj n/2
ℓ >ℓ =J (t w +z )≥ λm+1 .
µ µj µj µj,wj j µj,wj n |wj|2c,2∗ !
Thisimplies that|w |2∗ ≥ε>0forallj ∈N. Denote theclosureofY inL2∗(Θ)
j c,2∗ m
byY . Sincedim(Z )<∞,theprojectionY ⊕Z →Y iscontinuousinL2∗(Θ).
m m m m m
Hence, there is a positive constant A such that
0
e e e
ℓ ≤J (t w +z )
µ µ µ,wj j µ,wj
= t2µ,wj(1−µ|w |2 )+ 1( z 2 −µ z 2 )− 1 t w +z 2∗
2 j b,2 2 µ,wj a µ,wj b,2 2∗ µ,wj j µ,wj c,2∗
t2 t2∗ (cid:13) t2 (cid:13) (cid:12)t2∗ (cid:12) (cid:12) (cid:12)
≤ µ,wj −A µ,wj |w |2∗ ≤(cid:13) µ,w(cid:13)j −A (cid:12)ε µ,wj(cid:12) for al(cid:12)l j ∈N. (cid:12)
2 0 2∗ j c,2∗ 2 0 2∗
8 MO´NICACLAPP,ANGELAPISTOIA,ANDANDRZEJSZULKIN
Itfollowsthat(t )isbounded. Hence,(kz k )isboundedtoo. Consequently,
µ,wj µ,wj a
2
ℓ ≤J (t w +z )=J (t w +z )+(µ−µ ) t w +z
µ µ µ,wj j µ,wj µj µ,wj j µ,wj j µ,wj j µ,wj b,2
≤Jµj(tµj,wjwj +zµj,wj)+o(1)=ℓµj +o(1)≤sj∈uNpℓµj +(cid:12)(cid:12)o(1)<ℓµ+o(1)(cid:12)(cid:12).
This is a contradiction. It follows that the function λ7−→ℓ is continuous in T .
λ m
Finally, let µ ∈T be such that µ →λ . We have that
j m j m+1
0<ℓ ≤J (t e +z )
µj µj µj,em+1 m+1 µj,em+1
= t2µj,em+1(λ −µ )+ 1( z 2 −µ z 2 )
2 m+1 j 2 µj,em+1 a j µj,em+1 b,2
1 (cid:13) 2∗ (cid:13) (cid:12) (cid:12)
− t e +z (cid:13) (cid:13) (cid:12) (cid:12)
2∗ µj,em+1 m+1 µj,em+1 c,2∗
t2 (cid:12) t2∗ (cid:12)
≤ µj,em+1(cid:12)(λ −µ )−A µj,em(cid:12)+1 |e |2∗ .
2 m+1 j 0 2∗ m+1 c,2∗
It follows that (t ) is bounded and, hence, that
µj,em+1
t2
0<ℓ ≤ µj,em+1(λ −µ )=o(1).
µj 2 m+1 j
This proves that ℓ →0 as µ →λ from the left.
µj j m+1
(d): If λ ∈ T , λ ≤ λa,b,c, and w ∈ Σ were such that ℓ = I (w) then for
m m,∗ m λ λ
µ∈(λ,λa,b,c(Θ)) we would have that
m,∗
κa,c =ℓ ≤I (w)<I (w)=ℓ ,
µ µ λ λ
contradicting (a). It follows that ℓ is not attained if λ ∈ [λa,b,λa,b,c). Statement
λ m m,∗
(b) implies that ℓ is attained if λ∈(λa,b,c,λa,b ). (cid:3)
λ m,∗ m+1
Recall that a (PS) -sequence for J is a sequence (u ) in H1(Θ) such that
τ λ k 0
J (u )→τ and J′(u )→0 in H−1(Θ). The value ℓ is characterizedas follows.
λ k λ k λ
Corollary 2.3. ℓ =inf{τ >0: there exists a (PS) -sequence for J }.
λ τ λ
Proof. The argument given in the proof of statement (b) of Theorem 2.1 shows
thatthereexistsa(PS) -sequenceforJ .Toprovethatℓ isthe smallestpositive
ℓλ λ λ
numberwiththisproperty,wearguebycontradiction. Assumethatτ <ℓ andthat
λ
thereexistsa(PS) -sequenceforJ . Thenτ <κa,c andCorollary3.2in[4]asserts
τ λ
that (u ) contains a subsequence which converges to a critical point u of J with
k λ
J (u)=τ. If τ 6=0 then u∈N and, hence, ℓ ≤τ. This is a contradiction. (cid:3)
λ λ λ
For the classical Brezis-Nirenberg problem (1.2) (where a = b = c ≡ 1) with
n ≥ 4, it is known that λa,b,c = 0 and λa,b,c = λ , the m-th Dirichlet eigenvalue
0,∗ m,∗ m
of −∆ in Θ, for all m ∈ N. Moreover, ℓλ = n1Sn2 = κa,c for every λ ≤ 0, but
ℓλ < n1Sn2 for every λ>0 if n≥5, see [1, 11, 22].
Asweshallseebelow,thisisnottrueingeneral: Fortheproblem(℘#)inSection
λ
4 which arisesfrom the supercriticalone, one has that λa,b,c >0 in most cases, see
0,∗
Propositions 4.3 and 4.4. A special feature of that problem is that the value κa,c
is attained on the boundary of Θ. A different situation was considered by Egnell
[10] and Hadiji and Yazidi [13]. They showed for example that, if a attains its
minimum at an interior point x of Θ, b = 1 = c, and a is flat enough around x ,
0 0
then λa,b,c =0 for n≥4, as in the classical Brezis-Nirenberg case.
0,∗
SUPERCRITICAL PROBLEMS OF BREZIS-NIRENBERG TYPE 9
We do not know whether, in general, λa,b,c ≥ 0. But this will be true in the
0,∗
specialcaseweareinterestedin,seeProposition4.1. Theproofusesanonexistence
result for the supercritical problem, which we discuss in the following section.
3. Nonexistence of solutions to a supercritical problem
Let Θ be a bounded smooth domain in RN−k with Θ ⊂ (0,∞)×RN−k−1 and
0≤k ≤N −3. Set
Ω:={(y,z)∈Rk+1×RN−k−1 :(|y|,z)∈Θ}
and consider the problem
−∆u=λu+|u|p−2u in Ω,
(3.1)
u=0 on ∂Ω.
(cid:26)
Passaseo [16, 17] showed that, if Θ is a ball, problem (3.1) does not have a
nontrivialsolution for λ=0 and p≥2∗ := 2(N−k). In [5] it is shown that this is
N,k N−k−2
also true for doubly starshaped domains.
Definition 3.1. Θ is doubly starshaped if there exist two numbers 0 < t < t
0 1
such that t ∈ (t ,t ) for every (t,z) ∈ Θ and Θ is strictly starshaped with respect
0 1
to ξ :=(t ,0) and to ξ :=(t ,0), i.e.
0 0 1 1
hx−ξ ,ν (x)i>0 ∀x∈∂Θr{ξ }, i=0,1,
i Θ i
where ν is the outward pointing unit normal to ∂Θ.
Θ
We denote the first Dirichlet eigenvalue of −∆ in Ω by λ (Ω).
1
Theorem 3.2. If Θ is doubly starshaped, p≥2∗ and
N,k
2(p−2∗ )
λ≤ N,k λ (Ω),
2∗ (p−2) 1
N,k
then problem (3.1) does not have a nontrivial solution.
WepointoutthatthegeometricassumptiononΘcannotbedropped. Existence
ofmultiplesolutionstoproblem(3.1)forλ=0andp=2∗ insomedomainswhere
N,k
Θ is not doubly starshaped has been established in [4, 14, 23].
The proof of Theorem 3.2 follows the ideas introduced in [5, 16, 17]. Fix τ ∈
(0,∞) and let ϕ be the solution to the problem
ϕ′(t)t+(k+1)ϕ(t)=1, t∈(0,∞),
ϕ(τ)=0.
(cid:26)
Explicitly, ϕ(t) = 1 1−(τ)k+1 . Note that ϕ is strictly increasing in (0,∞).
k+1 t
For y 6=0 we define
(cid:2) (cid:3)
(3.2) χ (y,z):=(ϕ(|y|)y,z).
τ
Lemma 3.3. The vector field χ has the following properties:
τ
(a) divχ =N −k,
τ
(b) hdχ (y,z)[ξ],ξi ≤ max{1 − kϕ(|y|),1}|ξ|2 for every y ∈ Rk+1 r {0},
τ
z ∈RN−k−1, ξ ∈RN.
Proof. See [17, Lemma 2.3] or [5, Lemma 4.2]. (cid:3)
10 MO´NICACLAPP,ANGELAPISTOIA,ANDANDRZEJSZULKIN
Proposition 3.4. Assume there exists τ ∈(0,∞) such that |y|∈(τ,∞) for every
(y,z)∈Ω and hχ ,ν i>0 a.e. on ∂Ω. If p≥2∗ and
τ Ω N,k
2(p−2∗ )
λ≤ N,k λ (Ω),
2∗ (p−2) 1
N,k
then problem (3.1) does not have a nontrivial solution.
Proof. The variational identity (4) in Pucci and Serrin’s paper [19] implies that, if
u∈C2(Ω)∩C1(Ω) is a solution of (3.1) and χ∈C1(Ω,RN), then
1 1 λ 1
(3.3) |∇u|2hχ,ν idσ = (divχ) |u|p+ u2− |∇u|2 dx
2 Ω p 2 2
Z∂Ω ZΩ (cid:20) (cid:21)
+ hdχ[∇u],∇uidx,
ZΩ
where ν is the outward pointing unit normal to ∂Ω (in the notation of [19] we
Ω
have taken F(x,u,∇u)= 1|∇u|2− 1λu2− 1|u|p, h= χ and a= 0). Let χ :=χ .
2 2 p τ
Then, by Lemma 3.3,
divχ =N −k.
τ
Moreover, since 1−kϕ(t)<1 for t ∈(τ,∞), and |y|∈(τ,∞) for every (y,z)∈Ω,
Lemma 3.3 yields
hdχ (y,z)[ξ],ξi≤|ξ|2 ∀(y,z)∈Ω, ξ ∈RN.
τ
By assumption, hχ ,ν i > 0 a.e. on ∂Ω. Therefore, if u is a nontrivial solution of
τ Ω
(3.1) we have, using (3.3), that
1 1
0<(N −k) − |∇u|2−λu2 dx+ |∇u|2dx
p 2
(cid:18) (cid:19)ZΩh i ZΩ
1 1 1 1 1
=(N −k) − + |∇u|2dx−(N −k) − λ u2dx,
p 2 N −k p 2
(cid:18) (cid:19)ZΩ (cid:18) (cid:19) ZΩ
that is,
1 1 1 1
− λ u2dx> − |∇u|2dx.
2 p 2∗ p
(cid:18) (cid:19) ZΩ N,k !ZΩ
Therefore, if p≥2∗ and
N,k
2(p−2∗ ) |∇u|2dx 2(p−2∗ )
λ≤ N,k inf Ω = N,k λ (Ω),
2∗N,k(p−2)u∈H01(Ω)R Ωu2dx 2∗N,k(p−2) 1
u6=0
R
problem (3.1) does not have a nontrivial solution in Ω, as claimed. (cid:3)
The following result was proved in [5].
Proposition 3.5. If Θ is doubly starshaped then Θ ⊂ (t ,∞) × RN−k−1 and
0
hχ ,ν i>0 a.e. on ∂Θ, with t as in Definition 3.1.
t0 Θ 0
Proof. See the proof of (4.11) in [5]. (cid:3)
Proof of Theorem 3.2. The conclusion follows immediately from Propositions 3.4
and 3.5. (cid:3)