Download On the uniqueness and simplicity of the principal eigenvalue

A TTI A CCADEMIA N AZIONALE L INCEI C LASSE S CIENZE F ISICHE M ATEMATICHE N ATURALI
R ENDICONTI L INCEI
M ATEMATICA E A PPLICAZIONI
Marcello Lucia
On the uniqueness and simplicity of the
principal eigenvalue
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche,
Matematiche e Naturali. Rendiconti Lincei. Matematica e
Applicazioni, Serie 9, Vol. 16 (2005), n.2, p. 133–142.
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Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e
Naturali. Rendiconti Lincei. Matematica e Applicazioni, Accademia Nazionale dei
Lincei, 2005.
Rend. Mat. Acc. Lincei
s. 9, v. 16:133-142 (2005)
Equazioni a derivate parziali. Ð On the uniqueness and simplicity of the principal
eigenvalue. Nota di MARCELLO LUCIA, presentata (*) dal Socio A. Ambrosetti.
ABSTRACT. Ð Given an open set V of RN (N > 2), bounded or unbounded, and a function w 2 L 2 (V) with
w 6 0 but allowed to change sign, we give a short proof that the positive principal eigenvalue of the problem
N
‡
Du ˆ lw(x)u;
u 2 D01;2 (V);
is unique and simple. We apply this result to study unbounded Palais-Smale sequences as well as to give a priori
estimates on the set of critical points of functionals of the type
Z
Z
1
jruj2 dx
G(x; u)dx;
u 2 D01;2 (V);
I(u) ˆ
2
V
V
when G has a subquadratic growth at infinity.
KEY WORDS: Principal eigenvalue; Simple eigenvalue; Capacity; Palais Smale sequence.
RIASSUNTO. Ð Sull'unicitaÁ e la semplicitaÁ dell'autovalore principale. Dato un aperto connesso V di RN
N
(N > 2), limitato o illimitato, e una funzione w 2 L 2 (V) con w‡ 6 0 cui eÁ consentito cambiare segno, si dimostra
che l'autovalore principale positivo del problema
Du ˆ lw(x)u;
u 2 D01;2 (V);
eÁ unico e semplice. Tale risultato viene applicato allo studio delle successioni di Palais-Smale illimitate ed
utilizzato per costruire stime a priori sull'insieme dei punti critici di funzionali del tipo
Z
Z
1
jruj2 dx
G(x; u)dx;
u 2 D01;2 (V);
I(u) ˆ
2
V
V
dove G ha un andamento subquadratico all'infinito.
1. INTRODUCTION
Given V an open and connected subset of RN with N > 2, we consider the Beppo1;2
1
Levi space
Z D0 (V)
1=2defined as the closure of C0 (V) with respect to the norm
kuk :ˆ
jruj2
. In this space, we consider the following linear problem
…1:1†
V
4u ˆ lw(x)u;
u 2 D1;2
0 (V);
u 6 0;
where w satisfies
…1:2†
w 2 LN=2 (V)
and
w‡ 6 0:
We note that w is allowed to change sign on a domain V which may be unbounded.
(*) Nella seduta dell'11 marzo 2005.
134
M. LUCIA
Under assumption (1.2), we shall investigate the uniqueness and simplicity of positive
principal eigenvalues in the following sense:
DEFINITION 1.1. We say that l 2 R is a «principal eigenvalue» of Problem (1.1) if
there exists u 2 D1;2
0 (V) such that u 0 a.e. in V and (l; u) solves (1.1). We say that l 2 R
is a «simple eigenvalue» of Problem (1.1) if
fu 2 D1;2
0 (V) : (l; u) solves (1:1)g [ f0g
is of dimension 1.
For w satisfying (1.2), it is known by a result of Szulkin and Willem [15] that (1.1) has
a positive principle eigenvalue L. If w 6 0, by applying the result of [15] to the weight
function w, we deduce the existence of a negative principal eigenvalue of Problem (1.1). Moreover, under some additional assumption on w, they prove that such
eigenvalue is simple. The aim of this paper is to emphasize that condition (1.2) is actually
enough to ensure uniqueness and simplicity of the positive principal eigenvalue.
The study of linear problem with Dirichlet boundary conditions and weight allowed
to change sign as well as to have singularities was to our knowledge initiated by ManesMicheletti [13]. In their work, the simplicity of the first positive eigenvalue was proved
N
for weight w 2 L p (V) with p > , w‡ 6 0. By assuming moreover that w 2 L1 (V) and
2
the domain to be sufficiently regular, a proof of the uniqueness of the positive principal
eigenvalue can be found in [8, Proposition 1.15]. Similar results have been obtained for
elliptic operators having nondivergence form by Hess-Kato [11] and Berestycki,
Nirenberg, Varadhan [3], by assuming mainly the weight w to be bounded. When
V ˆ RN , sufficient conditions on w ensuring this existence of principal eigenvalues have
been given by Brown, Cosner and Fleckinger [5], Allegretto [1], Tertikas [16]. In [6],
beside the question of existence, Brown-Stavrakakis prove uniqueness and simplicity of
the positive principal eigenvalue for weight assumed to be sufficiently regular.
The paper is organized as follows.
In Section 2, we prove that the positive principal eigenvalue of Problem (1.1) is unique
and simple when w 2 LN=2 (V) and w‡ 6 0. Though we shall use classical technics, no proof
to our knowledge has been given to handle such weights. Let us also emphasize that the same
procedure allows to handle more general weights. But to keep the discussion short we are
just considering the case of weights of class LN=2 (V). In Section 3, we apply this uniqueness
property of the principal eigenvalue to analyze the behavior of unbounded Palais-Smale
sequences of a functional having a subquadratic growth at infinity. The same approach gives
a priori estimates for the set of critical points of such functional.
2. PROOF OF THE UNIQUENESS
For Problem (1.1), the question of existence has been studied by Szulkin and Willem.
By applying their result (see Theorem 2.2, [15]) we get the following:
ON THE UNIQUENESS AND SIMPLICITY OF THE PRINCIPAL EIGENVALUE
135
PROPOSITION 2.1. Let V be an open subset of RN , w 2 L 2 (V) be such that w‡ 6 0 and
consider
Z
Z
…2:3†
jrWj2 : wW2 ˆ 1 :
L :ˆ inf
N
W2D1;2
(V)
0
V
V
D1;2
0 (V),
F 0 solving the minimizing Problem (2.3). Hence, L is a
Then, there exists F 2
positive principal eigenvalue of Problem (1.1).
N
, then the strong maximum principle
2
applies. Therefore, for such weights w, any (l; u) solving (1.1) with u 0 has the property
u > 0 (see [14]). When w 2 LN=2 (V), we cannot exclude the existence of points where u
vanishes. However, we can use a result due to Ancona [2] (see also the paper of BrezisPonce [4]), that in our setting can be stated as follows:
Let us note that if w 2 Lploc (V) with p >
THEOREM 2.2. Assume w 2 LN=2 (V). Let (l; u) be a solution of (1.1) with u 0 and
consider its «precise representative» u~. Then, fx : u~(x) ˆ 0g is of H1 -capacity zero and in
particular of Lebesgue measure zero.
It is known that any element u of a Sobolev space has a «precise representative»,
whose main property is to be quasicontinuous (see [9]). Therefore, in the sequel, it will be
implicitely assumed that we work with such special representative.
Another difficulty to overcome when Problem (1.1) is considered with a function
solutions. For example, consider the
w 2 LN=2 (V) is the possible lack of regularity of the
1
function f(x) ˆ log (jxj) on the ball V :ˆ B 0; . A straightforward calculation shows
2
that f solves:
N
Df ˆ w(x)f;
2
. One check easily that w 2 LN=2 (V), but f 62 L1 (V).
jxj2 log jxj
Nevertheless, by refining slightly some argument found in [7], combined with
Theorem 2.2, we are able to prove the uniqueness of the positive principle eigenvalue.
with w(x) :ˆ
PROPOSITION 2.3. Assume w 2 LN=2 (V) and w‡ 6 0. Let (L; F) be a solution of the
minimizing Problem (2.3) and consider a solution (l; W) of (1.1) with l 0. Then, l L.
If furthermore W 0, then l ˆ L.
PROOF. From the definition of L, we deduce easily that l L. Assume now that (l; W)
solves (1.1) with
lL
and
W 0:
Hence, we have
…2:4†
…2:5†
DF ˆLwF; F 0;
DW ˆlwW; W 0:
136
M. LUCIA
For each k 0, let us consider the function
k
if F(x) k;
Fk (x) :ˆ
F(x) if F(x) 2 [0; k):
Clearly Fk 2 L1 (V) and it is well-known that Fk 2 D1;2
0 (V) (see [10]). Given e > 0, we
F2k
F2k
which is in D1;2
, are legitimate test
consider also the function
(V).
Since
F
and
k
0
W‡e
W‡e
function in (2.4), respectively (2.5), we get
! Z
Z
Z
Z
F2k
F2
rFrFk
rWr
lwW k ;
ˆ LwFFk
W‡e
W‡e
V
V
which is equivalent to
Z jrFk j2
…2:6†
V
rWr
V
F2k
W‡e
V
Z LwFFk
ˆ
lwW
V
F2k
:
W‡e
But, a direct calculation shows that the following «Picone's identity» holds:
2
2 Fk
Fk
2
)rW :
…2:7†
rWr
jrFk j
ˆ rFk (
W‡e
W‡e
By plugging (2.7) in (2.6), we get
2 Z Z F
0 rFk ( k )rW ˆ
LwFFk
…2:8†
W‡e
V
V
lwW
F2k
:
W‡e
Since by Theorem 2.2 the set fW ˆ 0g is of measure zero, (2.8) is equivalent to
2
Z Z F2k
Fk
rF
ˆ
lwW
LwFF
0
…2:9†
rW
:
k
k
W‡e
W‡e
fW>0g
fW>0g
Now, letting e ! 0 and k ! 1 in (2.9) and applying Lebesgue dominated Theorem
to the right handside, we get
Z
…2:10†
0 (L l) wF2 :
Z
V
2
w(x)F ˆ 1, (2.10) implies l ˆ L.
Since, l L and
p
V
We conclude this section by proving that the principle eigenvalue (2.3) has to be
simple.
PROPOSITION 2.4. Let w 2 LN=2 (V) be such that w‡ 6 0. Consider L defined by (2.3)
and the eigenspace V (L) :ˆ fF 2 D1;2
0 (V) : (L; F) solves (1:1)g [ f0g: Then,
1. For any F 2 V (L) n f0g, we have
…2:11†
F > 0 a:e: or
F < 0 a:e:
137
ON THE UNIQUENESS AND SIMPLICITY OF THE PRINCIPAL EIGENVALUE
2. dim V (L) ˆ 1.
Z
PROOF. (1) Let F 2 V (L) be such that
[15]) that
wF2 ˆ 1.We note (as in Theorem 2.5 of
V
‡
…2:12†
F ˆ F ‡ jFj 2 V (L):
Indeed, since
Z
V
jrFj2 ˆ
Z
jrjFjj2 ˆ L
Z
and
V
wjFj2 ˆ
V
Z
wF2 ˆ 1;
V
F and jFj solve the minimization Problem (2.3). In particular, jFj 2 V (L) and (2.12)
follows immediately. Now, Theorem 2.2 implies that either F‡ 0, or F‡ > 0 a.e.,
which proves (2.11).
(2) We know that dim V (L) 1 (by Theorem 2.1). The proof that the dimension is
exactly 1, can be done using the idea given in Lemma 7 of [13].
Let F1 ; F2 2 V (L). By part (1), we can assume without loss of generality that
F1 ; F2 > 0 a.e. Let us consider the set
T :ˆ ft 2 R : F1 ‡ tF2 > 0 a.e.g:
We shall show that F1 ‡ t0 F2 0 when t0 ˆ inf T.
Claim 1. T 6ˆ ;, inf T > 1.
Since 0 2 T, we see that T 6ˆ ;. To prove that this set is bounded from below, let us
consider for each d; M > 0, the set
Ad;M :ˆ fF1 < M; F2 > dg:
Since F2 > 0 a.e., there exists e
d > 0 such that jfF2 > e
dgj > 0. Moreover, since
e
e such that jA~ ~ j > 0. Now, for
[M>0 A~d;M ˆ fF2 > dg, we deduce the existence of M
d;M
e
M
, we get
any x 2 A~d;M~ and t <
e
d
e ‡ te
(F1 ‡ tF2 )(x) < M
d < 0:
e
M
, the function F1 ‡ tF2 is negative on a set of positive measure,
Hence, when t <
e
d
which proves inf T > 1.
Claim 2. Setting t0 :ˆ inf T, we have F1 ‡ t0 F2 0.
From part (1) of this proposition, we have the following alternatives:
(a) F1 ‡ t0 F2 > 0;
(b) F1 ‡ t0 F2 < 0;
(c) F1 ‡ t0 F2 0:
Assume (a) holds. By setting
Ed;M :ˆ fF1 ‡ t0 F2 > d; F2 < Mg;
e > 0. Thus, for any
and arguing as in claim (1), we prove jE~d;M~ j > 0 for some e
d; M
e
d
, we get
x 2 E~d;M~ and e 2 0;
e
M
(F1 ‡ (t0
e)F2 )(x) > e
d
e > 0:
eM
138
M. LUCIA
From part (1), we deduce that F1 ‡ (t0 e)F2 > 0 a.e., in contradiction with the
definition of t0 . A similar contradiction is reached if we assume (b). Therefore, the
alternative (c) holds, which concludes the proof of the proposition.
p
3. APPLICATION TO A PROBLEM ASYMPTOTICALLY LINEAR AT INFINITY
Given V an open set of RN (bounded or unbounded), we consider the functional:
Z
Z
1
I…u† ˆ
…3:13†
jruj2 dx
G…x; u†dx; u 2 D1;2
0 (V);
2
V
V
Zs
where G(x; s) ˆ
g(x; j)dj and g is assumed to satisfy the following assumptions.
0
(H1) g: V R ! R is a CaratheÂodory function and satisfies g(x; s) ˆ 0 a.e. x 2 V,
8s 0;
(H2) There exists g 2 LN=2 (V) such that jg(x; s)j g(x)s a.e. x 2 V; 8s 0;
(H3) The function g is «asymptotically linear» at infinity in the sense:
g(x; s)
exists a.e. x 2 V:
s!1
s
g1 (x) :ˆ lim
The existence of non-trivial critical points for I under such assumptions was done for
example in [17, 12]. An important step in those papers is to study the compactness of
Palais-Smale sequences. Actually, simple examples like
2
ls
if s 0;
G(x; s) ˆ
0
if s < 0;
show that under the assumptions (H1) to (H3), the functional I can have unbounded
Palais-Smale sequences. In [17] such analysis was done for positive and bounded g on a
bounded domain, and it was noticed in [12] that changing-sign nonlinearity could also be
considered. In this section, we show how those arguments can be extended to
nonlinearity, not necessarily positive, of class LN=2 (V) where V can be bounded or
unbounded. We start with the following Lemma:
p
LEMMA 3.1. Let p > 1 and consider a sequence (an ; bn ) 2 D1;2
0 (V) L (V) such that
an * a in D1;2
0 (V) and
Assume
bn * b in Lp (V):
1
1 1
:ˆ ‡ < 1, then an bn * ab in Lq (V).
q
p 2
PROOF. We check easily that an b n is bounded in Lq (V). Therefore, an bn * v in Lq (V).
e V.
To prove the lemma, it is enough to show that v ˆ ab on each open set V
139
ON THE UNIQUENESS AND SIMPLICITY OF THE PRINCIPAL EIGENVALUE
e On the one hand, we have
Let W 2 C01 (V).
Z
Z
…3:14†
an b n W ! vW
~
V
On the other hand,
Z
…3:15†
jan b n
~
V
Z
abjjWj ~
V
Z
jan
ajjbn Wj ‡
~
V
jb n
bjjafj
~
V
Now, we note that by assumption p0 < 2 . Hence,
…3:16†
kan
akLp (V)
~ ! 0
e Lp (V):
e
af 2 L2 (V)
and
0
From (3.15) and (3.16) we deduce that
Z
Z
…3:17†
an bn W ! abW:
~
V
~
V
From (3.14) and (3.17), we get
Z
Z
vW ˆ abW;
~
V
~
8W 2 C01 (V);
~
V
~
which shows that v ˆ ab on V.
p
In the sequel, we will denote by L(g1 ) the positive principal eigenvalue of the
problem: Df ˆ lg1 f.
PROPOSITION 3.2. Let (H1) to (H3) be satisfied. Assume the existence of a sequence
fun g such that
…3:18†
n
0
kDI(un ) k ! 0 in (D1;2
0 (V))
and
kun k ! 1:
un
* w in D1;2
Then,
0 (V); moreover, w > 0 a.e. and satisfies
kun k
Dw ˆ g1 (x)w in V:
In particular L(g1 ) ˆ 1.
PROOF. With the help of Lemma 3.1, we can adapt the arguments of [17]. Let us set
un
. Since kwn k ˆ 1, we have (up to a subsequence)
wn :ˆ
kun k
…3:19†
wn * w in D1;2
0 (V);
Claim 1. w 6 0
From (3.18), we have
wn * w in L2 (V);
R
supkWkˆ1 hrun ; rWi
V
kun k
R
V
wn ! w a.e.
g(; un )W
! 0:
140
M. LUCIA
In particular, we get
R
hrun ; rwn i
V
from which we deduce
kun k
1
…3:20†
R
Z
V
V
g(; un )wn ! 0;
g(; un ) wn ! 0:
kun k
On the other hand, from (H2), we have
Z
Z
Z
g(; un ) g jun j wn ˆ gw2 :
w
…3:21†
n
n
kun k
kun k
V
V
V
D1;2
0 (V).
N=2
Now, assume by contradiction wn * 0 in
Then, wn * 0 in L2 (V) and
2
2 =2
Lemma 3.1 implies that wn * 0 in L (V) ˆ (L (V))0 . Hence, since g 2 LN=2 (V),
the right handside of (3.21) should tend to zero. So,
Z
g(; un )
wn ! 0:
…3:22†
kun k
V
Since (3.22) contradicts (3.20), we must have wn * w 6 0 in D1;2
0 (V).
Claim 2. w > 0 a.e. on V.
Knowing that
DI(un ) (W)
! 0;
kun kkWk
we deduce
R
V
Since g(x; s) ˆ 0 for s 0;
R
V
…3:23†
By setting
8 W 2 D1;2
0 (V);
R
hrun ; rWidx
kun k
hrun ; rWidx
V
R
V
g(x; un )Wdx
! 0:
g(x; u‡
n )Wdx
kun k
8
‡
>
< g(x; un (x))
u‡
g~n (x) :ˆ
n (x)
>
:
0
we can rewrite (3.23) as follows
Z
…3:24†
hrwn ; rWidx
V
Z
V
! 0:
if u‡
n (x) 6ˆ 0;
if u‡
n (x) ˆ 0;
~
gn (x)w‡
n Wdx ! 0;
8W 2 D1;2
0 (V):
ON THE UNIQUENESS AND SIMPLICITY OF THE PRINCIPAL EIGENVALUE
141
Now, from (H2) and (3.19), we respectively get
g~n * ~g in LN=2 (V)
…3:25†
and
1;2
‡
w‡
n * w in D0 (V):
Therefore, by Lemma 3.1, we deduce
~
gn w ‡
gw‡ in Lp (V)
n *~
Hence,
Z
…3:26†
g~n w‡
nW
Z
!
V
with
1 2
1
1
ˆ ‡ ˆ 0:
p N 2
(2 )
~gw‡ W
8W 2 L2 (V):
V
From (3.24) and (3.26), we have
Z
Z
~g…x†w‡ Wdx ˆ 0
…3:27†
hrw; rWidx
V
V
Z
Choosing W ˆ w in (3.27) one gets
8W 2 D1;2
0 (V):
jrw j2 dx ˆ 0. Thus, w 0 and solves
V
…3:28†
Dw ˆ ~g…x†w
in V:
Since w 6 0 (by claim 1), the version of the strong maximum given in Theorem 2.2
implies w > 0 a.e.
Claim 3. ~g ˆ g1 and Dw ˆ g1 (x)w.
Since w > 0 a.e. (by claim 2), we get un ! ‡1 a.e. in V. Using (H3) and (3.25) we then
have
g(x; un )
! g1 (x) a.e.
un
and
g(x; un )
* ~g(x) in LN=2 (V):
un
This yields ~g ˆ g1 a.e., and using (3.28), we obtain
Dw ˆ g1 (x)w;
w > 0 a.e.
The positive principal eigenvalue beeing unique (Theorem 2.3), we get L(g1 ) ˆ 1.
p
From this proposition, we derive immediately the following corollary:
COROLLARY 3.3. Assume (H1) to (H3) and L(g1 ) 6ˆ 1. Then,
1. Every Palais-Smale sequence of the functional (3.13) is bounded;
2. The set of critical points of (3.13) is bounded in D1;2
0 (V).
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