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A note on uniqueness of solution for a class of semilinear nonselfadjoint elliptic problems
Nonlinear Analysis, Theory, Printed in Great Britain.
Methods
& Applicakw.s.
Vol.
12, No. 9, pp. 969-973,
1988. 0
0362-546X/88 $3.00 + .OO 1988 Pergamon Press plc
A NOTE ON UNIQUENESS OF SOLUTION FOR A CLASS OF SEMILINEAR NONSELFADJOINT ELLIPTIC PROBLEMS
Departamento
J. V. A. GONCALVES* de MatemBtica-IE, Universidade de Brasilia, 70910-Brasilia-DF, Brasil
(Received 15 February 1987; received for publication 8 July 1987) “i Key words and phrases: Maximum principles, uniqueness of solution, degree theory
INTRODUCTION LET Q C [WN(N 2 1) be a smooth
bounded domain, f: fi x R --, [w a Caratheodory function, p a real parameter and h a continuous function on S2. We are interested in the existence and uniqueness of solution for Lu=f(x,u)+ph
%u=O where
in on
i3Q N
N
LU = -
2
Q
UijU,,,,
+
i,] = 1
E
aiux,
+
'0'
1=1
is a uniformly strongly elliptic operator with smooth real valued coefficient functions U;j = aji, Uj, a0 3 0. Further 53 denotes either the Dirichlet on the Neumann boundary operator. Let A, be the principal eigenvalue of the eigenvalue problem 93u=Oond&
Lu=kuinQ,
obtained by the Krein-Rutman theory of positive operators (see [l]). As is well known A1 is real, positive and simple. (We will always assume when dealing with the Neuman condition that a,, is a nonconstant and strictly positive function.) The principal eigenfunction associated with A1 will be denoted by Q. It follows from the maximum principles that the outer unit normal derivative a$/a~ is negative on 8Q in the Dirichlet case and $J > 0 on S? in the Neumann case. The following hypotheses will be imposed on the nonlinearity f (fl) If@, s)] G a(x)lsl + b(x) on 52 x [w with 0 ~a~L”(Q),O~b~LP(Q)andsornep>N. (f2) ,tm= fs(x, 3) = m(x) f or each x E a and some function m : fi * R. We are now in position THEOREM
continuous
to state our result.
1. Suppose in addition and satisfies (m0)
to the above
m(x)
C Al
* Partially supported by CNPq. 969
hypotheses
on
fi,
that
m #Al
suplfS(x, SEU!
s)l E Lp(Q),
m
is
Jaylo aqy uo
u
U!
ue
uo
O=“@c
d 21+ (od ‘X)s=
07
.sawoDaq (1) uIaIqoJd (0~ d) d/n = 0 saIqeIJeA30 a8ueq~ aq4 kq 'pucq 'g OIU! dp3edtuoD ,,7sdwu s '2 %$$A 8wppaqura ayl 01 anp wqi IleDaJpue nie
uo
O=wg
‘0
u!
B=n7
33!n = 2s hq d? 3 2 Jo3 pau!jap JoleJadoJeauqaql &M td7:~hqalouaa.Cy< d,){a~ uog = ng :(~)“‘ZM 3 n} = $$M Ia1 pue‘( 1dq a]ouapaM UIJOU asoqM 0 uosuo~~~1n3snonuy1o~30 asedsaql aq(Q)s =g $a7 S.LLlflSEIH A~VNlt’JIl3lId
CINV
SNOIL.V.LON
‘1
'(S‘X)iJO q$MOJi? aql UO UO!]3'J$SaJ OU put? 'b 3 XJO3&LIJO3!Un
[p]JaUJcM-UepZey
‘[y > (~‘x)~jdns mG/:
$uaLuaJ!nbaJaql Japun ‘alduwxa ~03 ‘paU!WqO s! aXIa$s!xa aJaqM dq S$~nSaJaql I[BDaJaM (I) uraIqo.Id$LI!O~pl?3laSUOU aq$Ol pJE8aJ q$!M .uognIos
au0 LIeq$aJ0LI.I SEq (I) UIalqoJd $eql UMOqS aq ue3 I! u UO Iy> 2.~4 pW 'y< (O‘X)'~‘()= d 31 ‘aDuelsu!god yeJaua% ur lpz3s4InsaJssauanbyn q3ns 0 Jeau d ~03 ~aq~ uoguaw aM (un) '.suo~~~puo:, LJepunoq )alqD!xfa snoaua8ouroqJapun~uo~-3osanleAua8~aaq~a~ouap~~ .‘Yy‘. . ~‘Zy‘[yaJaH~yuopasodw! uoypuo3 a3ueuosaJuou atuos ~I!M Jayla804 ‘papnpxa r+yy= +ul= -u4 pue yy= +u4 = -tu sagqenba aql ql!~ ‘1 ey amos lo3 ‘I+?y 3 +zu ‘-2~syy pap!AoJd ‘&iJel0~ d ~03 uoy-qos C07+9 seq BI] 3 'ui = (S),s urq pw (s)j= (S ‘x)jwM (I) walqoJd talqyJ!a ayl 1~ [c]
au0 hlwxa
u! Joqlne ayl pue opaJ!anF?g ap-wo3
6q u~oqs uaaq seq lr v-
= 7 ase3 aql ur 0sIv (It) tv- =7despue~ XbUO
Iy>n ~(s‘x)sJ3~ (u)~ 3 y pue &Y 3 d yea ~03 uoynlos anbyn e seq (I) ura]qoJd Ialq$J!a aq$ $eq$ [z]u!a$sJauk&?H 01 anp suopenba [cJ8aw! uo IlnsaJpz!ssep r?LIIOJJ ~~01103 11 (1)
:JapIo u! aJe ase3 lu!oFpe3las aql 03 pawlar qDJeasaJ JawJo '"de Id\pap!AoJd uoyqos
pue IInsaJsy130 uos!JedLuo:, aq18uyJa3uo3 auo dp%xa
(.a’e)
seq (I) $ey$ qDns 0.~ (q)“d
cl 3 *
JO3 0 z WY
syJwuaJ auras
= md slsyxa alaqi
‘UaqL
(*) auJnssc ‘JaAoaJofl
.paJap!suoDBuraq s~‘d~aAp~adsaJ'uoypuo~ liJepunoq uueurnaNJ0
b uy u@ sa%ueq:, 029- 24.4
lalq3pra
aq3 olShpJo3x
(ZUI) JO
0 3 OX
aurosJO3
0<(%$44
(ILLI) Jaqlla pw
SXA-IV~NO+J
‘V
‘I\
O.l6
‘1
Semilinear nonselfadjoint
Now let F: R x E+- ZY be the nonlinear
operator
defined
rlf(x, 11-l r&9) F(rl, u(x))
= m(x)G)
It follows
from (fl)
that F has linear
From
operator.
rl #O,
XEQ
r]=o,
XEQ
x E Q (a.e.)
s
and continuous
for (7, V) E [w x E by
(f2) we get
fM =m(x)
lim
ISl+m So, F is a bounded
growth.
971
elliptic problems
(fi)
Let a: R x E + E be the map defined
for
(rl, u) E ~JJx E by a(~, So problem
(1,) can be studied
u) = u - SF(rl, u)
by looking
at the equation
Q(r), u) = Sh. The limiting
problem
associated
to (l,),
Lu =mu + h
(2)
namely in
93u=O
Q,
on
aS2
(14
will play an important role in the study of (2). We will use, for instance, that (lm) has a unique solution for each h E E. To get this we shall look at the elliptic eigenvalue problem %!u = po(x)u
in
93u=O
52,
on
as2
(3)
with weight function w E E. This problem has been studied by Hess-Kato [5] in the Dirichlet case (with Z8 = L) and by Hess-Senn [6] in the Neumann case (with % = L - ~1~).By making use of some of the results in those papers we can prove the following. LEMMA 2. Let
m E E be as in theorem Lu=mu
1. Then in
Q,
u = 0 is the only solution 93~=0
on
aR.
of the problem (4)
2. PROOFS
We will need in the course of the proof of theorem below, which will be proved later on.
1, an a priori estimate
given by the result
m be as in theorem 1 and suppose f satisfies ( fl), ( f $). Then there exist positive no = qo(h) and M = M(h) such that @(?I, u) # Sh whenever In] c v. and,]u] 2 M.
LEMMA 3. Let
constants
Proof of theorem 1. (Existence): we get from lemmata (2), (3) that Q(O, .) is a l-l map such that Sh E @(O, B,), where B,,, = B,(O) C E is the open ball of radius M. Actually, d(@(O,. ), BMM, Sh) = + 1 and from the homotopy invariance property of the Leray-Schauder degree we conclude that d( 0. (Uniqueness): assuming the contrary, there exists a sequence pk E R with ]pkl+ 00 and corresponding sequences u “, E E (a = 1, 2) with u: # u$ satisfying (lpk). By making
J. V.
972
A. GONCALVES
Pkl = qk we have @(qk, u”,) = Sh so that lui\ < M and further F(qk, u”,) is bounded in Lp(Q) if k is big enough. Now, from u”, = SF(q,, ~5) + Sh it follows that U$ --, u, in E ((u = 1,2) for convenient subsequences (we will not change notation when choosing subsequences). So u, = S(mu, + h) and from lemma 2 we have u1 = u2 = u. On the other hand, U: - U! = S(F(%‘> 0:) - F(rlk, 0’;)) =
qilu:) -f(x, %hi))
T]kS(fh
= s(fs(x~ek(x))(u:
- d>>
for some ok(x) between v;‘uf(x) and T]~‘u$(x), x E 52 (a.e.). Let uk = uf - u$ and ijk = u”/Iz? so that dk = Sfs(x, &)dk. By recalling that fs(x, 8k)bk is bounded in LJ’(Q) we have ijk+ ii in E, (1~7 = l), for some subsequence. Next, consider the following subsets of 52. n_:u
Q+:u>O
n,:u=o,
CLAIM. meas Q0 = 0 Indeed, from the estimates for elliptic equations we get v E l@*P (p > N) and by applying lemma A4 of [7, p. 531 we have uXiX,= 0 on Q, so that Lu = 0 on Q0 (a.e.) and this amounts meas Q, = 0 by using (*) and the equation u = S(mu + h). As a consequence of the claim, 8k(x)
+
&oo
x
E
52
(a.e.)
and further fs(x, @k(x)) uk+ m(x)fi(x) x E Q (a.e.). Hence, fs(x, 8k)dk+ m6 in LP(Q) so that L?= Smd, which is impossible since d = 0 by lemma 2. This ends the proof of Theorem 1. Now we proceed to the next proof. Proof of lemma 2. As we mentioned earlier we will make use of results by Hess-Kato [5] and Hess-Senn [6] concerning the eigenvalue problem (3). Let u be a solution of (4). (i) Dirichlet boundary condition. By applying the results of [5] to problem (3) with o = m and 2 = L and using (m,), there exists a principal eigenvalue PI(m) > 0 with eigenfunction say u1 > 0 in 9. Further from (mo) we get 1 = p,(n,) < ,ul(m). So u = 0 since there is no eigenvalue p of (3) with real part Re p between 0 and PI(m). (ii) Neumann boundary condition. We will apply the results of [6] to problem (3) with 2 = L - ao. At first, we intoduce an auxiliary function 111,which will be used later. For this purpose, let So : LP + W&P be the operator defined for g E LP by S& = u iff (Y+l)u=g
in
52,
g=O
on
aS2
We recall that S,, is a compact l-l map. Let 2: G&d($)C E + E be the linear operator defined through $ + 1 = S;‘. There exists, by the Krein-Rutman theorem, an element 111E E*, T,,? > 0, such that Ker(g*) = ($), where p* :9($*) C E* -+ E* is the Banach space adjoint of $. Moreover, 11,can be identified with a positive function, yet denoted by 6, in Lq(Q) (q > 1) (cf. [6, p. 4611 for details). On the other hand, it follows from
Semilinear nonselfadjoint
elliptic problems
973
and the maximum principles that (A, - a,,) changes sign in Q. Thus p = 1 is an eigenvalue of (3) with weight function o = A, - ao. Next, we have, by sucessively using the results of [6] that Jo (A1 - a,)$ < 0 and 1 = pI(A1 - ao), the principal eigenvalue of (3) with eigenfunction $ > 0 on a. Moreover, we also find by using (mo), (m2) that there exists a principal eigenvalue of (3), namely pr(m - ao) > 0 with eigenfunction u1 > 0 on fi satisfying 1 = pI(ilI - uo) < pI(m - uo). Hence u = 0, since there is no eigenvalue p of (3), (o = m - uo), with real part Re ~1between 0 and pl(m - uo). This proves lemma 2. Next we will give the following proof. Proof of lemma 3. Suppose, on the contrary, that there exist sequences with (ukl ---, 00 such that @(qk, u,J = Sh. Let Bk = uk/jukl. Then n _sF(Tlk, vk-
bki
uk)
qk-f
0 (nk # 0), uk E E
Sh ‘i6i-i
and by using the linear growth of F and the compactness of S: LP + E we find Ok+ D in E, (101 = l), for some subsequence. Moreover, F( qk, ok)1uklml is bounded in LP and + m(x)u(x) x E & (a.e.). So we get by passing to the limit that u = qkbki-*f(X> %?‘“k(x)> Smu which is imposible since u = 0 by lemma 2. This proves lemma 3.
REFERENCES M. G. & RUTMANM. A., Linear operators leaving invariant a cone in a Banach space, Am. math. .I&. Trunsl. Series 1 10, 199-325 (1962). 2. HAMMERSTEIN A., Nichtlineare integralgleichungen nebst anwendungen, Acta math. Srarkh. 54, 117-176 (1930). 3. COSTAD. G., DE FIGUEIREDOD. G. & GONCALVES J. V. A., On the uniqueness of solution for a class of semilinear elliptic problems, J. math. Analysis Applic. 123, 170-180 (1987) 4. KAZDANJ. L. C WARNERF. W., Remarks on some quasilinear elliptic equations. Communspure appl. Math. 28, 1.
KREIN
567-597 (1975). 5. HESS P. & KATO T., On some linear and nonlinear Commune partial diff. Eqns 5, 999-1030 (1980).
6.
HESS P. & SENN S.,
&positive
eigenvalue problems with an indefinite weight function.
solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions.
Math. Annln 258, 459-470 (1982).
7. KINDERLEHRER D. & STAMPACCHIA G., An Introduction to Variational Inequalities and theirApplications.Academic Press, New York (1980).
Methods
& Applicakw.s.
Vol.
12, No. 9, pp. 969-973,
1988. 0
0362-546X/88 $3.00 + .OO 1988 Pergamon Press plc
A NOTE ON UNIQUENESS OF SOLUTION FOR A CLASS OF SEMILINEAR NONSELFADJOINT ELLIPTIC PROBLEMS
Departamento
J. V. A. GONCALVES* de MatemBtica-IE, Universidade de Brasilia, 70910-Brasilia-DF, Brasil
(Received 15 February 1987; received for publication 8 July 1987) “i Key words and phrases: Maximum principles, uniqueness of solution, degree theory
INTRODUCTION LET Q C [WN(N 2 1) be a smooth
bounded domain, f: fi x R --, [w a Caratheodory function, p a real parameter and h a continuous function on S2. We are interested in the existence and uniqueness of solution for Lu=f(x,u)+ph
%u=O where
in on
i3Q N
N
LU = -
2
Q
UijU,,,,
+
i,] = 1
E
aiux,
+
'0'
1=1
is a uniformly strongly elliptic operator with smooth real valued coefficient functions U;j = aji, Uj, a0 3 0. Further 53 denotes either the Dirichlet on the Neumann boundary operator. Let A, be the principal eigenvalue of the eigenvalue problem 93u=Oond&
Lu=kuinQ,
obtained by the Krein-Rutman theory of positive operators (see [l]). As is well known A1 is real, positive and simple. (We will always assume when dealing with the Neuman condition that a,, is a nonconstant and strictly positive function.) The principal eigenfunction associated with A1 will be denoted by Q. It follows from the maximum principles that the outer unit normal derivative a$/a~ is negative on 8Q in the Dirichlet case and $J > 0 on S? in the Neumann case. The following hypotheses will be imposed on the nonlinearity f (fl) If@, s)] G a(x)lsl + b(x) on 52 x [w with 0 ~a~L”(Q),O~b~LP(Q)andsornep>N. (f2) ,tm= fs(x, 3) = m(x) f or each x E a and some function m : fi * R. We are now in position THEOREM
continuous
to state our result.
1. Suppose in addition and satisfies (m0)
to the above
m(x)
C Al
* Partially supported by CNPq. 969
hypotheses
on
fi,
that
m #Al
suplfS(x, SEU!
s)l E Lp(Q),
m
is
Jaylo aqy uo
u
U!
ue
uo
O=“@c
d 21+ (od ‘X)s=
07
.sawoDaq (1) uIaIqoJd (0~ d) d/n = 0 saIqeIJeA30 a8ueq~ aq4 kq 'pucq 'g OIU! dp3edtuoD ,,7sdwu s '2 %$$A 8wppaqura ayl 01 anp wqi IleDaJpue nie
uo
O=wg
‘0
u!
B=n7
33!n = 2s hq d? 3 2 Jo3 pau!jap JoleJadoJeauqaql &M td7:~hqalouaa.Cy< d,){a~ uog = ng :(~)“‘ZM 3 n} = $$M Ia1 pue‘( 1dq a]ouapaM UIJOU asoqM 0 uosuo~~~1n3snonuy1o~30 asedsaql aq(Q)s =g $a7 S.LLlflSEIH A~VNlt’JIl3lId
CINV
SNOIL.V.LON
‘1
'(S‘X)iJO q$MOJi? aql UO UO!]3'J$SaJ OU put? 'b 3 XJO3&LIJO3!Un
[p]JaUJcM-UepZey
‘[y > (~‘x)~jdns mG/:
$uaLuaJ!nbaJaql Japun ‘alduwxa ~03 ‘paU!WqO s! aXIa$s!xa aJaqM dq S$~nSaJaql I[BDaJaM (I) uraIqo.Id$LI!O~pl?3laSUOU aq$Ol pJE8aJ q$!M .uognIos
au0 LIeq$aJ0LI.I SEq (I) UIalqoJd $eql UMOqS aq ue3 I! u UO Iy> 2.~4 pW 'y< (O‘X)'~‘()= d 31 ‘aDuelsu!god yeJaua% ur lpz3s4InsaJssauanbyn q3ns 0 Jeau d ~03 ~aq~ uoguaw aM (un) '.suo~~~puo:, LJepunoq )alqD!xfa snoaua8ouroqJapun~uo~-3osanleAua8~aaq~a~ouap~~ .‘Yy‘. . ~‘Zy‘[yaJaH~yuopasodw! uoypuo3 a3ueuosaJuou atuos ~I!M Jayla804 ‘papnpxa r+yy= +ul= -u4 pue yy= +u4 = -tu sagqenba aql ql!~ ‘1 ey amos lo3 ‘I+?y 3 +zu ‘-2~syy pap!AoJd ‘&iJel0~ d ~03 uoy-qos C07+9 seq BI] 3 'ui = (S),s urq pw (s)j= (S ‘x)jwM (I) walqoJd talqyJ!a ayl 1~ [c]
au0 hlwxa
u! Joqlne ayl pue opaJ!anF?g ap-wo3
6q u~oqs uaaq seq lr v-
= 7 ase3 aql ur 0sIv (It) tv- =7despue~ XbUO
Iy>n ~(s‘x)sJ3~ (u)~ 3 y pue &Y 3 d yea ~03 uoynlos anbyn e seq (I) ura]qoJd Ialq$J!a aq$ $eq$ [z]u!a$sJauk&?H 01 anp suopenba [cJ8aw! uo IlnsaJpz!ssep r?LIIOJJ ~~01103 11 (1)
:JapIo u! aJe ase3 lu!oFpe3las aql 03 pawlar qDJeasaJ JawJo '"de Id\pap!AoJd uoyqos
pue IInsaJsy130 uos!JedLuo:, aq18uyJa3uo3 auo dp%xa
(.a’e)
seq (I) $ey$ qDns 0.~ (q)“d
cl 3 *
JO3 0 z WY
syJwuaJ auras
= md slsyxa alaqi
‘UaqL
(*) auJnssc ‘JaAoaJofl
.paJap!suoDBuraq s~‘d~aAp~adsaJ'uoypuo~ liJepunoq uueurnaNJ0
b uy u@ sa%ueq:, 029- 24.4
lalq3pra
aq3 olShpJo3x
(ZUI) JO
0 3 OX
aurosJO3
0<(%$44
(ILLI) Jaqlla pw
SXA-IV~NO+J
‘V
‘I\
O.l6
‘1
Semilinear nonselfadjoint
Now let F: R x E+- ZY be the nonlinear
operator
defined
rlf(x, 11-l r&9) F(rl, u(x))
= m(x)G)
It follows
from (fl)
that F has linear
From
operator.
rl #O,
XEQ
r]=o,
XEQ
x E Q (a.e.)
s
and continuous
for (7, V) E [w x E by
(f2) we get
fM =m(x)
lim
ISl+m So, F is a bounded
growth.
971
elliptic problems
(fi)
Let a: R x E + E be the map defined
for
(rl, u) E ~JJx E by a(~, So problem
(1,) can be studied
u) = u - SF(rl, u)
by looking
at the equation
Q(r), u) = Sh. The limiting
problem
associated
to (l,),
Lu =mu + h
(2)
namely in
93u=O
Q,
on
aS2
(14
will play an important role in the study of (2). We will use, for instance, that (lm) has a unique solution for each h E E. To get this we shall look at the elliptic eigenvalue problem %!u = po(x)u
in
93u=O
52,
on
as2
(3)
with weight function w E E. This problem has been studied by Hess-Kato [5] in the Dirichlet case (with Z8 = L) and by Hess-Senn [6] in the Neumann case (with % = L - ~1~).By making use of some of the results in those papers we can prove the following. LEMMA 2. Let
m E E be as in theorem Lu=mu
1. Then in
Q,
u = 0 is the only solution 93~=0
on
aR.
of the problem (4)
2. PROOFS
We will need in the course of the proof of theorem below, which will be proved later on.
1, an a priori estimate
given by the result
m be as in theorem 1 and suppose f satisfies ( fl), ( f $). Then there exist positive no = qo(h) and M = M(h) such that @(?I, u) # Sh whenever In] c v. and,]u] 2 M.
LEMMA 3. Let
constants
Proof of theorem 1. (Existence): we get from lemmata (2), (3) that Q(O, .) is a l-l map such that Sh E @(O, B,), where B,,, = B,(O) C E is the open ball of radius M. Actually, d(@(O,. ), BMM, Sh) = + 1 and from the homotopy invariance property of the Leray-Schauder degree we conclude that d(
J. V.
972
A. GONCALVES
Pkl = qk we have @(qk, u”,) = Sh so that lui\ < M and further F(qk, u”,) is bounded in Lp(Q) if k is big enough. Now, from u”, = SF(q,, ~5) + Sh it follows that U$ --, u, in E ((u = 1,2) for convenient subsequences (we will not change notation when choosing subsequences). So u, = S(mu, + h) and from lemma 2 we have u1 = u2 = u. On the other hand, U: - U! = S(F(%‘> 0:) - F(rlk, 0’;)) =
qilu:) -f(x, %hi))
T]kS(fh
= s(fs(x~ek(x))(u:
- d>>
for some ok(x) between v;‘uf(x) and T]~‘u$(x), x E 52 (a.e.). Let uk = uf - u$ and ijk = u”/Iz? so that dk = Sfs(x, &)dk. By recalling that fs(x, 8k)bk is bounded in LJ’(Q) we have ijk+ ii in E, (1~7 = l), for some subsequence. Next, consider the following subsets of 52. n_:u
Q+:u>O
n,:u=o,
CLAIM. meas Q0 = 0 Indeed, from the estimates for elliptic equations we get v E l@*P (p > N) and by applying lemma A4 of [7, p. 531 we have uXiX,= 0 on Q, so that Lu = 0 on Q0 (a.e.) and this amounts meas Q, = 0 by using (*) and the equation u = S(mu + h). As a consequence of the claim, 8k(x)
+
&oo
x
E
52
(a.e.)
and further fs(x, @k(x)) uk+ m(x)fi(x) x E Q (a.e.). Hence, fs(x, 8k)dk+ m6 in LP(Q) so that L?= Smd, which is impossible since d = 0 by lemma 2. This ends the proof of Theorem 1. Now we proceed to the next proof. Proof of lemma 2. As we mentioned earlier we will make use of results by Hess-Kato [5] and Hess-Senn [6] concerning the eigenvalue problem (3). Let u be a solution of (4). (i) Dirichlet boundary condition. By applying the results of [5] to problem (3) with o = m and 2 = L and using (m,), there exists a principal eigenvalue PI(m) > 0 with eigenfunction say u1 > 0 in 9. Further from (mo) we get 1 = p,(n,) < ,ul(m). So u = 0 since there is no eigenvalue p of (3) with real part Re p between 0 and PI(m). (ii) Neumann boundary condition. We will apply the results of [6] to problem (3) with 2 = L - ao. At first, we intoduce an auxiliary function 111,which will be used later. For this purpose, let So : LP + W&P be the operator defined for g E LP by S& = u iff (Y+l)u=g
in
52,
g=O
on
aS2
We recall that S,, is a compact l-l map. Let 2: G&d($)C E + E be the linear operator defined through $ + 1 = S;‘. There exists, by the Krein-Rutman theorem, an element 111E E*, T,,? > 0, such that Ker(g*) = ($), where p* :9($*) C E* -+ E* is the Banach space adjoint of $. Moreover, 11,can be identified with a positive function, yet denoted by 6, in Lq(Q) (q > 1) (cf. [6, p. 4611 for details). On the other hand, it follows from
Semilinear nonselfadjoint
elliptic problems
973
and the maximum principles that (A, - a,,) changes sign in Q. Thus p = 1 is an eigenvalue of (3) with weight function o = A, - ao. Next, we have, by sucessively using the results of [6] that Jo (A1 - a,)$ < 0 and 1 = pI(A1 - ao), the principal eigenvalue of (3) with eigenfunction $ > 0 on a. Moreover, we also find by using (mo), (m2) that there exists a principal eigenvalue of (3), namely pr(m - ao) > 0 with eigenfunction u1 > 0 on fi satisfying 1 = pI(ilI - uo) < pI(m - uo). Hence u = 0, since there is no eigenvalue p of (3), (o = m - uo), with real part Re ~1between 0 and pl(m - uo). This proves lemma 2. Next we will give the following proof. Proof of lemma 3. Suppose, on the contrary, that there exist sequences with (ukl ---, 00 such that @(qk, u,J = Sh. Let Bk = uk/jukl. Then n _sF(Tlk, vk-
bki
uk)
qk-f
0 (nk # 0), uk E E
Sh ‘i6i-i
and by using the linear growth of F and the compactness of S: LP + E we find Ok+ D in E, (101 = l), for some subsequence. Moreover, F( qk, ok)1uklml is bounded in LP and + m(x)u(x) x E & (a.e.). So we get by passing to the limit that u = qkbki-*f(X> %?‘“k(x)> Smu which is imposible since u = 0 by lemma 2. This proves lemma 3.
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KREIN
567-597 (1975). 5. HESS P. & KATO T., On some linear and nonlinear Commune partial diff. Eqns 5, 999-1030 (1980).
6.
HESS P. & SENN S.,
&positive
eigenvalue problems with an indefinite weight function.
solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions.
Math. Annln 258, 459-470 (1982).
7. KINDERLEHRER D. & STAMPACCHIA G., An Introduction to Variational Inequalities and theirApplications.Academic Press, New York (1980).