- Email: [email protected]
NEUMANN PROBLEMS OF A CLASS OF ELLIPTIC EQUATIONS WITH DOUBLY CRITICAL SOBOLEV EXPONENTS
2004,24B (4) :633-638
NEUMANN PROBLEMS OF A CLASS OF ELLIPTIC EQUATIONS WITH DOUBLY CRITICAL SOBOLEV EXPONENTS 1 Han Pigong (
#;£.~./) )
Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China E-mail: [email protected]
Abstract
This paper deals with the Neumann problem for a class of semilinear elliptic
equations -~u + u = luj2* - 2 U + JLlul q - 2U in n, g~ = luI" - 2 U on an, where 2· = ;':2' s: = 2<:_-21 ) , 1 < q < 2, N ~ 3, JL > 0" denotes the unit outward normal to boundary an. By variational method and dual fountain theorem, the existence of infinitely many solutions with negative energy is proved. Key words Neumann problem, semilinear elliptic equation, Sobolev exponent 2000 MR Subject Classification
1
(PS)~
condition, critical
35J65
Introduction
Let n be a bounded domain in R N with C 1 boundary, N ~ 3. In this paper, we study multiplicity of solutions for the following semilinear elliptic equation (1.1) where 2' = ;::'2 ' s" = 2r:~21) , 1 < q < 2, /-l > 0, "'( denotes the unit outward normal to boundary an. U E H1(n) is said to be a weak solution of (1.1) if
r(V'u. V'v + uv -luI 2*-2
Jn
UV -
/-lluI Q-2uv)dx -
r IU(-2 uvds
Jan
= 0,
"tv E H1(n),
(1.2)
(t;:; ,t
xu2 ' ••• , a~: ), V'U . V'v denotes the inner product of V'u and V'v. where V'u = Hence the nontrivial solutions of problem (1.1) are equivalent to the nonzero critical points of the functional
J(u)
r(lV'ul
= "21 J
n
1Received July 11,2002.
2
2
1
2*
+ lui - 2,l ul -
JL 1 qlulQ)dx - s'
r luiS * ds Jan
u E H1(D).
(1.3)
634
ACTA MATHEMATICA SCIENTIA
Vol.24 Ser.B
Note that both 2* and s* are critical Sobolev exponents for embeddings H 1(D) Y L Z" (D) and H 1(D) Y Y"(aD) respectively. So J(u) fails to satisfy (P.S.)c condition, and difficulty may arise from such a failture. The Dirichlet problem of the equation -~u = luIZ"-Zu + f(x, u), where f(x, u) is a lowerorder perturbation of lul z' -zu in the sense that lim fl(X;':) = 0, was studied by H. Brezis and L. lul~oo u1 Nirenbergl'", and by many other authors later. In 1991, Garcia and Peral[10] proved the existence of multiplicity of solutions with negative energy, when f(x, u) = Jllulq-Zu, 0 < Jl < Jl*, Jl* being a positive constant. F. Escobar[9] considered the best constant S(a, b) in the Sobolev inequality alulz.,R~
+ bluls.,8R~
(1.4)
~ S(a, b)lV'ulz,R~
and the attainable function 'I/J(x) = (1 + Ix'i z + IXN + x~IZ)- N:;2 , where a, b are nonnegative constants with a + b > 0, x~ is a constant depending only on a, b, N . By using Escobar's results and variational method, Y. Deng and X. Wang[8] obtained a nontrivial positive solution of Neumann problem (1.1) where Jllulq-Zu is replaced by f(x, u) which is superlinear at u = 0 and subcritical at u = 00. Lots of works have appeared on the combined effects of concave and convex nonlinearities in recent years, see for example [1, 2], in which Dirichlet boundary value problems have been studied. In this paper, we use some ideas in [8, 10] to prove that problem (1.1) has infinitely many solutions with negative energy for Jl > 0 small enough. We denote D 1 ,Z(R -f. ) = {u E LZ·(R-f.) IIV'ul E LZ(R-f.n, x' = (Xl,XZ,oo',XN_l), R-f. = RN n {XN > O} and Co, C 1 , C z , ' " denote (possibly different) positive constants. For simplicity, we will always write
Our main result is the following: Theorem 1.1 There exists Jl* > 0 such that, for every 0 < Jl < Jl*, problem (1.1) has a sequence of solutions (Uk) C H 1(D) such that J(Uk) < 0 and J(Uk) -+ 0 as k -+ 00.
2
Proof of the Main Result
The following lemma is due to T. Bartsch and M. Willem[3], which plays an important role in the proof of Theorem 1.1. Let X be a Hilbert space with a finite dimensional subspace approximation in the sense that X = Y n EB Zn, where Y" = EBj'=1 Xj, Zn = EB~n X, and Xj(j = 1,2,00') are all finite dimensional subspaces. The functional J E C 1P;, R) is said to satisfy the (P.S.)~ condition (with respect to (}~,)) if any sequence (un;) C X such that n J· -+ 00
,
Un'}
E Y~ J , J(u n 'J· )
-+ C
,
JI~ (u w] ) 1 n i
-+ 0
contains a subsequence convergent to a critical point of J. Moreover, the (P.S.)~ condition implies the (P,S')c condition. Lemma 2.1 Let J E C 1(X, R) be an even functional. If for every k ~ k o , there exists Pk
> Tk > 0 such that
635
Han: NEUMANN PROBLEMS OF A CLASS OF ELLIPTIC EQUATIONS
No.4
(A 1 ) ak := (A 2 ) bk := (A 3 ) dk :=
inf
UEZk ,lIull=Pk
max
UEYk ,llull=rk
inf
uEzdull~Pk
J(u)
~
OJ
J(u) < OJ J(u) -+ 0 as k -+ OOj
(A 4 ) J satisfies the (P.S.)~ condition for every c E [dko'O). Then J has a sequence of negative critical values convergent to O. The following lemma is from [9).
Lemma 2.2
The infimum
S:=
lV'ul~ RN
inf UEDl.2(R~)\{O} lul~, ,RN+
' +
(2.1)
+ lul 2s,, aRN+
is achieved by the function
'l/J,(x)
= C2 + Ix'I2 + ~XN + €X~12 )
N-2 ,€
-2-
> 0,
(2.2)
where x~ is a constant depending only on N.
Lemma 2.3 and c satisfying
Let S be defined by (2.1), then there exists k
1
~
-oo
N-2
> 0 such
that for any f.l
_2_
-kf.l2- q
>0
(2.3)
,
the functional J satisfies (P.S.)~ condition. Proof Consider a sequence unj C H 1 (0) such that nj
-+ 00, Un J
E
YnJ , J(u n·) -+ c, JI~nj (un·) -+ 0, J J
where c satisfies (2.3). That is, 1 2 1 2 Jro(21V'unj I + 2!u nj I -
1 2' 2* IU nj I
-
f.l q 1 qlunj I )dx - s*
Jr
ao
s'_
IUnj I ds -
c+ 0(1),
r (V'unj·V'V+unjV-IUnjI2'-2UV-f.llulq-2Unjv)dx- Jrao IUnjIS'-2Uvds = o(lIvll),
Jo
Let v 1 (-2 -
= u nj
in (2.5), we obtain from (2.4) and (2.5)
-!.) r(IV'U nj 12 + IUnj 12 )dx + (-!.s* s* Jo
1) 2 *
lu ,.1 Jr o n
2' dx - f.l( ~ -
:S C(l + Ilunj II)
q
(2.5)
(2.6)
Unj ----' U weakly in
H1(0),
unj ----' U weakly in
£2'(0),
Un; ----' U weakly in
U' (an),
Un; -+ U strongly in
Lq(n),
u nj -+ U strongly in
L 2(n),
n.
\Iv E H1(0).
-!.) rIU n.Iqdx s* Jo '
Note that 1 < q < 2 and from (2.6), it therefore follows that lIun; II :S subsequence, if necessary, we may assume that
u nj -+ U a.e. on
(2.4)
c.
By choosing a
636
ACTA MATHEMATICA SCIENTIA
It follows that
r(V'u.
U
satisfies
r
V'v + uv - luI2*-2uv - J.lluIQ-2 uv)dx -
k
ho
Vol.24 Ser.B
lul s*-2 uv ds
=0
"Iv E HI (0,),
(2.7)
and (2.8)
Set vn j
=u
n j
-
u. From [4]' we get
L
2 IU n j 1 * dx =
r IU i:
n j
(ds
=
L
2* IV n j 1 dx
r IV i;
n j
+
L
+
r lul;* ds + 0(1). i:
(ds
2* lul dx
+ 0(1),
Obviously,
Hence we obtain J(u)
2
1
+ -2 lV'vn· 12 0 J
Since (J' (unJ, un)
~
0 as
l
nj
1 2* 1 s* -2* Ivn·12* 0 - -lvn·ls* J, s* ] ,an u
~ 00,
2
= C + 0(1).
(2.9)
we deduce that s·
2*
IV'VnJ120 - IV n J 12* 0 - IV n J ·Is* , , ao 1
(2.10) = -(J'(u),u) = O.
We may therefore assume that
where b, b1 , bz are nonnegative constants satisfying b = b1 + b2 . Now we use the following inequality which can be found in [8]. Let 0, be a bounded domain in R N with C 1 boundary. Then for every
€
> 0,
where C< is a positive constant depending only on the diameter of the domain 0, and Therefore, from (2.11) we obtain
2
Hence, we derive that b 2: (S - €)(bf 2
€,
b 2: S(bf
2
+ b{).
2
+ b[).
Furthermore, due to the arbitrariness of
Therefore, b
b = b1 + b2 < (-)"2 - S· 2*
L
bi »
+ (_)"2 . S
637
Han: NEUMANN PROBLEMS OF A CLASS OF ELLIPTIC EQUATIONS
No.4
If b = 0, the proof is complete. Assume b > 0, after an elementary calculation, we obtain
b > _1_ S()1 - 2N - 2 It follows from
(2.12)
(2.8)-(2.12) that b
C
+ 4S - I)N-2.
bl b2 2* - s*
= J(u) + 2" 1
2
1
1
2'
1
1
q
2(N _ 1) (b + lIull ) + (s* - 2J(bl + /UI2' ,0) - /-l( q - s* )lulq,o
1
~
2: 2N _l (N _ l ) S(v l + 4S - 1) Now we choose k
N-2
1 2 1 1 1-£ q +2(N_l)lluI1 -/-l(q- sJlnl 2!1U!l.
> 0 by (2.13)
then obtain a contradiction. Remark 2.1 It is easy to verify that the positive constant k in (2.13) is independent of Proof of Theorem 1.1 We choose an orthonormal basis (ej) of HI (n) and define X j := Re, and consider the antipodal action of Z /2 on HI (n), then it suffices to verify assumptions of Lemma 2.1. sup lulq,o and so (3k -+ 0 as k -+ 00. In addition, lIull=l
As in [11]' we define (3k :=
UEZk,
.
4 {3q
1
we choose Pk := ( : l qk ) 2- q , and it follows that Pk -+ 0 as k -+ 00. Hence, for k 2: ko(ko is certain positive integer), u E Zk and !lull = Pk, we get J(u) 2: 0, and relation (AI) is proved. Since on the finite dimensional space Yk , all norms are equivalent, for u E Y k and !lull = rk < Pk, we have as k large enough 1
1
2
2'
J(u) :S -211ull - -2* lul2' ,0
-
/-l
q
-Iul q q ,0 + Collull
s'
< O.
Thus, relation (A 2 ) is proved. For k > k o , u E
z; ,
J(u)
Ilull:S Pb
2: 2Clilul1 2 - C2(lluI1 2' + !Iul() - /-l(3% Ilull q
q
q
2: -/-l(3% Ilull 2: -/-l(3% Pk . q
q
Since (3k -+ 0 , Pk -+ 0, as k -+ 00, relation (A 3 ) is satisfied. We choose 11* > 0 by 2" l(N_l)S()1 + 4S - I)N-2 - kl1*2: q = 0, then, by Lemma 2.3, for every 0 < 11 < 11* and c < 0, the functional J satisfies the (P.S.); condition.
638
ACTA MATHEMATICA SCIENTIA
Vol.24 Ser.B
References
2 3 4 5 6 7 8 9 10 11
Adimurthi, Pacella F, Yadava S L. On the number of positive solutions of some semilinear Dirichlet problem in a ball. Diff Integral Equats, 1999, 10: 1157-1170 Ambrosetti A, Brezis H, Cerami G. Combined effects of concave and convex nonlinearities in some elliptic problems. J Funct Anal, 1994, 122: 519-543 Bartsch T, Willem M. On an elliptic equation with concave and convex nonlinearity. Proc Amer Math Soc, 1995, 123: 3555-3561 Brezis H, Lieb E. Relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc, 1983, 88: 486-490 Brezis H, Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent. Comm Pure Appl Math, 1983,36: 437-478 Cherrier I;. Meillewres constants dans les inegalites relatives awx espaces de Sobolev. Bull Sc Math, 2< Serie, 1984, 108: 225-262 Cherrier P. Problemas de Neumann non lineaires sur lesvarietes Riemanniennes. J Funct Anal, 1984, 54: 154-206 Deng Y, Wang X. Neumann problem of elliptic equations with limit nonlinearity in boundary condition. Chinese Ann Math, 1994, 15B: 299-310 Escobar F. Uniqueness theorems of conformal deformation of metrics,Sobolev inequalities, and an eigenvalue estimate. Comm Pure Appl Math, 1990,43: 857-884 Garcia J, Peral I. Multiplicity of solutions for elliptic problems with critical exponent or with nonsymmetric term. Trans Amer Math Soc, 1991, 323: 877-895 Willem M. Minimax Theorems. Boston: Birkhauser, 1996
NEUMANN PROBLEMS OF A CLASS OF ELLIPTIC EQUATIONS WITH DOUBLY CRITICAL SOBOLEV EXPONENTS 1 Han Pigong (
#;£.~./) )
Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China E-mail: [email protected]
Abstract
This paper deals with the Neumann problem for a class of semilinear elliptic
equations -~u + u = luj2* - 2 U + JLlul q - 2U in n, g~ = luI" - 2 U on an, where 2· = ;':2' s: = 2<:_-21 ) , 1 < q < 2, N ~ 3, JL > 0" denotes the unit outward normal to boundary an. By variational method and dual fountain theorem, the existence of infinitely many solutions with negative energy is proved. Key words Neumann problem, semilinear elliptic equation, Sobolev exponent 2000 MR Subject Classification
1
(PS)~
condition, critical
35J65
Introduction
Let n be a bounded domain in R N with C 1 boundary, N ~ 3. In this paper, we study multiplicity of solutions for the following semilinear elliptic equation (1.1) where 2' = ;::'2 ' s" = 2r:~21) , 1 < q < 2, /-l > 0, "'( denotes the unit outward normal to boundary an. U E H1(n) is said to be a weak solution of (1.1) if
r(V'u. V'v + uv -luI 2*-2
Jn
UV -
/-lluI Q-2uv)dx -
r IU(-2 uvds
Jan
= 0,
"tv E H1(n),
(1.2)
(t;:; ,t
xu2 ' ••• , a~: ), V'U . V'v denotes the inner product of V'u and V'v. where V'u = Hence the nontrivial solutions of problem (1.1) are equivalent to the nonzero critical points of the functional
J(u)
r(lV'ul
= "21 J
n
1Received July 11,2002.
2
2
1
2*
+ lui - 2,l ul -
JL 1 qlulQ)dx - s'
r luiS * ds Jan
u E H1(D).
(1.3)
634
ACTA MATHEMATICA SCIENTIA
Vol.24 Ser.B
Note that both 2* and s* are critical Sobolev exponents for embeddings H 1(D) Y L Z" (D) and H 1(D) Y Y"(aD) respectively. So J(u) fails to satisfy (P.S.)c condition, and difficulty may arise from such a failture. The Dirichlet problem of the equation -~u = luIZ"-Zu + f(x, u), where f(x, u) is a lowerorder perturbation of lul z' -zu in the sense that lim fl(X;':) = 0, was studied by H. Brezis and L. lul~oo u1 Nirenbergl'", and by many other authors later. In 1991, Garcia and Peral[10] proved the existence of multiplicity of solutions with negative energy, when f(x, u) = Jllulq-Zu, 0 < Jl < Jl*, Jl* being a positive constant. F. Escobar[9] considered the best constant S(a, b) in the Sobolev inequality alulz.,R~
+ bluls.,8R~
(1.4)
~ S(a, b)lV'ulz,R~
and the attainable function 'I/J(x) = (1 + Ix'i z + IXN + x~IZ)- N:;2 , where a, b are nonnegative constants with a + b > 0, x~ is a constant depending only on a, b, N . By using Escobar's results and variational method, Y. Deng and X. Wang[8] obtained a nontrivial positive solution of Neumann problem (1.1) where Jllulq-Zu is replaced by f(x, u) which is superlinear at u = 0 and subcritical at u = 00. Lots of works have appeared on the combined effects of concave and convex nonlinearities in recent years, see for example [1, 2], in which Dirichlet boundary value problems have been studied. In this paper, we use some ideas in [8, 10] to prove that problem (1.1) has infinitely many solutions with negative energy for Jl > 0 small enough. We denote D 1 ,Z(R -f. ) = {u E LZ·(R-f.) IIV'ul E LZ(R-f.n, x' = (Xl,XZ,oo',XN_l), R-f. = RN n {XN > O} and Co, C 1 , C z , ' " denote (possibly different) positive constants. For simplicity, we will always write
Our main result is the following: Theorem 1.1 There exists Jl* > 0 such that, for every 0 < Jl < Jl*, problem (1.1) has a sequence of solutions (Uk) C H 1(D) such that J(Uk) < 0 and J(Uk) -+ 0 as k -+ 00.
2
Proof of the Main Result
The following lemma is due to T. Bartsch and M. Willem[3], which plays an important role in the proof of Theorem 1.1. Let X be a Hilbert space with a finite dimensional subspace approximation in the sense that X = Y n EB Zn, where Y" = EBj'=1 Xj, Zn = EB~n X, and Xj(j = 1,2,00') are all finite dimensional subspaces. The functional J E C 1P;, R) is said to satisfy the (P.S.)~ condition (with respect to (}~,)) if any sequence (un;) C X such that n J· -+ 00
,
Un'}
E Y~ J , J(u n 'J· )
-+ C
,
JI~ (u w] ) 1 n i
-+ 0
contains a subsequence convergent to a critical point of J. Moreover, the (P.S.)~ condition implies the (P,S')c condition. Lemma 2.1 Let J E C 1(X, R) be an even functional. If for every k ~ k o , there exists Pk
> Tk > 0 such that
635
Han: NEUMANN PROBLEMS OF A CLASS OF ELLIPTIC EQUATIONS
No.4
(A 1 ) ak := (A 2 ) bk := (A 3 ) dk :=
inf
UEZk ,lIull=Pk
max
UEYk ,llull=rk
inf
uEzdull~Pk
J(u)
~
OJ
J(u) < OJ J(u) -+ 0 as k -+ OOj
(A 4 ) J satisfies the (P.S.)~ condition for every c E [dko'O). Then J has a sequence of negative critical values convergent to O. The following lemma is from [9).
Lemma 2.2
The infimum
S:=
lV'ul~ RN
inf UEDl.2(R~)\{O} lul~, ,RN+
' +
(2.1)
+ lul 2s,, aRN+
is achieved by the function
'l/J,(x)
= C2 + Ix'I2 + ~XN + €X~12 )
N-2 ,€
-2-
> 0,
(2.2)
where x~ is a constant depending only on N.
Lemma 2.3 and c satisfying
Let S be defined by (2.1), then there exists k
1
~
-oo
N-2
> 0 such
that for any f.l
_2_
-kf.l2- q
>0
(2.3)
,
the functional J satisfies (P.S.)~ condition. Proof Consider a sequence unj C H 1 (0) such that nj
-+ 00, Un J
E
YnJ , J(u n·) -+ c, JI~nj (un·) -+ 0, J J
where c satisfies (2.3). That is, 1 2 1 2 Jro(21V'unj I + 2!u nj I -
1 2' 2* IU nj I
-
f.l q 1 qlunj I )dx - s*
Jr
ao
s'_
IUnj I ds -
c+ 0(1),
r (V'unj·V'V+unjV-IUnjI2'-2UV-f.llulq-2Unjv)dx- Jrao IUnjIS'-2Uvds = o(lIvll),
Jo
Let v 1 (-2 -
= u nj
in (2.5), we obtain from (2.4) and (2.5)
-!.) r(IV'U nj 12 + IUnj 12 )dx + (-!.s* s* Jo
1) 2 *
lu ,.1 Jr o n
2' dx - f.l( ~ -
:S C(l + Ilunj II)
q
(2.5)
(2.6)
Unj ----' U weakly in
H1(0),
unj ----' U weakly in
£2'(0),
Un; ----' U weakly in
U' (an),
Un; -+ U strongly in
Lq(n),
u nj -+ U strongly in
L 2(n),
n.
\Iv E H1(0).
-!.) rIU n.Iqdx s* Jo '
Note that 1 < q < 2 and from (2.6), it therefore follows that lIun; II :S subsequence, if necessary, we may assume that
u nj -+ U a.e. on
(2.4)
c.
By choosing a
636
ACTA MATHEMATICA SCIENTIA
It follows that
r(V'u.
U
satisfies
r
V'v + uv - luI2*-2uv - J.lluIQ-2 uv)dx -
k
ho
Vol.24 Ser.B
lul s*-2 uv ds
=0
"Iv E HI (0,),
(2.7)
and (2.8)
Set vn j
=u
n j
-
u. From [4]' we get
L
2 IU n j 1 * dx =
r IU i:
n j
(ds
=
L
2* IV n j 1 dx
r IV i;
n j
+
L
+
r lul;* ds + 0(1). i:
(ds
2* lul dx
+ 0(1),
Obviously,
Hence we obtain J(u)
2
1
+ -2 lV'vn· 12 0 J
Since (J' (unJ, un)
~
0 as
l
nj
1 2* 1 s* -2* Ivn·12* 0 - -lvn·ls* J, s* ] ,an u
~ 00,
2
= C + 0(1).
(2.9)
we deduce that s·
2*
IV'VnJ120 - IV n J 12* 0 - IV n J ·Is* , , ao 1
(2.10) = -(J'(u),u) = O.
We may therefore assume that
where b, b1 , bz are nonnegative constants satisfying b = b1 + b2 . Now we use the following inequality which can be found in [8]. Let 0, be a bounded domain in R N with C 1 boundary. Then for every
€
> 0,
where C< is a positive constant depending only on the diameter of the domain 0, and Therefore, from (2.11) we obtain
2
Hence, we derive that b 2: (S - €)(bf 2
€,
b 2: S(bf
2
+ b{).
2
+ b[).
Furthermore, due to the arbitrariness of
Therefore, b
b = b1 + b2 < (-)"2 - S· 2*
L
bi »
+ (_)"2 . S
637
Han: NEUMANN PROBLEMS OF A CLASS OF ELLIPTIC EQUATIONS
No.4
If b = 0, the proof is complete. Assume b > 0, after an elementary calculation, we obtain
b > _1_ S()1 - 2N - 2 It follows from
(2.12)
(2.8)-(2.12) that b
C
+ 4S - I)N-2.
bl b2 2* - s*
= J(u) + 2" 1
2
1
1
2'
1
1
q
2(N _ 1) (b + lIull ) + (s* - 2J(bl + /UI2' ,0) - /-l( q - s* )lulq,o
1
~
2: 2N _l (N _ l ) S(v l + 4S - 1) Now we choose k
N-2
1 2 1 1 1-£ q +2(N_l)lluI1 -/-l(q- sJlnl 2!1U!l.
> 0 by (2.13)
then obtain a contradiction. Remark 2.1 It is easy to verify that the positive constant k in (2.13) is independent of Proof of Theorem 1.1 We choose an orthonormal basis (ej) of HI (n) and define X j := Re, and consider the antipodal action of Z /2 on HI (n), then it suffices to verify assumptions of Lemma 2.1. sup lulq,o and so (3k -+ 0 as k -+ 00. In addition, lIull=l
As in [11]' we define (3k :=
UEZk,
.
4 {3q
1
we choose Pk := ( : l qk ) 2- q , and it follows that Pk -+ 0 as k -+ 00. Hence, for k 2: ko(ko is certain positive integer), u E Zk and !lull = Pk, we get J(u) 2: 0, and relation (AI) is proved. Since on the finite dimensional space Yk , all norms are equivalent, for u E Y k and !lull = rk < Pk, we have as k large enough 1
1
2
2'
J(u) :S -211ull - -2* lul2' ,0
-
/-l
q
-Iul q q ,0 + Collull
s'
< O.
Thus, relation (A 2 ) is proved. For k > k o , u E
z; ,
J(u)
Ilull:S Pb
2: 2Clilul1 2 - C2(lluI1 2' + !Iul() - /-l(3% Ilull q
q
q
2: -/-l(3% Ilull 2: -/-l(3% Pk . q
q
Since (3k -+ 0 , Pk -+ 0, as k -+ 00, relation (A 3 ) is satisfied. We choose 11* > 0 by 2" l(N_l)S()1 + 4S - I)N-2 - kl1*2: q = 0, then, by Lemma 2.3, for every 0 < 11 < 11* and c < 0, the functional J satisfies the (P.S.); condition.
638
ACTA MATHEMATICA SCIENTIA
Vol.24 Ser.B
References
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