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A quasilinear elliptic problem with a discontinuous nonlinearity
NonlinearAnah,sLL Theory, Methods & Applications, Vol. 25, No. I 1, pp. 1 1 1 5 - 1 1 2 1 , 1995 C o p y r i g h t © 1995 E l s e v i e r Science Ltd P r i n t e d in G r e a t Britain. All r i g h t s r e s e r v e d 0362 5 4 6 X / 9 5 $9.50 + .00
Pergamon
0362-546X(94)00233-9 A
QUASILINEAR ELLIPTIC PROBLEM WITH DISCONTINUOUS NONLINEARITY
A
SIDI. M O H A M E D B O U G U I M A Departement de Mathematiques, Univesit6 de Tlemcen. B.P. 119, Tlemcen, 13000, Algeria
(Received 24 January 1994: received in revisedJorm 30 July 1994; receioedfor publication 26 August 1994) Key words and phrases: Quasilinear elliptic problems, discontinuous nonlinearities, upper and lower solutions
1. I N T R O D U C T I O N
In this paper, we investigate the existence of solutions of a quasilinear elliptic problem with a discontinuous nonlineariW. More specifically, we consider the following Dirichlet problem
-Apu = f ( u ) + h
in II
u =0
(1)
on 3 ~ ,
where Apu = div([~TulP 27u), p > 2 and tl is a bounded open subset of ~", n >_ 2 with a C z boundary; f: ~ ~ [R is discontinuous at the origin, and h is a function or a distribution on fl. We suppose that lira, . 0+f(0 _+ e) exist and f(0) ~ [f(0 - ),f(0 + )], where f(0-)=
lim f ( 0 - E ) < f ( 0 + ) =
lim f ( 0 + ~ ) .
Let F(s) = .l~f(t)dt. We shall assume that F satisfies (H)
pF(s~
lim inf s ~
, < tz',
+~
si P
where / x ' = tx'(~, p) is given by ¢ __
11"
r a1 p - 1 pp
R ( ~ )p
fi(1/p,1 - l/p),
R(f~) is the radius of the smallest ball B containing 1), and /3 is the usual Beta function, i.e. for a, b > 0 /3(a,b)=
f
l
t '~ I ( 1 - t ) b ldt.
*t 0
Note that we assume no growth condition on f. Problem (1) has been considered in [1] under the assumption limsup
2F(s) s2
where h 1 is the first eigenvalue of - A in H~l(ll). 1115
andp=2,
1116
S.M. B O U G U I M A
Also, when f is continuous, problem (1) has been investigated in [2]. Our objective is to extend the ideas developed in [2] to the case where f is discontinuous. In the process, we generalize the results in [3, 4]. Finally, we note that problems with discontinuous nonlinearities have been considered by many authors (see for instance [3, 5-7] and the references therein). It was pointed out in [6] that elliptic problems with discontinuous nonlinearities have been examined in connection with modeling plasma physics and thermal conduction problems.
The main result In this section we state and prove our main result. THEOREM 1. Assume h • L~(I~) and the condition ( H ) is satisfied. Then problem (1) has a solution u • W2"P(I)), p > n.
Remark. By solution we mean a function u • W0~'P(ID satisfying -Aeu=f(u)+h
a.e. inl),
The proof of theorem 1 is based on the notion of lower and upper solutions.
Definitions. (i) A function c~ • Wz'P(I)) is a lower solution of (1) if - - A p a
-Ap/3>_f(/3)+ha.e. in~,/3>_Oondl2.
Remark. The above inequalities should be understood in the sense of distributions. PROPOSITION ]. If problem (1) has a lower solution o~ and an upper solution /3 such that a < 13 in f~, then (1) has a solution u • W2P(II) such that a < u
Proof Let p(x, u) = max( c~,min{u,/3}) for x ~ 1/; i.e.
( ~(x) p(x,u) = ,'~u (x)
u < ~(x) ~(x)/3(x).
Let l(x, u) =f(p(x, u)) + h(x) and set L(x, u) = f~'l(x, s)ds. Note that there exist constants a 0, /30 > 0 such that
l(x,u) < aolsjP 1 +/3o
with a 0 < A 1
(A 1 is the first eigenvalue of - A p in W~]'P(ID). Consequently, the functional I defined by
l(u)= l / P ! l V u l P d x -
faL(x,u)dx
is coercive and lower semi-continuous. Hence the functional I has a minimum u in w0m'e(l~). If ,9I(u) is the generalized gradient of I at u then 0 • c~I(u) (see [7]). Also, following [7] we can show that
M(u) =Au - ;)J(u),
A quasilinear elliptic problem with a discontinuous nonlinearity
1117
where A , J : WO "+' ~ (W~]4') ' are the operators defined by
(.4u,u) = / I v . l e 2VuVv d x (J(u),u)
£L(x,uIdx.
=
Moreover, a J ( u ) c [l(x, u - ), l(x, u + )]. The fact that 0 ~ aI(u) implies the existence of a w ~ ? J ( u ) such that A u = w a.e. in f/. We have that w ~ L p' c(WoI'P) ', where p ' is the H61der conjugate of p. H e n c e A u ~ L p' and (Au,v)
Vu e
= (w,u)
wol+(~).
F r o m this we conclude that -Apu = w with
a.e. in ~
{l(x,u} wE[(x,u)=
(2)
ifu~0
[l(x,O-t,[l(x,O+)]ifu=O.
Let us show now that the Lebesgue measure of the set {x ~ ~ ;
u ( x ) = 0}
is zero.
Let F := { x e Xl;
u ( x ) = 0}.
F r o m (2) we have -@u~[l(x,O
i.e.
l(x,O + ) ]
),
- Apu
-
h(x)
c
a.e. in F,
[f(O-),f(O +)1.
Now, using a result in [8], we can show that a.e. i n F
-Apu=O
and-h(x)~[f(0-),f(0+)].
A s s u m e that the Lebesguc measure of the set (x e F; - h ( x )
#:f(O + )}
is strictly greater than 0.
Let & be the eigenfunction corresponding to the eigenvalue A1 such that &(x) > 0 Vx c fL Then we have 0 <
lim
l ( u + ~&) - l ( u )
e~0*
= L
E
- , x..
= - f[f
.IF
+) +h(x)]&(x)dx
<0.
1118
S.M. BOUGUIMA
This contradiction shows that meas{x ~ F; - h ( x ) ~f(O + )} = 0. Similarly, we have meas{x ~ F; - h ( x ) ¢f(O - )} = 0. Consequently, the Lebesgue measure of the set F = {x c F; - h ( x ) f(0 + )}U{x E F; - h ( x ) ¢f(O - )} is zero. We conclude that u satisfies -Apu =l(x,u)
a.e. in ~ .
Moreover, a ( x ) < u(x) <_[3(x) Vx ~ 12. In fact, since /3 is an upper solution, multiplying the differential equation in (2)by (u - / 3 ) + = max{(u - / 3 ), 0} and integrating by parts the resulting equation, we obtain that u(x) <_/3(x) Vx ~ 12. Similarly, we show that a ( x ) < u(x) V x ~ ~ . The fact that u ~ WZ'P(ll), p > n is a consequence of the regularity theory. In order to complete the proof of theorem 1, we must show that (1) has a positive upper solution, and a negative lower solution. • The following lemmas are needed. LEMMA 1. Let M ~ IR + and n ~ N. Then for each nonnegative real number d, the initial-value problem -(Iv'lP-
l
2v')
-
';lv'l P
2v'
=f(v) +M
v(O) = d
(3)
v'(O) = 0
has a positive solution v0, which is defined on a maximal interval [0, R d [. Moreover, v0 ~ C1([0, Rd[) and ~v~lP--°v~~ C1([0, Rd[). Proof First, we construct the solution in a neighborhood of O. Let rt > O. Set g(s) = f ( s ) + M if s > O, and g(O) =f(O + ) + M. On C([O, rt]; E), we consider the compact operator T defined by (Tv)(t) =d-
[(Hv)(s)Ip'-2(Hv)(s)ds,
where (Hv)(s) =s n
rng(iv(r)l)dr
fo '~
and p' denotes the H61der conjugate of p. There exists a 6 > 0 such that Ig(s+d)-g(d)l<_l
for 0_
Hence, if Iv(r) - dl _< 6, then ](Tv)(t) -dl<_
s~
< (Ig(d)l + -
n + l
r'(lg(d)l+ 1)dr
l)P'-lt P', < 8 p
-
ds
A quasilinear elliptic problem with a discontinuous nonlinearity
1119
for all t E [ 0 , r/1], where r/1 > 0 is small enough. Consequently, T has a fixed point v* in C([0, r h ]; ~). Moreover, we can choose a nonnegative n u m b e r T12 such that 0 < ~2 < ~1, and
f ' f s-" ~
' + l )1 r"(g(lv*(r)t) d r ( '
p'-'te' -fir
Vt ~ [0, rh]. In this case v * ( r ) = (Tv* Xr) > 0 for all r ~ [0, rt2]. This shows that v* is a solution of (3). T h e rest of the l e m m a follows from the theory of ordinary differential equations. • LEMMA 2. Let B be the smallest ball containing ~ , and let M ~ N + , then the p r o b l e m
-ApV=f(V) + M V> 0
in B
(4)
on OB
has a C t radial solution.
Proof Set V(x o + x ) = v(lxl), where x 0 is the center of B. Let r = [xl, and let R be the radius of B. T h e n v satisfies
[
_(Ic~,]p 2u,),_ n - l l v ' l p
2v'=f(v)+M"
t
(5)
c,(0) = d v ' ( 0 ) = 0. Following l e m m a 1, p r o b l e m (5) has a solution v, defined on a maximal interval [0, Rd[, such that v(0) = d. Moreover, v(r) > 0 for all r ~ [0, Ra[. Now, an a r g u m e n t in [6, p. 9], shows that condition ( H ) implies that, it is possible to choose d, in such a way that R a > R. This yields that v is a solution of (5). Hence, V(x) = v(lx -x0l) is a radial solution of (4). • PROPOSITION 2. P r o b l e m (1) has a positive upper solution /3.
Proof (i)
If f is b o u n d e d from below in E + , we take M > Ilhl]~. Let V be the solution of
- ~ p V = f ( V ) +M
in B,
V_> 0 on
OB.
Let /3 = V . T h e n /3 is a positive u p p e r solution of (1). (ii) If f is not b o u n d e d from below, then any /3 e N* such that f ( / 3 ) _< Ilhll~ is an u p p e r solution of (1). Similarly, we can prove the following proposition. PROPOSITION 3. P r o b l e m (1) has a negative lower solution ~. This completes the p r o o f of t h e o r e m 1.
Remarks. (1) We have assumed f discontinuous at the origin. If f is discontinuous at another point, we can reduce the p r o b l e m to a discontinuity at the origin, by translation. (2) T h e case f(0 +) < f ( 0 ) can be treated similarly.
1120
S.M. B O U G U I M A
Application. Assume 11 = B(0, R) and h = 0. Suppose f satisifes ( f l ) f is nondecreasing, (f2) f • C ( N \ { a } ; N) with f(u)-OVu•[-~,a],f(a+)=b>O, (f3) b(I/p
1)
- - a
~l],,p
1
[
>p'meas(t~)
14,11pj ] ] l/p
( f~
]p/p 1
["
, '
t where q5~ is an eigenfunction corresponding to the first eigenvalue A1 of - A p in H(~(fD, meas(.) denotes the n dimensional Lebesgue measure and p' is the H61der conjugate of p. Assume, further that ( H ) is satisfied, then we have the following. THEOREM 2. Under the above assumptions, the problem
-Au=f(u)
infl
u = 0
on c?11
(6)
has a solution u • W,]'P(tl)p > n. Moreover, u ~- 0 and meas({x ~ ti;
Proof
where
u(x)
= a}) = 0.
As in the proof of proposition 1, consider the functional
L(x,u)= j~;'l(x,s)ds and If(a(x)) l(x,s) = I f ( s ) If(~(x))
s < a(x) a ( x )/3(x).
Here, a and /3 are lower and upper solutions of (6), respectively. The functional 1 has a minimizing function u ~ W~,P(11), solution of the problem - A p u = f ( u ) a.e. in 1~. CLAIM 1. There exists ~r • JR, such that l(~r&~ ) < 0.
Proof
Let l/p
O'~
I
A quasilinear elliplic problem with a discontinuous nonlineariW F r o m c o n d i t i o n s fl a n d f2, we h a v e L ( x , s ) - L ( x , a ) > _ b ( s - a ) w i t h l ( , r ~ b ~ ) = crP [
_< - -
IV~,I:,-
I V 4 , ~ I ~'
P
[L(x,~r&,
b~r
1121
L ( x , a ) = - O in ~ . T h e n
)dx
cbl + ab m e a s ( f ~ ) < O.
T h i s c o m p l e t e s the p r o o f o f the c l a i m 1. C l a i m 1 s h o w s that u ¢ (). W c p r o c e e d as in t h e p r o o f o f p r o p o s i t i o n m e a s ( { x ~ ~); u ( x ) = a}) = 0. T h i s c o m p l e t e s the p r o o f o f t h e o r e m 2. •
1 to s h o w t h a t
R e m a r k . T h e o r e m 2 is m o t i v a t e d by a similar result o f A r c o y a a n d C a l a h o r r a n o [3]. In this w o r k , t h e a u t h o r s use an a d d i t i o n a l c o n d i t i o n o n J'. In fact t h e y s u p p o s e t h a t t h e r e exist c~, / 3 > 0 such t h a t f ( s ) < cxlsl p ~ +/3 w i t h ~ < A~, w h e r e ,~ is t h e first e i g e n v a l u e o f - A p . Acknowledgements
The author wishes to thank the reieree for interesting remarks. REFERENCES
l. FONDA A., GOSSEZ J. P. & ZANOLIN F., On a nonrcsonance condition for a semilinear elliptic problem, Diff. Integral Eqns 4, 945 951 (19911. 2. EL HACHIMI A. & GOSSEZ J. P., A note on a resonance condition for a quasilinear elliptic problem, Nonlinear Analvsis (to appear). 3. ARCOYA D. & CALAH()RRANO M., Some discontinuous problems with a quasilinear operator, Scu. norm. sup. Pisa, preprint ( 1991 ). 4. BOUGUIMA S. M., A scmilinear elliptic problem with discontinuous nonlinearities, Communs appl. Nonlinear Analysis (to appear). 5. BOUCHERIF A. & BOUGUIMA S. M, Perturbation m ~t frcc boundary problem, J. math. Analysis Applic. 183, 430-437 (1994). 6. BOUGUIMA S. M. & BOUCHERIF A., A discontxnuous semilinear elliptic problem without a growth condition, Dyn. Syst. Applic. 2, 183 188 (1993). 7. CHANG K. C., Variational methods for non diflerentiablc functionals and their applications to Partial. Diff. Eq., J. math. Analysis.4pplic 80, 1112 129 (19811. 8. MORREY C. B., Multiple Integrals in the Calculu.s O/ ~:anatton~. Swinger, Berlin (1966).
Pergamon
0362-546X(94)00233-9 A
QUASILINEAR ELLIPTIC PROBLEM WITH DISCONTINUOUS NONLINEARITY
A
SIDI. M O H A M E D B O U G U I M A Departement de Mathematiques, Univesit6 de Tlemcen. B.P. 119, Tlemcen, 13000, Algeria
(Received 24 January 1994: received in revisedJorm 30 July 1994; receioedfor publication 26 August 1994) Key words and phrases: Quasilinear elliptic problems, discontinuous nonlinearities, upper and lower solutions
1. I N T R O D U C T I O N
In this paper, we investigate the existence of solutions of a quasilinear elliptic problem with a discontinuous nonlineariW. More specifically, we consider the following Dirichlet problem
-Apu = f ( u ) + h
in II
u =0
(1)
on 3 ~ ,
where Apu = div([~TulP 27u), p > 2 and tl is a bounded open subset of ~", n >_ 2 with a C z boundary; f: ~ ~ [R is discontinuous at the origin, and h is a function or a distribution on fl. We suppose that lira, . 0+f(0 _+ e) exist and f(0) ~ [f(0 - ),f(0 + )], where f(0-)=
lim f ( 0 - E ) < f ( 0 + ) =
lim f ( 0 + ~ ) .
Let F(s) = .l~f(t)dt. We shall assume that F satisfies (H)
pF(s~
lim inf s ~
, < tz',
+~
si P
where / x ' = tx'(~, p) is given by ¢ __
11"
r a1 p - 1 pp
R ( ~ )p
fi(1/p,1 - l/p),
R(f~) is the radius of the smallest ball B containing 1), and /3 is the usual Beta function, i.e. for a, b > 0 /3(a,b)=
f
l
t '~ I ( 1 - t ) b ldt.
*t 0
Note that we assume no growth condition on f. Problem (1) has been considered in [1] under the assumption limsup
2F(s) s2
where h 1 is the first eigenvalue of - A in H~l(ll). 1115
andp=2,
1116
S.M. B O U G U I M A
Also, when f is continuous, problem (1) has been investigated in [2]. Our objective is to extend the ideas developed in [2] to the case where f is discontinuous. In the process, we generalize the results in [3, 4]. Finally, we note that problems with discontinuous nonlinearities have been considered by many authors (see for instance [3, 5-7] and the references therein). It was pointed out in [6] that elliptic problems with discontinuous nonlinearities have been examined in connection with modeling plasma physics and thermal conduction problems.
The main result In this section we state and prove our main result. THEOREM 1. Assume h • L~(I~) and the condition ( H ) is satisfied. Then problem (1) has a solution u • W2"P(I)), p > n.
Remark. By solution we mean a function u • W0~'P(ID satisfying -Aeu=f(u)+h
a.e. inl),
The proof of theorem 1 is based on the notion of lower and upper solutions.
Definitions. (i) A function c~ • Wz'P(I)) is a lower solution of (1) if - - A p a
-Ap/3>_f(/3)+ha.e. in~,/3>_Oondl2.
Remark. The above inequalities should be understood in the sense of distributions. PROPOSITION ]. If problem (1) has a lower solution o~ and an upper solution /3 such that a < 13 in f~, then (1) has a solution u • W2P(II) such that a < u
Proof Let p(x, u) = max( c~,min{u,/3}) for x ~ 1/; i.e.
( ~(x) p(x,u) = ,'~u (x)
u < ~(x) ~(x)
Let l(x, u) =f(p(x, u)) + h(x) and set L(x, u) = f~'l(x, s)ds. Note that there exist constants a 0, /30 > 0 such that
l(x,u) < aolsjP 1 +/3o
with a 0 < A 1
(A 1 is the first eigenvalue of - A p in W~]'P(ID). Consequently, the functional I defined by
l(u)= l / P ! l V u l P d x -
faL(x,u)dx
is coercive and lower semi-continuous. Hence the functional I has a minimum u in w0m'e(l~). If ,9I(u) is the generalized gradient of I at u then 0 • c~I(u) (see [7]). Also, following [7] we can show that
M(u) =Au - ;)J(u),
A quasilinear elliptic problem with a discontinuous nonlinearity
1117
where A , J : WO "+' ~ (W~]4') ' are the operators defined by
(.4u,u) = / I v . l e 2VuVv d x (J(u),u)
£L(x,uIdx.
=
Moreover, a J ( u ) c [l(x, u - ), l(x, u + )]. The fact that 0 ~ aI(u) implies the existence of a w ~ ? J ( u ) such that A u = w a.e. in f/. We have that w ~ L p' c(WoI'P) ', where p ' is the H61der conjugate of p. H e n c e A u ~ L p' and (Au,v)
Vu e
= (w,u)
wol+(~).
F r o m this we conclude that -Apu = w with
a.e. in ~
{l(x,u} wE[(x,u)=
(2)
ifu~0
[l(x,O-t,[l(x,O+)]ifu=O.
Let us show now that the Lebesgue measure of the set {x ~ ~ ;
u ( x ) = 0}
is zero.
Let F := { x e Xl;
u ( x ) = 0}.
F r o m (2) we have -@u~[l(x,O
i.e.
l(x,O + ) ]
),
- Apu
-
h(x)
c
a.e. in F,
[f(O-),f(O +)1.
Now, using a result in [8], we can show that a.e. i n F
-Apu=O
and-h(x)~[f(0-),f(0+)].
A s s u m e that the Lebesguc measure of the set (x e F; - h ( x )
#:f(O + )}
is strictly greater than 0.
Let & be the eigenfunction corresponding to the eigenvalue A1 such that &(x) > 0 Vx c fL Then we have 0 <
lim
l ( u + ~&) - l ( u )
e~0*
= L
E
- , x..
= - f[f
.IF
+) +h(x)]&(x)dx
<0.
1118
S.M. BOUGUIMA
This contradiction shows that meas{x ~ F; - h ( x ) ~f(O + )} = 0. Similarly, we have meas{x ~ F; - h ( x ) ¢f(O - )} = 0. Consequently, the Lebesgue measure of the set F = {x c F; - h ( x ) f(0 + )}U{x E F; - h ( x ) ¢f(O - )} is zero. We conclude that u satisfies -Apu =l(x,u)
a.e. in ~ .
Moreover, a ( x ) < u(x) <_[3(x) Vx ~ 12. In fact, since /3 is an upper solution, multiplying the differential equation in (2)by (u - / 3 ) + = max{(u - / 3 ), 0} and integrating by parts the resulting equation, we obtain that u(x) <_/3(x) Vx ~ 12. Similarly, we show that a ( x ) < u(x) V x ~ ~ . The fact that u ~ WZ'P(ll), p > n is a consequence of the regularity theory. In order to complete the proof of theorem 1, we must show that (1) has a positive upper solution, and a negative lower solution. • The following lemmas are needed. LEMMA 1. Let M ~ IR + and n ~ N. Then for each nonnegative real number d, the initial-value problem -(Iv'lP-
l
2v')
-
';lv'l P
2v'
=f(v) +M
v(O) = d
(3)
v'(O) = 0
has a positive solution v0, which is defined on a maximal interval [0, R d [. Moreover, v0 ~ C1([0, Rd[) and ~v~lP--°v~~ C1([0, Rd[). Proof First, we construct the solution in a neighborhood of O. Let rt > O. Set g(s) = f ( s ) + M if s > O, and g(O) =f(O + ) + M. On C([O, rt]; E), we consider the compact operator T defined by (Tv)(t) =d-
[(Hv)(s)Ip'-2(Hv)(s)ds,
where (Hv)(s) =s n
rng(iv(r)l)dr
fo '~
and p' denotes the H61der conjugate of p. There exists a 6 > 0 such that Ig(s+d)-g(d)l<_l
for 0_
Hence, if Iv(r) - dl _< 6, then ](Tv)(t) -dl<_
s~
< (Ig(d)l + -
n + l
r'(lg(d)l+ 1)dr
l)P'-lt P', < 8 p
-
ds
A quasilinear elliptic problem with a discontinuous nonlinearity
1119
for all t E [ 0 , r/1], where r/1 > 0 is small enough. Consequently, T has a fixed point v* in C([0, r h ]; ~). Moreover, we can choose a nonnegative n u m b e r T12 such that 0 < ~2 < ~1, and
f ' f s-" ~
' + l )1 r"(g(lv*(r)t) d r ( '
p'-'te' -fir
Vt ~ [0, rh]. In this case v * ( r ) = (Tv* Xr) > 0 for all r ~ [0, rt2]. This shows that v* is a solution of (3). T h e rest of the l e m m a follows from the theory of ordinary differential equations. • LEMMA 2. Let B be the smallest ball containing ~ , and let M ~ N + , then the p r o b l e m
-ApV=f(V) + M V> 0
in B
(4)
on OB
has a C t radial solution.
Proof Set V(x o + x ) = v(lxl), where x 0 is the center of B. Let r = [xl, and let R be the radius of B. T h e n v satisfies
[
_(Ic~,]p 2u,),_ n - l l v ' l p
2v'=f(v)+M"
t
(5)
c,(0) = d v ' ( 0 ) = 0. Following l e m m a 1, p r o b l e m (5) has a solution v, defined on a maximal interval [0, Rd[, such that v(0) = d. Moreover, v(r) > 0 for all r ~ [0, Ra[. Now, an a r g u m e n t in [6, p. 9], shows that condition ( H ) implies that, it is possible to choose d, in such a way that R a > R. This yields that v is a solution of (5). Hence, V(x) = v(lx -x0l) is a radial solution of (4). • PROPOSITION 2. P r o b l e m (1) has a positive upper solution /3.
Proof (i)
If f is b o u n d e d from below in E + , we take M > Ilhl]~. Let V be the solution of
- ~ p V = f ( V ) +M
in B,
V_> 0 on
OB.
Let /3 = V . T h e n /3 is a positive u p p e r solution of (1). (ii) If f is not b o u n d e d from below, then any /3 e N* such that f ( / 3 ) _< Ilhll~ is an u p p e r solution of (1). Similarly, we can prove the following proposition. PROPOSITION 3. P r o b l e m (1) has a negative lower solution ~. This completes the p r o o f of t h e o r e m 1.
Remarks. (1) We have assumed f discontinuous at the origin. If f is discontinuous at another point, we can reduce the p r o b l e m to a discontinuity at the origin, by translation. (2) T h e case f(0 +) < f ( 0 ) can be treated similarly.
1120
S.M. B O U G U I M A
Application. Assume 11 = B(0, R) and h = 0. Suppose f satisifes ( f l ) f is nondecreasing, (f2) f • C ( N \ { a } ; N) with f(u)-OVu•[-~,a],f(a+)=b>O, (f3) b(I/p
1)
- - a
~l],,p
1
[
>p'meas(t~)
14,11pj ] ] l/p
( f~
]p/p 1
["
, '
t where q5~ is an eigenfunction corresponding to the first eigenvalue A1 of - A p in H(~(fD, meas(.) denotes the n dimensional Lebesgue measure and p' is the H61der conjugate of p. Assume, further that ( H ) is satisfied, then we have the following. THEOREM 2. Under the above assumptions, the problem
-Au=f(u)
infl
u = 0
on c?11
(6)
has a solution u • W,]'P(tl)p > n. Moreover, u ~- 0 and meas({x ~ ti;
Proof
where
u(x)
= a}) = 0.
As in the proof of proposition 1, consider the functional
L(x,u)= j~;'l(x,s)ds and If(a(x)) l(x,s) = I f ( s ) If(~(x))
s < a(x) a ( x )
Here, a and /3 are lower and upper solutions of (6), respectively. The functional 1 has a minimizing function u ~ W~,P(11), solution of the problem - A p u = f ( u ) a.e. in 1~. CLAIM 1. There exists ~r • JR, such that l(~r&~ ) < 0.
Proof
Let l/p
O'~
I
A quasilinear elliplic problem with a discontinuous nonlineariW F r o m c o n d i t i o n s fl a n d f2, we h a v e L ( x , s ) - L ( x , a ) > _ b ( s - a ) w i t h l ( , r ~ b ~ ) = crP [
_< - -
IV~,I:,-
I V 4 , ~ I ~'
P
[L(x,~r&,
b~r
1121
L ( x , a ) = - O in ~ . T h e n
)dx
cbl + ab m e a s ( f ~ ) < O.
T h i s c o m p l e t e s the p r o o f o f the c l a i m 1. C l a i m 1 s h o w s that u ¢ (). W c p r o c e e d as in t h e p r o o f o f p r o p o s i t i o n m e a s ( { x ~ ~); u ( x ) = a}) = 0. T h i s c o m p l e t e s the p r o o f o f t h e o r e m 2. •
1 to s h o w t h a t
R e m a r k . T h e o r e m 2 is m o t i v a t e d by a similar result o f A r c o y a a n d C a l a h o r r a n o [3]. In this w o r k , t h e a u t h o r s use an a d d i t i o n a l c o n d i t i o n o n J'. In fact t h e y s u p p o s e t h a t t h e r e exist c~, / 3 > 0 such t h a t f ( s ) < cxlsl p ~ +/3 w i t h ~ < A~, w h e r e ,~ is t h e first e i g e n v a l u e o f - A p . Acknowledgements
The author wishes to thank the reieree for interesting remarks. REFERENCES
l. FONDA A., GOSSEZ J. P. & ZANOLIN F., On a nonrcsonance condition for a semilinear elliptic problem, Diff. Integral Eqns 4, 945 951 (19911. 2. EL HACHIMI A. & GOSSEZ J. P., A note on a resonance condition for a quasilinear elliptic problem, Nonlinear Analvsis (to appear). 3. ARCOYA D. & CALAH()RRANO M., Some discontinuous problems with a quasilinear operator, Scu. norm. sup. Pisa, preprint ( 1991 ). 4. BOUGUIMA S. M., A scmilinear elliptic problem with discontinuous nonlinearities, Communs appl. Nonlinear Analysis (to appear). 5. BOUCHERIF A. & BOUGUIMA S. M, Perturbation m ~t frcc boundary problem, J. math. Analysis Applic. 183, 430-437 (1994). 6. BOUGUIMA S. M. & BOUCHERIF A., A discontxnuous semilinear elliptic problem without a growth condition, Dyn. Syst. Applic. 2, 183 188 (1993). 7. CHANG K. C., Variational methods for non diflerentiablc functionals and their applications to Partial. Diff. Eq., J. math. Analysis.4pplic 80, 1112 129 (19811. 8. MORREY C. B., Multiple Integrals in the Calculu.s O/ ~:anatton~. Swinger, Berlin (1966).