New York Journal of Mathematics
New York J. Math. 6 (2000) 95{106.
An Eigenvalue Problem for Elliptic Systems
Marco Squassina
Abstract. By means of non-smooth critical point theory we prove existence
of weak solutions for a general nonlinear elliptic eigenvalue problem under
natural growth conditions .
Contents
1. Introduction
2. Recollections from non-smooth critical point theory
3. The Palais{Smale condition
4. Final remarks
References
1.
95
98
100
104
106
Introduction
Let be a bounded and open subset of R and N 1 . Existence and multiplicity results for quasilinear eigenvalue problems of the type :
n
8 P
P P
><;
(a (x u) ki ) + 21
j
>: k ==11 : : : N (u ) 2 M R=1 =1
n
ij
@
ij
@x
n
@u
@x
ij
N
h
on the submanifold of H01 ( R )
M
N
= u 2 H01 ( R ) :
N
@aij
@ uk
(x u)
@ uh @ uh
@ xi @xj
=
@G
@ uk
(x u) in Z
G(x u) dx = 1 have been rstly studied in 1983 by M. Struwe 15] and recently by G. Arioli 1]
via techniques of non-smooth critical point theory.
The goal of this paper is to study the following more general eigenvalue problem
(1) ;div(r L(x u ru)) + r L(x u ru) = r G(x u) (u ) 2 M R s
s
Received February 23, 2000.
Mathematics Subject Classication. 35J50, 47J10, 58E05.
Key words and phrases. Nonlinear eigenvalue problems, Non-smooth critical point theory.
95
ISSN 1076-9803/00
Marco Squassina
96
on the submanifold of W01 ( R )
p
N
Z
M = u 2 W01 ( R ) :
(2)
p
N
G(x u) dx = 1 :
1
We shall consider functionals f : W0 ( R ) ! R dened by
p
N
Z
f (u) = L(x u ru) dx with 1 < p < n . In general, f is not even locally Lipschitzian unless L does not
depend on u or n = 1, so that classical critical point theory fails.
(3)
To overcome this diculty, we shall use the non-smooth critical point theory
developed in 8, 9, 10, 11, 12] and also the subdierential for continuous functions
recently introduced in 6].
We shall prove that problem (1) admits a nontrivial weak solution in M R by
restricting f to M and looking for constrained critical points .
We assume that M 6= , that L : R R ! R is measurable in x for
all (s ) 2 R R of class C 1 in (s ) for a.e. x 2 and L(x s ) is strictly
convex. Moreover, we shall assume that:
L1 ] There exist > 0 such that for each " > 0 there is a 2 L1() and b 2 R
with
(4)
j j L(x s ) a (x) + "jsj + b j j
for a.e. x 2 and for all (s ) 2 R R , where p denotes the critical
Sobolev's exponent.
L2 ] There exists b 2 R such that for each " > 0 there exists a 2 L1 () with
(5)
jr L(x s )j a (x) + "jsj + bj j
for a.e. x 2 and for all (s ) 2 R R . Moreover there is a1 2 L 0 ()
with
p
jr L(x s )j a1 (x) + bjsj p0 + bj j ;1
(6)
for a.e. x 2 and for all (s ) 2 R R .
L3 ] For a.e. x 2 and for all (s ) 2 R R
(7)
r L(x s ) s 0 :
N
N
nN
nN
"
p
p
"
N
"
p
"
nN
"
s
p
p
"
N
nN
p
p
N
nN
N
nN
s
L4 ] If N > 1, there exists a bounded Lipschitz function : R ! R such that
(8)
r L(x s ) exp (r s) + r L(x s ) r exp (r s ) 0
for a.e. x 2 , for all 2 R , 2 f;1 1g and r s 2 R where
(exp (r s)) := exp (
(r ) ; (s ))]
and
r exp (r s )] := ; exp (
(r ) ; (s ))]
0 (s )
for each h = 1 : : : N and i = 1 : : : n .
s
nN
h
hi
N
h
h
h
N
h
h
h
h
h
h
i
An Eigenvalue Problem for Elliptic Systems
97
G1 ] G(x s) is measurable in x and of class C 1 in s with G(x 0) = 0npa.e. in . If
g(x s) denotes r G(x s) for every " > 0 there exists a 2 L n p; p () such
that
(9)
jg(x s)j a (x) + "jsj ;1
s
(
"
1)+
p
"
for a.e. x 2 and all s 2 R .
G2 ] For a.e. x 2 and for each s 6= 0 we have g(x s) s > 0 .
Under the preceding assumptions, the following is our main result.
Theorem 1. The eigenvalue problem
(10)
;div(r L(x u ru)) + r L(x u ru) = g(x u) (u ) 2 M R has at least one nontrivial weak solution (u ) 2 M R .
In the vectorial case (N > 1), to my knowledge, problem (1) has only been
considered in 15] and in 1] in the particular case
N
s
L(x s ) = 21
(11)
X X
n
ij
N
=1
hk
=1
a (x s) hk
ij
h
i
k
j
for coecients a : R ! R of the type a (x s) = (x s).
In 15, Theorem 3.2] the statement is essentially of perturbative nature, since it
says that if for each k 2 N there exists a % > 0 with
(12)
jr (x s)j < % for a:e: x 2 for all s 2 R then the problem has at least k distinct weak solutions:
(u ) 2 H01 ( R ) R ` = 1 : : : k :
In other words, the less the coecients (x s) vary in s, the more solutions we
get.
In 1] a new technical condition is introduced to be compared with (8). It is
assumed that there exist K > 0 and an increasing bounded Lipschitz function from 0 +1 to 0 +1 with (0) = 0, 0 non-increasing, (s) ! K as s ! +1
and such that
hk
ij
N
2
n
hk
ij
hk
ij
k
s
ij
N
k
`
N
`
ij
(13)
X X @
2e;4 0 (jsj) X (x s) (
x
s
)
=1 =1 @s
=1
n
N
n
ij
i
ij
K
j
ij
k
k
i
j
ij
for a.e. x 2 , for all 2 R and for all r s 2 R .
The proof itself of 2, Lemma 6.1] shows that this condition implies assumption
(8) in the case of integrands L like (11). On the other hand, if N 2, the two
conditions look quite similar. However, our condition (8) seems to be preferable,
because when N = 1 and L is given by (11), it reduces to the inequality
n
N
X @ (x s) 2
0(s) X (x s) =1 @s
=1
n
n
ij
i
ij
j
ij
ij
i
j
Marco Squassina
98
which is not so restrictive in view of the ellipticity of , while (13) is in this
case much stronger. For a general Lagrangian L, in the case N = 1, condition (8)
reduces to
jD L(x s )j 0 (s)r L(x s ) for a.e. x 2 and all (s ) 2 R R . This assumption has already been considered
in literature in jumping problems (see e.g. 7]) .
In Remarks 2 and 3 we will show examples of L not of the form (11) and satisfying
(8). Finally, we point out that (13) and (8) are not easily comparable to (12).
ij
s
n
2.
Recollections from non-smooth critical point theory
In this section, we want to recall the relationship between weak solutions to (1)
and constrained critical points of f to M: Let a0 2 L1 (), b0 2 R, a1 2 L1 ()
and b1 2 L1 () be such that for a.e. x 2 and for all (s ) 2 R R
np
(14)
jL(x s )j a0 (x) + b0 jsj n;p
+ b0 j j N
nN
loc
loc
p
np
jr L(x s )j a1 (x) + b1 (x)jsj n;p
+ b1 (x)j j (15)
p
s
np
jr L(x s )j a1 (x) + b1 (x)jsj n;p
+ b1 (x)j j :
Conditions (15) and (16) imply that for every u 2 W01 ( R ) we have
r L(x u ru) 2 L1 ( R ) r L(x u ru) 2 L1 ( R ) :
Therefore for every u 2 W01 ( R ) we have
;div (r L(x u ru)) + r L(x u ru) 2 D0 ( R ) :
(16)
p
p
nN
p
N
s
loc
N
loc
N
N
s
We shall now recall two denitions from 6], where a new notion of subdierential
for continuous functionals on normed spaces has been recently introduced by M.
Degiovanni and I. Campa.
Denition 1. Let X be a real normed space and C X . For each u 2 C , we
denote with T (u) the set of all v 2 X such that for each " > 0, there exist > 0
and
: (B (u ) \ C ) ]0 ] ! B (v ")
continuous with
+ t ( t) 2 C
when 2 B (u ) \ C and t 2]0 . T (u) is said the cone tangent to C at u.
Denition 2. For each u 2 X set
@f (u) := f 2 X : ( ;1) 2 Nepi (u f (u))g where
Nepi (u) := f 2 X : h vi 0 for all v 2 Tepi (u)g :
@f (u) is said to be the subdierential of f at u.
C
C
f
f
f
An Eigenvalue Problem for Elliptic Systems
99
Via @f (u) we shall connect critical points for functionals of calculus of variations
(3) constrained to M , with weak solutions to the related eigenvalue problem. Dene
the submanifold of W01 ( R )
n
o
M = u 2 W01 ( R ) : (u) = 1 p
N
p
N
where : W01 ( R ) ! W ;1 0 ( R ) is of class C 1 M 6= 0 62 M and
moreover r(u) 6= 0 for each u 2 M .
We now recall the fundamental denition of weak slope (see, 8, 9, 10, 11, 12]) .
Denition 3. Let (X d) be a metric space, f : X ! R a continuous function and
u 2 X . We denote by jdf j(u) the supremum of 2 0 +1 such that there exist
> 0 and a continuous map
H : B (u) 0 ] ;! X
such that for all (v t) 2 B (u) 0 ]
d(H(v t) v) t f (H(v t)) f (v) ; t:
We say that the extended real number jdf j(u) is the weak slope of f at u.
It is easy to prove that the map fu 7! jdf j(u)g is lower semicontinuous .
Denition 4. Let (X d) be a metric space, f : X ! R a continuous function and
u 2 X . We say that u is a critical point of f if jdf j(u) = 0 .
Denition 5. Let (X d) be a metric space, c 2 R and f : X ! R a continuous
function. A sequence (u ) X is said to be a Palais{Smale sequence at level c
((PS ) {sequence, in short) for f , if f (u ) ! c and jdf j(u ) ! 0 . We say that
f satises the Palais{Smale condition ((PS ) in short) at level c if every (PS ) {
sequence admits a convergent subsequence.
We now come to the case when X = W01 ( R ) and f : W01 ( R ) ! R
given by (3) . From (14) it follows that f is well dened and continuous.
is metric space endowed with the metric of W01 ( R ) the weak slope
dfSince
(u)Mand
the (PS ) {condition for fjM may of course be dened .
jM
Theorem 2. For every u 2 W01 ( R ) there exists 2 R such that
df (u) sup rf (u)(v) ; r(u)(v) : v 2 C 1 ( R ) kvk 1 :
1
jM
In particular, for each (PS ) {sequence (u ) for fjM there exists ( ) R such that
lim sup rf (u )(v) ; r(u )(v) : v 2 C 1 ( R ) kvk1 1 = 0:
p
N
p
N
h
c
h
h
c
c
p
p
N
p
N
N
c
p
N
N
p
c
c
h
h
h
h
h
h
N
c
p
Proof. By conditions (15) and (16), for every u 2 M and v 2 C 1( R ) there
exists
Z
Z
c
N
f 0 (u)(v) = r L(x u ru) rv dx + r L(x u ru) v dx
and the function fu 7! f 0 (u)(v)g is continuous from M into R. Now, let us extend
fjM to the functional f : W01 ( R ) ! R f+1g given by
(
f (u) = f (u) if u 2 M
(17)
+1 if u 62 M:
s
p
N
Marco Squassina
100
We may assume that jdfjM j(u) < +1: Consequently jdf j(u) = jdfjM j(u) so that
by 6, Theorem 4.13] there exists ! 2 @f (u) with jdf j(u) k!k;1 0 : Moreover,
by 6, Corollary 5.4] we have
@f (u) @f (u) + Rr(u) :
Finally, by 6, Theorem 6.1], we get @f (u) = fg where
p
Z
Z
h vi = r L(x u ru) rv dx + r L(x u ru) v dx = rf (u)(v)
for each v 2 C 1 ( R ) and the proof is complete.
By the preceding result, each critical point u 2 W01 ( R ) of fjM is a weak
s
N
c
p
N
solution to the eigenvalue problem :
rf (u) = r(u) (u ) 2 M R :
3.
The Palais{Smale condition
Recall rst a very useful conseguence of Brezis{Browder's Theorem 5].
Proposition 1. Let T 2 L1loc( R ) \ W ;1 0 ( R ), v 2 W01 ( R ) and 2
L1() with T v . Then T v 2 L1 () and
N
p
hT vi =
Proof. Argue as in 13, Lemma 3] .
Z
p
N
N
T v dx
As a consequence of assumption L1 ] and convexity of L(x s ), for each " > 0 there
exists a 2 L1() such that
(18)
r L(x s ) j j ; a (x) ; "jsj
for a.e. x 2 and for all (s ) 2 R R .
We now come to one of the main results of this paper, i.e., the local compactness
property for (PS ) {sequences.
Theorem 3. Let (u ) be a bounded sequence in W01 ( R ) and set
"
p
N
p
"
nN
c
Z
Z
h
hw v i =
(19)
h
p
N
r L(x u ru ) rv dx + r L(x u ru ) v dx
h
h
s
h
h
for all v 2 C 1 ( R ). If (w ) is strongly convergent to some w in W ;1 0 ( R )
then (u ) admits a strongly convergent subsequence in W01 ( R ).
Proof. Since (u ) is bounded in W01 ( R ) we nd a u in W01 ( R ) such
that, up to subsequences,
ru * ru in L ( R ) u ! u in L ( R ) u (x) ! u(x) a:e: x 2 :
By 4, Theorem 2.1], up to a subsequence, we have
ru (x) ! ru(x) a:e: x 2 :
Therefore, by (6) we get
r L(x u ru ) * r L(x u ru) in L 0 ( R ):
N
c
p
h
p
N
h
p
h
p
N
p
h
p
N
h
N
h
h
h
h
p
N
nN
N
An Eigenvalue Problem for Elliptic Systems
We now want to prove that we have
Z
hw ui =
(20)
101
Z
r L(x u ru) ru dx + r L(x u ru) u dx:
s
Let be as in L4 ] and test equation (19) with the following functions
v = '(1 expf1 (
(u1 ) ; (u1 ))g : : : expf (
(u ) ; (u ))g)
where ' 2 C 1 (), ' 0 and = 1 for all i = 1 : : : N . By direct computation
we obtain for a.e. x 2 D v = ( D ' + (
0 (u )D u ; 0 (u )D u )') exp (
(u ) ; (u ))]
for each i = 1 : : : N and j = 1 : : : n. Therefore, with the notation
expf(
(u) ; (u ))g
0 (u )ru ] := expf (
(u ) ; (u ))g
0 (u )D u
for each i = 1 : : : N and j = 1 : : : n, we get
h
hi
i
j
i
h
Z
N
N
N
h
i
c
j
N
h
j
i
h
hi
j
h ij
hi
i
i
i
i
hi
hi
hi
j
hi
r L(x u ru ) r' + 0 (u)ru'] expf(
(u) ; (u ))g dx
; hw ' expf(
(u) ; (u ))gi
Z n
+
r L(x u ru ) expf(
(u) ; (u ))g dx
h
h
h
h
h
s
h
h
o
h
; r L(x u ru ) expf(
(u) ; (u ))g
0 (u )ru ' dx = 0 :
Observe that if v = (1 ' : : : ') we have
lim hw ' expf(
(u) ; (u ))gi = hw vi:
Since u * u in W01 ( R ), we have
Z
lim r L(x u ru ) r' + 0 (u)ru'] expf(
(u) ; (u ))g dx
h
h
h
h
h
N
p
h
h
h
h
h
h
N
Z
Z
h
h
r L(x u ru) rv dx + r L(x u ru) 0 (u)ru' dx:
Note now that by assumption (8) for each h 2 N we have
r L(x u ru ) expf(
(u) ; (u ))g
; r L(x u ru ) expf(
(u) ; (u ))g
0 (u )ru 0:
=
s
h
h
h
h
h
h
h
h
Therefore, Fatou's Lemma implies that
lim sup
(Z
r L(x u ru ) expf(
(u) ; (u ))g' dx
s
h
h
h
Z
h
; r L(x u ru ) expf(
(u) ; (u ))g
0 (u )ru ' dx
Z
h
h
Z
h
h
r L(x u ru) v dx ; r L(x u ru) 0 (u)ru' dx:
s
Combining the previous inequalities we get
Z
h
Z
r L(x u ru) rv dx + r L(x u ru) v dx hw vi
)
s
Marco Squassina
102
for each v = (1 ' : : : ') with ' 2 C 1 () '
with ;v, we obtain
N
Z
(21)
0: Since we may exchange v
c
Z
r L(x u ru) rv dx + r L(x u ru) v dx = hw vi:
s
for each v = (1 ' : : : ') with ' 2 C 1 () and ' 0. Since each v 2
C 1 ( R ) is a linear combination of such functions, taking into account Proposition 1, we obtain relation (20). The nal step is to prove that (u ) goes to u in
W01 ( R ): To this aim, let us rst get the following inequality
N
c
N
c
h
p
N
(22)
lim sup
Z
h
r L(x u ru ) ru dx h
h
h
Because of (7) Fatou's Lemma yields
Z
(23)
Z
r L(x u ru) u dx lim inf
s
Z
h
r L(x u ru) ru dx :
r L(x u ru ) u dx :
s
h
h
h
Combining this fact with (20) and taking into account that
hw u i ! hw ui as h ! +1
we deduce
h
lim sup
h
Z
h
r L(x u ru ) ru dx
h
h
h
Z
= lim sup ;
Z
h
r L(x u ru ) u dx + hw u i
s
h
h
h
; r L(x u ru) u dx + hw ui
=
Z
s
h
r L(x u ru) ru dx :
In particular, again by Fatou's Lemma, we have
Z
Z
r L(x u ru) ru dx lim inf r L(x u ru ) ru dx
Z
h
h
h
h
lim sup r L(x u ru ) ru dx
that is
lim
h
Z
Z
h
h
h
h
h
h
r L(x u ru) ru dx
r L(x u ru ) ru dx =
Z
h
r L(x u ru) ru dx
which gives convergence in L1(). Therefore, by (18) we conclude that :
lim
h
Z
jru j dx =
h
p
Z
jruj dx
p
which gives convergence of (u ) to u in W01 ( R ) .
p
h
N
h
An Eigenvalue Problem for Elliptic Systems
103
Corollary 1. Let (u ) be a bounded sequence in W01 ( R ) ( ) a sequence in
R and set for all v 2 C 1 ( R )
Z
Z
h w vi = r L(x u ru ) rv dx + r L(x u ru ) v dx :
0
;
1
If (w ) converges to some w =
6 0 in W ( R ) then (u ) admits a strongly
convergent subsequence in W01 ( R ) R:
Proof. By density, we can nd 2 C 1( R ) such that
lim hw i = hw i > 0:
p
h
c
h
h
h
h
s
p
h
p
h
N
h
h
h
N
h
h
Since of course the sequence
h
N
c
Z
N
N
Z
r L(x u ru ) r dx + r L(x u ru ) dx
h
h
s
h
h
is bounded, ( ) is also bounded and the assertion follows by Theorem 3.
In the next result we prove that f satises (PS ) {condition.
Lemma 1. Let c 2 R . Then, for each (PS ) {sequence (u ) for fjM there exists
u 2 M and 2 R such that, up to subsequences, u ! u in W01 ( R ) and
! in R: In particular, we have
h
c
c
Z
h
h
p
h
Z
N
Z
r L(x u ru) rv dx + r L(x u ru) v dx = g(x u) v dx
for each v 2 C 1 ( R ):
Proof. Let (u ) be a (PS ) {sequence for fjM : Since by (4) (u ) is bounded in
W01 ( R ), up to a subsequence (u ) weakly goes to a u 2 M . Moreover,0 since by
G1 ], g is completely continuous as mapping from W01 ( R ) to W ;1 ( R ),
s
N
c
h
p
c
h
N
h
p
N
p
N
up to a further subsequence, we have
g(x u ) ! g(x u) in W ;1 0 ():
Now, by Theorem 2, there exists a sequence ( ) R with
p
h
sup
nZ
r L(x u ru ) rv dx +
h
;
h
Z
h
Z
h
r L(x u ru ) v dx
s
h
h
o
g(x u ) v dx : v 2 C 1 ( R ) kvk1 1 ! 0
N
h
p
c
as h ! +1: Hence, by applying Corollary 1 to
w = g(x u ) + ! 0 in W ;1 0 ( R )
up to subsequences (u ) converges to (u ) in W01 ( R ) R:
We may now prove the main result of this paper.
Proof of Theorem 1. By assumption G2] and G(x 0) = 0 we easily see that
0 62 M and g(x u) 6 0 for each u 2 M . Since f is bounded from below, there exists
a (PS ) {sequence (u ) for fjM at the level
c = inf
f (u) :
2
h
h
h
p
h
p
h
c
h
h
u
M
N
N
Marco Squassina
104
Indeed, let (u ) a sequence of minimizers for f in M . Of course we have f (u ) ! c.
Moreover, if jdf j(u ) 6! 0 we would nd a > 0 such that jdf j(u ) . Then by
8, Theorem 1.1.11] there exists a continuous deformation
: M 0 ] ! M
for some > 0 such that for all t 2 0 ] and h 2 N
f ((u t)) f (u ) ; t :
This easily yields the contradiction f ((u t)) < c for suciently large values of
h 2 N . Thus (u ) is a (PS ) {sequence for fjM . Lemma 1 now provides a weak
solution (u ) 2 M R to (1). Of course u 6 0 .
h
h
h
h
h
h
h
h
4.
c
Final remarks
We refer the reader to 2] for some concrete examples where the condition (8) is
fullled for an integrand L like (11) .
Remark 1. Assume that there exists R > 0 such that
jsj R =) r L(x s ) = 0
for a.e. x 2 and for all (s ) 2 R R and
X
1 r L(x s ) jsj R =) jD k L(x s )j 4eR
=1
for a.e. x 2 and for all (s ) 2 R R . Then (8) holds for a dened by
(
(24)
(s) = 41 if 0 s R
4 if s R :
Indeed, (8) is implied by the following condition:
There exist K > 0 and an increasing bounded Lipschitz function : 0 +1!
0 +1 with (0) = 0, 0 non-increasing, (t) ! K as t ! +1 and such that
s
N
nN
N
s
k
N
nN
s
R
(25)
X
N
k
=1
jD k L(x s )j e;4 0 (jsj)r L(x s ) K
s
for a.e. x 2 , for all 2 R and for all r s 2 R .
It is easy to verify that the dened in (24) satises (25) .
We now exhibit an example of L satisfying (8) and not of quadratic type.
Remark 2. Let L : R R ! R be dened by
p p
L(x s ) = 1p ( + arctan jsj2 )j j e N ( 3 + )
n
N
N
nN
p
for a.e. x 2 and for all (s ) 2 R R . By following 2, Example 9.2] it
is possible to show that there exist K > 0 and an increasing bounded Lipschitz
function : 0 +1! 0 +1 with (0) = 0, 0 non-increasing, (s) ! K as
s ! +1 given by
p 4 (3 = s
if s 2 0 3;1 4]
p4
(s) = Ne
R
3
;1 4 +1
4 + 3; = 1+ d if s 2 3
N
K
nN
3 4
=
s
1 4
4
=
An Eigenvalue Problem for Elliptic Systems
105
such that L satises (25). Therefore (8) is fullled.
Remark 3. Since condition (8) does not look very nice and it is not clear how
to describe the class of systems that satisfy this assumption, it seems natural to
look for some classes of quasilinear systems with a more particular structure but
requiring a simpler hypothesis . To this aim, consider the eigenvalue problem
8> ;
1 0
;2
< ;div .A1(u1)jru1 j . ru1. + A1(u1)jr. u1j = g1(x u)
.. ..
..
>: ; ..
1
;div A (u )jru j ;2 ru + A0 (u )jru j = g (x u)
p
p
(26)
p
N
N
N
p
N
p
N
N
N
p
N
in in :
In a variational setting, the weak solutions u = (u1 u ) of (26) are the
critical points of f j where f : W01 ( R ) ! R is given by
N
M
p
N
XZ
1
f (u ) = p
A (u )jru j dx :
N
k
k
=1 k
k
p
and M is as in (2). Consider the following assumptions (k = 1 : : : N ):
A1 ] A 2 C 1 (R) with a A a for some a a > 0 A2 ] A0 (s)s 0 for each s 2 R A3 ] there exists a bounded Lipschitz function : R ! R such that
k
k
k
k
k
k
k
(27)
A (s)e; ( ( ); ( )) A (t) A (s)e ( ( ); ( ))
for each s t 2 R with s t .
k
p t
s
k
p t
k
s
Under the previous assumptions, by exploiting the proof of Theorem 3 it is
possible to see that for system (26) assumption (8) may be replaced by (27). Indeed,
(27) immediately implies that
8k = 1 : : : N 8s 2 R : jA0 (s)j pA (s)
0 (s) :
Of course (27) looks much simpler and more understandable. In some sense,
this condition says that for each s 2 R and k xed, A (t) must remain within the
\exponential cone" determined by
k
k
k
n
o
o
n
t 7! A (s)e; ( ( ); ( )) and t 7! A (s)e ( ( ); ( ))
k
p t
s
k
p t
s
for each t s .
Remark 4. Condition (8) in only needed when N > 1, since in the case N = 1
Theorem 3 may be substituted by 14, Theorem 3.4] where no condition like (8) is
requested in order to get the compactness property of (PS ) {sequences .
Secondly, we remark that in the case N = 1 condition (7) can be assumed only
for large values of jsj, that is, there exists R > 0 such that
(28)
jsj R =) D L(x s ) s 0
c
s
for a.e. x 2 and for all (s ) 2 R R
N
nN
(see again 14, Theorem 3.4]) .
106
Marco Squassina
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Dipartimento di Matematica, Universita degli studi di Milano, Via C. Saldini 50, I
20133 Milano, Italy.
squassin@mat.unimi.it http://www.dmf.bs.unicatt.it/~squassin
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