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Some remarks on a quasilinear elliptic boundary value problem
.LbnLnear Andw. Theory. .Uethodr & Pruned m Great Bntain.
.-lppi,canom.Vol. 8. No. 7. pp
SOME REMARKS
Department
1984.
ON A QUASILINEAR ELLIPTIC VALUE PROBLEM
BOUNDARY
NGUYEN PHUONG CAc of Mathematics, The University of Iowa, Iowa City, Iowa 52242, U.S.A (Receioed
Key words and phrases:
LET R BE
597-709.
in revised form
20 July 1983)
Upper solutions, lower solutions. first eigenvalue.
a bounded domain in the Euclidean space Rv (N 3 2) with smooth boundary aS2.
Let
(AlI where the usual summation convention is used: if an index is repeated then summation over that index from 1 to N is implied, and a!,( . ) E C(Q),
i,j=l,...,
h( *) E L-;(Q),
N.
We assume that L is uniformly elliptic, i.e., gv> 0 such that &j(x) (I <, 2 4 C12, (A2)
f( . ) E Lp( R)
c( . ) E LP(Q),
vc E IRv.
VXEQ,
for some p > N.
(Al) and (A2) constitute the minimum hypothesis throughout the paper unless otherwise is clearly indicated. Furthermore, let b : R x Iw-, W be nonnegative and Caratheodory, i.e., for each f E W, b(. , t) : 51--s, W is measurable, and for almost all (a.a.) x c 52, b(x, .) : R + W is continuous. We assume that for some q 3 p > N we have for every I> 0: 6$(X)= spq b(x, r) E Lyn).
(1)
tr
We discuss the solvability of the boundary value problem (abbreviated Lu = cu + F(x) +f(x,
u=O
u, Vu)
to BVP in the sequel)
in S2,
onan,
(2)
in the Sobolev space W’,P(Q), where the Caratheodory R X R X RN satisfies the growth condition lfk
4 r7)l c G,
r)1711Z-~
function
f(x, t, n) defined
on
(3)
for a constant E, 0 6 ES 1. Special cases of (2) have been considered by other authors as prototypes on which various general existence theories for quasilinear equations can be 697
695
N. P. CAC
checked 4. One and a b(x, r)$
out, see for example [5]. Section 3, example 1 where f(x, r, rl) = $; also [l], lemma way of proving the solvability of (2) is to show the existence of an upper solution v, lower solution q with Q:G v/. This approach is adopted in [5] with f‘(x. r, 7) = : b is continuous in all of its variables. the function 191defined in (1) belongs to L”(R) and c( .) is constant. Recently, Pohoiaev [lo] proved that the method of upper and lower solutions is still applicable if 0l E Lq(Q) with 4 2 p > N provided that E = ;y/q. Except for very simple cases (see, e.g. [lo]), with E = 0 the construction of upper and lower solutions in 12.5) under suitable conditions relies on the positive eigenfunction corresponding to the first eigenvalue of the elliptic operator L. In this paper we describe alternative constructions not based on such an eigenfunction. We then obtain a number of results on the solvability of (2) depending on c( .) and F( .). From these solvability results we derive a lower bound for the first eigenvalue of -A with Dirichlet boundary condition. For the definition of upper and lower solutions as well as the notations for Sobolev’s spaces used here, we refer to [lo].
Definition. Let Ai = sup {A 2 0 : Lu - Au = 1
c(. ) is bounded
from above
(i) in the case /Ii < x, there is a number (ii) in the case Ai = =, c( .) is essentially We shall need
solution
in
for any finite t > 1).
w 2.‘( Q) n W;,‘(Q) We say that the function
has a nonnegatire
(4)
away from Ai if
CY> 0 such that C(X) G & - N for a.a. x E Q; bounded from above on Q.
the following
THEOREM 0. (i) J.i is finite if and only if the spectrum is the first eigenvalue of L.
of L is nonempty.
If J.i is finite then it
(ii) Suppose that c( .) is bounded from the above away from Ai and c( .) E L’(Q) with s = N/2 if 1 < r < 1V/2, s > N/2 if r = N/2 and s = r if r > N/2. Then for every H( .) E Lr(S2), H 5 0 in R, the equation
Lu = cu + H has a unique solution of H such that
u E I&“~‘(52) f’ W,‘3’(S2), u 2 0 in R and there is a constant
y independent
(5)
~I&+QJ(Q~s r]iHlir~n,.
This result is proved in [3]. In the case J.i < = and c E L%(Q), it can be found in [7] with an entirely different proof. See also [4]. We shall assume throughout the rest of the paper, unless otherwise is clearly indicated, that c( .) is bounded from above away from J.i and that the function f satisfies the Lipschitz condition 643) for a.a. x ER,
lf(x, r, rl) - f(X, t, 5) V(r, E, 7) E 8” x W’v where
I s bo(x, t, E, rl) *117 - El
bo(x, r, c, r]) is a Caratheodory
function
such that
Some remarks on a quasilinear elliptic boundary value problem
for any
699
I> 0. sup {b”(. , t. :, 17)~/t, s 1. izi s I, 1q) s I} E L.p(Q)
We first prove
a result
that motivates
function
p > A’.
our later investigation:
, . . .) satisfies (3) where the nonnegative
THEOREM 1. Let E = N/p withp
Caratheodory
with
> hi. Suppose thatf(. b satisfies the condition
with 0( .) E LP(R). 6 2 0 and c(x) + 6 G j? for some constant p< 0 for a.a. x E Q. V’rE 2. Then equation (2) has a solution u E W’+(Q) n Wd,P(R) for any F( .) E L”(O), Proof. We first construct Sobolev imbedding theorem
a nonnegative we have
upper
solution
i# of the BVP (2). By (5) and the
here and in the sequel ~~(i = 1.2,. . .) denotes a constant which may not always be the same. Since F, 8 E LP(Q) we can find a number kl > 0 sufficiently large such that :&t& where xk, is the characteristic
function
+ @)jllp(n) s 1,
of the set
Rk, = {x E n/F(x) Let ui be the nonnegative
function
in w’,p(Q)
nIY,‘J’(Q)
LUl = (C f 6)~
Since for a.a. x E R, c(x) + 6 G /3 for some sufficiently large such that (c(x) + 6)kz+
+ e(x) 2 k,}.
-I- ,Q,(Fi-
constant
klsO
satisfying 0).
,0 < 0, we can find a number
fora.a.
xEQ.
k: > 0
(7)
Then it can be verified that v = u1 + kz is a nonnegative upper solution of the BVP (2). In fact, on the closure S=Jof R, t# > 0. Furthermore, in Q - &, because of (7) we have for a.a. x,
Ly = Lu, = (c + 6)rJ ,s (c + 6)(u1+
kz) + k,
qc+6)7p+F+ez=q+F+b(x,yJ) a cy + F f b(st r#yI’-Ea because
cv + F+ f(x? v, ty),
[Vql = lVui/ < 1 by (6). On the other Lip=
Lu1=
hand,
(c + 6)Ulf
on Rk, we have F+
8.
700
N. P. CAC
Since c(x) - 6 < 0 on Q, L~>(c’6)(ul+k?)+F;e~(c+6)~~
F-e
3 cy + F + b(x, y) ~Vwi’-‘a We now construct we have
Therefore
a nonpositive
lower solution
it suffices to find a nonpositive
cy -I- F + f(x, I,U,Gy). q of the BVP (2). We note
lower solution
Lu = cu + F + [-e(x)
that for r < 0
q of the BVP
+ &L]~CU~‘-~
in 9,
(8)
onan.
u=O We fix ml > 0 so large that if
= {X E R / F(x) - e(x) s -m,}
k, and h,, is the characteristic
function
of ii,,,, then we have
~ll%&O where y2 is the constant M’dlp(Q) such that
involved
in (6).
- F))IiLP(oJ Let
G 1,
w1 be the
unique
function
in R”.f’(Q)
n
LWi = (c + 6)Wi + ;im,(F - e). Then
by theorem
0, wi s 0. Next we fix ml > 0 so large that (c(x) + 6)m, + ml < 0
Let q(x)
= wi(x)
- mz. Then
for a.a.
x E R
q(x) < 0 on 0. If x E R - di,, we have
Lg, = L(wl - m?) = LwI = (c + 6)wl S (c + 6)(wl-
ml) - ml.
Lcp~(c+6)q+F-e=cq+F-e+6~ L+ because
-e(
cq + F+
p-e+
6q]jvg,j’-~,
.) + 6~( .) < 0 and by (6) and the choice
Thus Q, is a nonpositive lower solution of the BVP n conclude that the BVP (2) has a solution.
of ml:
(8). By theorem
3.1 of [lo] we then
Note. In the case L = -A, 6(.) = 1, E= 0 and c(e) is constant, theorem 1 is a direct consequence of the general theory established in [5] (see [S], theorem 4.5). However, as noted above, [5] relies essentially on the existence of a positive eigenfunction corresponding to the first eigenvalue of L in constructing upper and lower solutions.
Soms remarks on a quasilinear
elliptic
boundary
value problem
701
In the next theorem we shall see that if, instead of requiring that c( .) be bounded from above away from 0 as in theorem 1, we only require that it is bounded from above away from Ai, then to insure the solvability of the BVP (2), F( .) has to be restricted to an order interval of the space Lq(Q). 2. Suppose that f satisfies (3) where the function @(.) defined in (1) belongs to Lq(Q) for some 4 Sp > N, that E = ‘v/q and c( .) is bounded from above away from Ai. Then there exists a function in Lq(Q)r( .) > 0 on S2 such that if IF(x)1 < r((x) for a.a. p E Q then the BVP (2) is solvable in W’+(Q). THEOREM
Proof. Let, as in (l),
We first show that if t( .) > 0 is sufficiently small, then the BVP in Q,
Lu = cu + t(x) + 81(,~)IVul’-~ u=O
onaS2,
(9)
has a nonnegative upper solution v with 1~1s 1 on S2. Let u E W’+‘(Q) n HJ’~~~(Q)be the nonnegative solution of Lu = cu + 1 + e,(x). Then we know that there are constants yl and yl such that m;x luC~)lc ~1, ,uu is a nonnegative
rnp iVu(x)l s y2.
upper solution of (9) with ,UUG 1 if 0 < p < llyi and
LU 2 cu + i + e,(,+PpU12-t
(10)
Let us further require that for some k > 1, p satisfies the additional condition ,$-Ey”z--E< -1 k’ i.e.
1
1 1 ,U= min ;’ ki,(‘-E)y$Z-E)/(I-$ := duo. Then (10) is satisfied if r((x)4,4[l+(1-j!)BL(~)]
fora.a.xER.
It is not difficult to see that IJJ= ,hu is a nonnegative upper solution of the BVP (2) if IF(X)/ < r(x) for a.a. x E R. Furthermore, p?= - $J is a nonpositive lower solution of the BVP (2) in that situation. In fact, since v, is an upper solution of the BVP (9), LIZ
cv + t+ e1(x)pv(2-t
702
Ii. P. C.AC
Hence Lqs bearing
cr,c - t-
81(x)jCq;/2-Ec
cp: T F-
b(x, q)sTp2-E
in mind that 1q(x) / c 1 on 0. Thus Lg: s cy, + F-t
We deduce a solution.
from theorem n
3.1 in conjunction
f(x, q, =i’q:).
with corollary
1.3 in [lo] that the BVP (2) has
Nore. (i) If in addition to the hypotheses concerning the functions f and c of theorem 2 we assume that f(x, I,7)3 0 for a.a. x E Q, V(t, 7) E IR x R,’ then for any F(.) E LP(Q) the solution of the BVP Lu = cu - F-
in R,
u=O
ondQ,
where F-(x) = max[ - F(x), 0] is a nonpositive lower solution of the BVP (2). Thus in this case the BVP (2) is solvable if F(x) s s(x) for a.a. x E 51. (ii) It is not difficult to see that, with obvious modification in the case of theorem 1, theorems 1 and 2 remain valid if instead of (3) fsatisfies
if(x.t, q)l =Sb(x.
t)[l T ~q1’-‘]
fora.a.x
E Q.
V(t, n) E R x 2,,v.
We now give a simple description of r(.) in the case f(~, t, n) = n2 and from it we shall deduce a lower bound for the first eigenvalue of -A subject to 0-Dirichlet boundary condition. We note that for ,u > 0, ;LU is an upper solution of Lu = cu + F + iVuj2-E u=O if u is an upper
solution
in R,
onaC?,
(11)
of LLl = cu
+ t
+ ,,$-Ep(Z-E
in R,
,U
u=o onm, where t= IIF]/ L=(o) and F(X) = max[F(x), W’.P(Q) fl w&P(Q) of the equation
(12)
O]. Let u be the unique
Lu = cu + 1 and let K = rn? This u is an upper
solution
/VU].
of (12) if 4 + Ui-EKZ-Ec .u ’
1.
The function p(p) = ; +
,ul-KZ--E- 1
nonnegative
solution
in
Some remarks
on a quasilinear
elliptic
boundary
value problem
703
attains its minimum when ~1= ;ll where
and the value of that minimum is t2-E P(Ul) = -- 1 ,ut 1 - E Since we need p(,ur) 6 0, we require that l--E tS,U,-=2- E
l--E 1 2 - &(I - #Q-E)K
r’/(’ - @
or $1 - E)iC- d
s
2
1 -
E
r< i 2_c ts
(1
1
1-E
_
E(l
-
-
,)li(2-4K
(2-d/(1--E)
J -
&)[(2
1 l/(1 (1 - &)
-
E) K(’
,)K](‘-?)I(l-E)
- E)/( 1 - E)
:=
q).
(13)
So, if F(x) s to for a.a. x E R, then ,ulu is an upper solution of (11) and (11) is solvable. It has been pointed out in [5] that the BVP Au = F(x) + ]Vu/* in R u=O
0naQ
is not solvable if F(x) 2 Al ( = first eigenvalue of -A) for a.a. x E Q. Combining this fact with (13) in the case E = 0 (i.e., taking 4 = r: in theorem 2), c( .) = 0 we obtain THEOREM 3. Let u. be the unique function such that - Au0 = 1 in R, u. = 0 on aR, and K =max ]Vucj. Then the first eigenvalue Al of -A subject to 0-Dirichlet boundary condition R satisfies
To put theorem 3 in perspective we make a digression at this point and obtain an alternative estimate for J.r. This dimension dependent estimate is better than theorem 3 for lower dimensions but becomes worse than it is for higher dimensions. For dimension N = 2 it can be deduced from [8,9]; see also [12]. THEOREM4. With the same notations as in theorem 3 we have Al > ;+.
Proof. Let q1 be the positive,
normalized
eigenfunction
of -A corresponding
to its first
70-l
N. P. C.AC
eigenvalue
?.1. Since
-Au0
= 1 we have
J-1 LQ~TJ~dx = iR
‘F~dx.
Therefore (1-t) We now make the crucial observation (which for N = 2 was used by Payne [8,9] who attributed it to Miranda; see also [ 121) that the expression E(x) = 1Vu~(x) (’ + ,ULIO(X)is subharmonic with u = 2/N. Then by the maximum principle we obtain IVUOI’+
iuo dR
c max jVu$
= max Itr~~l~ =
R
K'.
and hence
iv
max rlo < -
2
Q
From
K'.
(13
(14) and (15) we deduce 2 1
/II >
-7.
N
K-
To show that E(x) =~VuO(x)l’ -I- JUI~(X) is subharmonic computation: (Di = a/&X,. i = 1, . . . , iv)
for u = 2/N we proceed
by direct
,v
DiiE= 2,zl{(D~jUo)' + D,u~.Dllit~o} + ,UD,iUo. Taking
into account
the fact that AE = 2 ii1
Since AK,J= -1, Suppose
AEaOif
now that instead If+,
,uL2/N.
-Au0
= 1 we have
(D,jUo)' - ,U 3 2 lil (D~~LLo)' - U. n
of (3) f satisfies
the condition
t, rj) j G 6(x)1771 for a.a.
x E Q,
V(t, r]) E R X Rv,
(16)
of where .!$ .) ELJ’(R), d 3 0, p > N. The correspondin g BVP (2) is not in the framework the problems discussed in theorems 1 and 2 because there E = N/q with q 3 p > N. We study it briefly here since the technique of constructing upper and lower solutions for the BVP (2) as described in the proof of theorem 2 and in the computation leading to (13) is not valid when E= 1. Despite the fact that under condition (16) the BVP (2) is not of the type considered in [lo], it can still be proved that if it has an upper solution v and a lower solution CQwith q s r/j then
Some remarks on a quasilinear elliptic boundary value problem
it has a solution
u with q G u 6 r+ In fact, it suffices
exists a constant
y1 depending
on the coefficients
to show that if rn?
705
lui GM
then there
of L, c, F, d and M such that
m,ax IVuj s ;I:
(17)
for any solution u of the BVP (2). because it can be seen that the other steps in the proof of solvability in [lo] (i.e. lemma 3.1 and lemma 3.2) can be carried through with little modification. But estimate (17) is an immediate consequence of well-known results on linear partial differential equations (cf., e.g. [6], lemma 11.1, page 192) because equation (2) can be written Lu = c(x)u + F(x) + CO&) $,
(18)
1
wherefori=l,...,N, 04(x) = f(x, U, Vu) . /VU(X) /-‘g
if jVu(x) / f 0,
(19)
if jVu(x) / = 0.
(20)
‘ q(x) Since f(. , -) satisfies
= 0
(16) we have II4
* >IlLw2) s
IN *%Yn~~
i=l,...,N.
Therefore theorem 1 is applicable: If f(. , . , *) satisfies (16) and c(x) G &u< 0 for a.a. x E R for some number ?Xthen the BVP (2) is solvable for any F( .) E LP(Q). Actually, we can improve this result and prove 5. Suppose that f( . , . , .) satisfies (16) and c(x) G 0 for a.a. x E R. Then (2) has a solution in W’*p(Q) nw,‘.p(Q) for any F( .) E LP(S2). We first establish the following
THEOREM
LEMMA.
Suppose
the BVP
that d( .) E Lp(Q) (p > A’) and c(x) < 0 for a.a. x E R, then the BVP Lu = c(x)u + F(x) + d(x) IVul u=O
in R,
(21)
onan,
has a unique solution u E w*J’(Q) fl W,‘,p(Q) f or any F( *) E LP(Q). The conclusion if in the equation above h(x)]Vul is replaced by - d(x)/Vul. Proof of the lemma. To prove Letting w = u1 - uz we have
uniqueness
suppose
that there
are two solutions
still holds
ul and 4.
Lw = cw + b(x)[lVu*l - jVuzl] 3 cw - d(x)~Vwj or Lw 2 cw - f&(x) g I
where
oi(-),
i = 1,. . . , N,
are
defined
as in (19),
(20)
with
f(x, r, r]) = 8(x)/171. Since
706
N. P. CAC
wt( . ) E LP(Q). i = 1,. . A’. the maximum principle in the space W’+(Q) (cf., e.g. [4,7]) gives w 3 0, i.e., u1 a u:. on R. Interchanging ul and u2 we see that rll C U: on Q also. To prove solvability, for any aE [O, I] consider the BVP Lu = cu T F(x) A d(x)lT’n)
in R,
u=O
0naR.
(22)
Writing the equation in the form (18) we see (by [4], theorem 2; [7]. theorem 1.1) that there is a constant y depending only on the coefficients of L, on the LP(R)-norms of c( .) and d(e) such that every solution u of the BVP (22) satisfies the apriori estimate I]+~.P(*) c y.
(23)
On the other hand, for every u E C’(G) we know that the BVP Lu = cu f F(x)
+ d(x)lVu(x)/
u=O
in R,
ondR
has a unique solution (again by our theorem 0 above or [4,7]) which we denote by So and there is a constant y1 independent of u such that IlS~ll@P(R)=5 Y1(lll~ulllUG2, +
llav2))~
Therefore the mapping S of C,(Q) into itself is compact. It is not difficult to see that it is also continuous. From (23) we deduce, by using Leray-Schauder’s theorem, that S has a fixed point. This is the solution of the BVP (21). From the proof above it is not difficult to see that in (21) we replace d(x)]Vu( by -d(x)]Vu] then the existence and uniqueness of a solution still hold. n Proof of theorem W,‘J’(Q) such that
5. By the lemma Lp=q--
Furthermore,
just proved
F--t$x)iVc+ccp+
there
is (a unique)
v, EW’J’(R)
n
F+f(x,q,Vq).
since Lq-cy=F-+
&x)]V+=O
and c(x) s 0 a.a. x E 52, the maximum principle gives J$ b 0. Thus q is a nonnegative solution of the BVP (2). Similarly, there is (a unique) Q,E H’*J’(Q) rl w,lP(Q) such that
upper
Lrp = cq - F- - d(x)]Vg;/ G ccp + F + f (x, Q?,Vq).
Again, by the maximum principle, Q,G 0. Thus 97is a nonpositive (2). We then deduce that the BVP (2) is solvable. l
lower solution of the BVP
THEOREM 6. Suppose that f(. , . , -) satisfies (16) with b(v) E L”(R). F( .) E LP(Q) even if c( .) assumes positive values. To see this we prove
Suppose
also that
6. Suppose that f(. , . , .) satisfies (16) with b(e) E L”(R). Suppose also that c( *) E Lx(R) and there is a number & such that c(x) < &< Ai - ]]h( *)]]L-co,fl for a.a. x E R where Al is the first eigenvalue of -A subject to 0-Dirichlet boundary condition on THEOREM
Someremarks
on a quasihear
aQ. Then for every F(a) ELP(R),
elliptic
boundary
value problem
707
p ,Z 2, the BVP
-Au = cu f F(x) +f(x, u, Cu)
u=o
in 52,
0naQ,
has a soiution u E w’.p(Q). Proof. The operator u-+ Au = -Au - cu -f (x, K, vu) from 14’h,2(Q) into its dual H’-‘.?(Q) is pseudo-monotone (see, e.g. [13], Chapter 2, Section 2). Therefore ‘to prove the theorem it suffices to show that it is coercive. With (e, .) denoting the pairing between w;,‘(Q) and its dual, we have for any u E W$2(S2): (Au, u) = i, (1Vu/’ - cu* - f(x, u, Vu)u}dx,
(Au,ub+Vulza2 - II~(~)hdVul lull dx,
we deduce from the last inequality that (Au, u) 2
1 -t-!-j
jn~~u~‘dx
for Vu E W$‘(Q). Thus under the hypotheses of the theorem the operator A is coercive and therefore our BVP is solvable in W;.‘(Q) for every F( .) E LP(Q), p s 2. Using the familiar bootstrap argument we can deduce that its solution actually belongs to W’~“(Q). n Note. If the diameter of Q is sufficiently small, we shall have Ai - II@0) IIL=(oJV’& > 0 and under the hypotheses of theorem 6, c( .) can assume positive values. In view of theorem 6 it is natural to ask whether the BVP considered there is still solvable, at least for suitably chosen F( .), even if c is no longer smaller than A1- //a(. )]/L=(Q)d/;i; on R. We provide a partial answer to this question by proving the following theorem for the case f(x, r>II) = 1111. THEOREM 7. Let R = sup{lxl . x E 52). Suppose that c( .) is bounded from above away from At. Then the BVP Lu = cu + F(x) f [Vu I in R, is solvable if c G l/R and F =S2( N - R) on R. Before proving the theorem we make the following
u=O
ortaR,
705
N. P. c\c
,%‘ote. This theorem is of interest oniy if both of the following conditions are met: (i) N> R. In fact, if F s 0 on R then our BVP is solvable if c( .) is merely bounded above away from J.i: For in that case 0 is an upper solution of it and a nonpositive solution is provided by the solution of the BVP Lu = cu f F (ii) l/R 2 AI - fl,
otherwise
in 52,
it is already
u=O
contained
from lower
0nJQ. in theorem
6.
We give an example for which both of these conditions are satisfied. Take N = 2. The first eigenvalue of -A subject to 0-Dirichlet boundary condition on the disc of radius R is [ll] A, = (2.404825556)’ R’ Then
if we take R = 1.95 < N = 2 we certainly
’
shall have Ai > l/R 3 J.1 - G.
Proof of theorem 7. As pointed out in the discussion following theorem 4 and preceding theorem 5, it suffices to show that the BVP considered has an upper solution v, a lower solution QI with q c y on R. Since c is bounded from above away from Ai, we can find a nonpositive lower solution q by taking the solution of the BVP Lu = cu - FWe shall verify that under solution. Let 1x1 = r,
the conditions c-(x)
computations
u=O
of the theorem,
= max [c(x), 01.
E = Ilcf( .)ljY(cq, Elementary
in R,
F-(x)
on&?. q(x) = R’ - /xi’ is a positive = max [F(x), 01,
r: = llF_( .)iiL’(R,.
give:
cy+F+~Vt/,l~E(R’-?)+~+2r
forr
-AI& = 2N. Let p(r) = E(R’ - ;) value is
upper
+ p + 2r. The local maximum
of p is attained
when r = l/c : = ro and its
p(ro) = f + CR’ + i? On the other
hand, p(O) = ?R’ + F,
Therefore if e < l/R, the maximum solution is 2N 3 2R + l? n
p(R)
= 2R + I?
of p(r) for 0 < r =s R is 2R + F and by (24), y is an upper
Acknowledgements-I wish to thank the referee for raising thoughtful questions that led to improvements in the paper and to Professor Michael G. Crandall to whom this paper was submitted. H. F. Weinberger for I am grateful to Professor J. B. Serrin for sendin, 0 me a copy of [12] and to Professor informing me that in the case N = 2 a better estimate than theorem 3 (namely, theorem 4 with N = 2) can be deduced from [8, 91. iMy thanks also go to Professor Catherine Bandle for discussions about bounds for the eigenvalue 1,.
Some remarks
on a quastlinear
ellipnc
boundary
value problem
709
REFERENCES 1. A.LIANS H. & CRXND.+LL M. Cr.. On some existence theorems for semi-linear elliptic equations, Indiana L’nio. .Llarh. J. 27, 779-790 (1978). 2. BR~ZIS H. & NIRENBERG L.. Characterizarions of the ranges of some nonlinear operators and apphcations to boundary value problems. rlnnali Scu. norm. sup. Pisn Serie IV. V. 225-326 (1978). 3. C.X NGtiY’& P.. On an elliptic boundary value problem not in divergence form, Proc. rim. marh. Sot. 88. 17-52 (1983) 4. CHICCO M., Solvability of the Dirichiet problem in H’+‘(Q) for a class of linear second order elliptic partial differential equations. Boll. tin. mar. iral. 4. 371387 (19711. 5. QZDAN J. & KRAMER R.. Invariant criteria for ehstence of solutions to second order quasilinear elliptic equations, Commrcns pure uppl. Math. 31, 619-645 (1978). 6. LADYZHENSKAYA 0. A. & UR..XL’TSEVAiv.. Linear and Quasilinenr Elliptic Equafions, Academic Press. New York (1968). 7. LIONS P. L., Problemes elliptiques du Z&me ordre non sons forme divergence, Proc. R. Sot. Edinb. 84& 263271 (1979). 8. PAYNE L. E., New isoperimetric inequalities for eigenvalues and other physical quantities, Communs pure appl. Math. 9. 531-542 (1956). 9. PAYNE L. E., Bounds for the maximum stress in the St. Venant torsion problem, Indian J. .Mech. Jfarh. Special Issue 6. 51-59 (1968). 10. POHO~AEV S. f.. 6; equations of the form Au = f(x, u, Du),~,,$Marh. USSR Sbornik,,41, 269-280 (1982). Am. math. Sot. Transl. 11. OLIVER F. (ed.). Royal Society Mathemarical Tables. Vol. 7, Bessel Funcrions. Part III, Zeros and Associared V&es, Cambridge Univ. Press (1960). 12. WEINBERGER H. F. & SERRIN J. B., Optimal shapes for brittle beams under torsion, in Complex Analysis and Applications: VOL.dedicated to I. N. Vekua. Nauk, Moscow, pp. 88-91 (1978). 13. LIONS J. L., Quelques mPthodes de rholurion des problemes aux limires non lineaires, Dunod. Gauthier-Villars, Paris (1969).
.-lppi,canom.Vol. 8. No. 7. pp
SOME REMARKS
Department
1984.
ON A QUASILINEAR ELLIPTIC VALUE PROBLEM
BOUNDARY
NGUYEN PHUONG CAc of Mathematics, The University of Iowa, Iowa City, Iowa 52242, U.S.A (Receioed
Key words and phrases:
LET R BE
597-709.
in revised form
20 July 1983)
Upper solutions, lower solutions. first eigenvalue.
a bounded domain in the Euclidean space Rv (N 3 2) with smooth boundary aS2.
Let
(AlI where the usual summation convention is used: if an index is repeated then summation over that index from 1 to N is implied, and a!,( . ) E C(Q),
i,j=l,...,
h( *) E L-;(Q),
N.
We assume that L is uniformly elliptic, i.e., gv> 0 such that &j(x) (I <, 2 4 C12, (A2)
f( . ) E Lp( R)
c( . ) E LP(Q),
vc E IRv.
VXEQ,
for some p > N.
(Al) and (A2) constitute the minimum hypothesis throughout the paper unless otherwise is clearly indicated. Furthermore, let b : R x Iw-, W be nonnegative and Caratheodory, i.e., for each f E W, b(. , t) : 51--s, W is measurable, and for almost all (a.a.) x c 52, b(x, .) : R + W is continuous. We assume that for some q 3 p > N we have for every I> 0: 6$(X)= spq b(x, r) E Lyn).
(1)
tr
We discuss the solvability of the boundary value problem (abbreviated Lu = cu + F(x) +f(x,
u=O
u, Vu)
to BVP in the sequel)
in S2,
onan,
(2)
in the Sobolev space W’,P(Q), where the Caratheodory R X R X RN satisfies the growth condition lfk
4 r7)l c G,
r)1711Z-~
function
f(x, t, n) defined
on
(3)
for a constant E, 0 6 ES 1. Special cases of (2) have been considered by other authors as prototypes on which various general existence theories for quasilinear equations can be 697
695
N. P. CAC
checked 4. One and a b(x, r)$
out, see for example [5]. Section 3, example 1 where f(x, r, rl) = $; also [l], lemma way of proving the solvability of (2) is to show the existence of an upper solution v, lower solution q with Q:G v/. This approach is adopted in [5] with f‘(x. r, 7) = : b is continuous in all of its variables. the function 191defined in (1) belongs to L”(R) and c( .) is constant. Recently, Pohoiaev [lo] proved that the method of upper and lower solutions is still applicable if 0l E Lq(Q) with 4 2 p > N provided that E = ;y/q. Except for very simple cases (see, e.g. [lo]), with E = 0 the construction of upper and lower solutions in 12.5) under suitable conditions relies on the positive eigenfunction corresponding to the first eigenvalue of the elliptic operator L. In this paper we describe alternative constructions not based on such an eigenfunction. We then obtain a number of results on the solvability of (2) depending on c( .) and F( .). From these solvability results we derive a lower bound for the first eigenvalue of -A with Dirichlet boundary condition. For the definition of upper and lower solutions as well as the notations for Sobolev’s spaces used here, we refer to [lo].
Definition. Let Ai = sup {A 2 0 : Lu - Au = 1
c(. ) is bounded
from above
(i) in the case /Ii < x, there is a number (ii) in the case Ai = =, c( .) is essentially We shall need
solution
in
for any finite t > 1).
w 2.‘( Q) n W;,‘(Q) We say that the function
has a nonnegatire
(4)
away from Ai if
CY> 0 such that C(X) G & - N for a.a. x E Q; bounded from above on Q.
the following
THEOREM 0. (i) J.i is finite if and only if the spectrum is the first eigenvalue of L.
of L is nonempty.
If J.i is finite then it
(ii) Suppose that c( .) is bounded from the above away from Ai and c( .) E L’(Q) with s = N/2 if 1 < r < 1V/2, s > N/2 if r = N/2 and s = r if r > N/2. Then for every H( .) E Lr(S2), H 5 0 in R, the equation
Lu = cu + H has a unique solution of H such that
u E I&“~‘(52) f’ W,‘3’(S2), u 2 0 in R and there is a constant
y independent
(5)
~I&+QJ(Q~s r]iHlir~n,.
This result is proved in [3]. In the case J.i < = and c E L%(Q), it can be found in [7] with an entirely different proof. See also [4]. We shall assume throughout the rest of the paper, unless otherwise is clearly indicated, that c( .) is bounded from above away from J.i and that the function f satisfies the Lipschitz condition 643) for a.a. x ER,
lf(x, r, rl) - f(X, t, 5) V(r, E, 7) E 8” x W’v where
I s bo(x, t, E, rl) *117 - El
bo(x, r, c, r]) is a Caratheodory
function
such that
Some remarks on a quasilinear elliptic boundary value problem
for any
699
I> 0. sup {b”(. , t. :, 17)~/t, s 1. izi s I, 1q) s I} E L.p(Q)
We first prove
a result
that motivates
function
p > A’.
our later investigation:
, . . .) satisfies (3) where the nonnegative
THEOREM 1. Let E = N/p withp
Caratheodory
with
> hi. Suppose thatf(. b satisfies the condition
with 0( .) E LP(R). 6 2 0 and c(x) + 6 G j? for some constant p< 0 for a.a. x E Q. V’rE 2. Then equation (2) has a solution u E W’+(Q) n Wd,P(R) for any F( .) E L”(O), Proof. We first construct Sobolev imbedding theorem
a nonnegative we have
upper
solution
i# of the BVP (2). By (5) and the
here and in the sequel ~~(i = 1.2,. . .) denotes a constant which may not always be the same. Since F, 8 E LP(Q) we can find a number kl > 0 sufficiently large such that :&t& where xk, is the characteristic
function
+ @)jllp(n) s 1,
of the set
Rk, = {x E n/F(x) Let ui be the nonnegative
function
in w’,p(Q)
nIY,‘J’(Q)
LUl = (C f 6)~
Since for a.a. x E R, c(x) + 6 G /3 for some sufficiently large such that (c(x) + 6)kz+
+ e(x) 2 k,}.
-I- ,Q,(Fi-
constant
klsO
satisfying 0).
,0 < 0, we can find a number
fora.a.
xEQ.
k: > 0
(7)
Then it can be verified that v = u1 + kz is a nonnegative upper solution of the BVP (2). In fact, on the closure S=Jof R, t# > 0. Furthermore, in Q - &, because of (7) we have for a.a. x,
Ly = Lu, = (c + 6)rJ ,s (c + 6)(u1+
kz) + k,
qc+6)7p+F+ez=q+F+b(x,yJ) a cy + F f b(st r#yI’-Ea because
cv + F+ f(x? v, ty),
[Vql = lVui/ < 1 by (6). On the other Lip=
Lu1=
hand,
(c + 6)Ulf
on Rk, we have F+
8.
700
N. P. CAC
Since c(x) - 6 < 0 on Q, L~>(c’6)(ul+k?)+F;e~(c+6)~~
F-e
3 cy + F + b(x, y) ~Vwi’-‘a We now construct we have
Therefore
a nonpositive
lower solution
it suffices to find a nonpositive
cy -I- F + f(x, I,U,Gy). q of the BVP (2). We note
lower solution
Lu = cu + F + [-e(x)
that for r < 0
q of the BVP
+ &L]~CU~‘-~
in 9,
(8)
onan.
u=O We fix ml > 0 so large that if
= {X E R / F(x) - e(x) s -m,}
k, and h,, is the characteristic
function
of ii,,,, then we have
~ll%&O where y2 is the constant M’dlp(Q) such that
involved
in (6).
- F))IiLP(oJ Let
G 1,
w1 be the
unique
function
in R”.f’(Q)
n
LWi = (c + 6)Wi + ;im,(F - e). Then
by theorem
0, wi s 0. Next we fix ml > 0 so large that (c(x) + 6)m, + ml < 0
Let q(x)
= wi(x)
- mz. Then
for a.a.
x E R
q(x) < 0 on 0. If x E R - di,, we have
Lg, = L(wl - m?) = LwI = (c + 6)wl S (c + 6)(wl-
ml) - ml.
Lcp~(c+6)q+F-e=cq+F-e+6~ L+ because
-e(
cq + F+
p-e+
6q]jvg,j’-~,
.) + 6~( .) < 0 and by (6) and the choice
Thus Q, is a nonpositive lower solution of the BVP n conclude that the BVP (2) has a solution.
of ml:
(8). By theorem
3.1 of [lo] we then
Note. In the case L = -A, 6(.) = 1, E= 0 and c(e) is constant, theorem 1 is a direct consequence of the general theory established in [5] (see [S], theorem 4.5). However, as noted above, [5] relies essentially on the existence of a positive eigenfunction corresponding to the first eigenvalue of L in constructing upper and lower solutions.
Soms remarks on a quasilinear
elliptic
boundary
value problem
701
In the next theorem we shall see that if, instead of requiring that c( .) be bounded from above away from 0 as in theorem 1, we only require that it is bounded from above away from Ai, then to insure the solvability of the BVP (2), F( .) has to be restricted to an order interval of the space Lq(Q). 2. Suppose that f satisfies (3) where the function @(.) defined in (1) belongs to Lq(Q) for some 4 Sp > N, that E = ‘v/q and c( .) is bounded from above away from Ai. Then there exists a function in Lq(Q)r( .) > 0 on S2 such that if IF(x)1 < r((x) for a.a. p E Q then the BVP (2) is solvable in W’+(Q). THEOREM
Proof. Let, as in (l),
We first show that if t( .) > 0 is sufficiently small, then the BVP in Q,
Lu = cu + t(x) + 81(,~)IVul’-~ u=O
onaS2,
(9)
has a nonnegative upper solution v with 1~1s 1 on S2. Let u E W’+‘(Q) n HJ’~~~(Q)be the nonnegative solution of Lu = cu + 1 + e,(x). Then we know that there are constants yl and yl such that m;x luC~)lc ~1, ,uu is a nonnegative
rnp iVu(x)l s y2.
upper solution of (9) with ,UUG 1 if 0 < p < llyi and
LU 2 cu + i + e,(,+PpU12-t
(10)
Let us further require that for some k > 1, p satisfies the additional condition ,$-Ey”z--E< -1 k’ i.e.
1
1 1 ,U= min ;’ ki,(‘-E)y$Z-E)/(I-$ := duo. Then (10) is satisfied if r((x)4,4[l+(1-j!)BL(~)]
fora.a.xER.
It is not difficult to see that IJJ= ,hu is a nonnegative upper solution of the BVP (2) if IF(X)/ < r(x) for a.a. x E R. Furthermore, p?= - $J is a nonpositive lower solution of the BVP (2) in that situation. In fact, since v, is an upper solution of the BVP (9), LIZ
cv + t+ e1(x)pv(2-t
702
Ii. P. C.AC
Hence Lqs bearing
cr,c - t-
81(x)jCq;/2-Ec
cp: T F-
b(x, q)sTp2-E
in mind that 1q(x) / c 1 on 0. Thus Lg: s cy, + F-t
We deduce a solution.
from theorem n
3.1 in conjunction
f(x, q, =i’q:).
with corollary
1.3 in [lo] that the BVP (2) has
Nore. (i) If in addition to the hypotheses concerning the functions f and c of theorem 2 we assume that f(x, I,7)3 0 for a.a. x E Q, V(t, 7) E IR x R,’ then for any F(.) E LP(Q) the solution of the BVP Lu = cu - F-
in R,
u=O
ondQ,
where F-(x) = max[ - F(x), 0] is a nonpositive lower solution of the BVP (2). Thus in this case the BVP (2) is solvable if F(x) s s(x) for a.a. x E 51. (ii) It is not difficult to see that, with obvious modification in the case of theorem 1, theorems 1 and 2 remain valid if instead of (3) fsatisfies
if(x.t, q)l =Sb(x.
t)[l T ~q1’-‘]
fora.a.x
E Q.
V(t, n) E R x 2,,v.
We now give a simple description of r(.) in the case f(~, t, n) = n2 and from it we shall deduce a lower bound for the first eigenvalue of -A subject to 0-Dirichlet boundary condition. We note that for ,u > 0, ;LU is an upper solution of Lu = cu + F + iVuj2-E u=O if u is an upper
solution
in R,
onaC?,
(11)
of LLl = cu
+ t
+ ,,$-Ep(Z-E
in R,
,U
u=o onm, where t= IIF]/ L=(o) and F(X) = max[F(x), W’.P(Q) fl w&P(Q) of the equation
(12)
O]. Let u be the unique
Lu = cu + 1 and let K = rn? This u is an upper
solution
/VU].
of (12) if 4 + Ui-EKZ-Ec .u ’
1.
The function p(p) = ; +
,ul-KZ--E- 1
nonnegative
solution
in
Some remarks
on a quasilinear
elliptic
boundary
value problem
703
attains its minimum when ~1= ;ll where
and the value of that minimum is t2-E P(Ul) = -- 1 ,ut 1 - E Since we need p(,ur) 6 0, we require that l--E tS,U,-=2- E
l--E 1 2 - &(I - #Q-E)K
r’/(’ - @
or $1 - E)iC- d
s
2
1 -
E
r< i 2_c ts
(1
1
1-E
_
E(l
-
-
,)li(2-4K
(2-d/(1--E)
J -
&)[(2
1 l/(1 (1 - &)
-
E) K(’
,)K](‘-?)I(l-E)
- E)/( 1 - E)
:=
q).
(13)
So, if F(x) s to for a.a. x E R, then ,ulu is an upper solution of (11) and (11) is solvable. It has been pointed out in [5] that the BVP Au = F(x) + ]Vu/* in R u=O
0naQ
is not solvable if F(x) 2 Al ( = first eigenvalue of -A) for a.a. x E Q. Combining this fact with (13) in the case E = 0 (i.e., taking 4 = r: in theorem 2), c( .) = 0 we obtain THEOREM 3. Let u. be the unique function such that - Au0 = 1 in R, u. = 0 on aR, and K =max ]Vucj. Then the first eigenvalue Al of -A subject to 0-Dirichlet boundary condition R satisfies
To put theorem 3 in perspective we make a digression at this point and obtain an alternative estimate for J.r. This dimension dependent estimate is better than theorem 3 for lower dimensions but becomes worse than it is for higher dimensions. For dimension N = 2 it can be deduced from [8,9]; see also [12]. THEOREM4. With the same notations as in theorem 3 we have Al > ;+.
Proof. Let q1 be the positive,
normalized
eigenfunction
of -A corresponding
to its first
70-l
N. P. C.AC
eigenvalue
?.1. Since
-Au0
= 1 we have
J-1 LQ~TJ~dx = iR
‘F~dx.
Therefore (1-t) We now make the crucial observation (which for N = 2 was used by Payne [8,9] who attributed it to Miranda; see also [ 121) that the expression E(x) = 1Vu~(x) (’ + ,ULIO(X)is subharmonic with u = 2/N. Then by the maximum principle we obtain IVUOI’+
iuo dR
c max jVu$
= max Itr~~l~ =
R
K'.
and hence
iv
max rlo < -
2
Q
From
K'.
(13
(14) and (15) we deduce 2 1
/II >
-7.
N
K-
To show that E(x) =~VuO(x)l’ -I- JUI~(X) is subharmonic computation: (Di = a/&X,. i = 1, . . . , iv)
for u = 2/N we proceed
by direct
,v
DiiE= 2,zl{(D~jUo)' + D,u~.Dllit~o} + ,UD,iUo. Taking
into account
the fact that AE = 2 ii1
Since AK,J= -1, Suppose
AEaOif
now that instead If+,
,uL2/N.
-Au0
= 1 we have
(D,jUo)' - ,U 3 2 lil (D~~LLo)' - U. n
of (3) f satisfies
the condition
t, rj) j G 6(x)1771 for a.a.
x E Q,
V(t, r]) E R X Rv,
(16)
of where .!$ .) ELJ’(R), d 3 0, p > N. The correspondin g BVP (2) is not in the framework the problems discussed in theorems 1 and 2 because there E = N/q with q 3 p > N. We study it briefly here since the technique of constructing upper and lower solutions for the BVP (2) as described in the proof of theorem 2 and in the computation leading to (13) is not valid when E= 1. Despite the fact that under condition (16) the BVP (2) is not of the type considered in [lo], it can still be proved that if it has an upper solution v and a lower solution CQwith q s r/j then
Some remarks on a quasilinear elliptic boundary value problem
it has a solution
u with q G u 6 r+ In fact, it suffices
exists a constant
y1 depending
on the coefficients
to show that if rn?
705
lui GM
then there
of L, c, F, d and M such that
m,ax IVuj s ;I:
(17)
for any solution u of the BVP (2). because it can be seen that the other steps in the proof of solvability in [lo] (i.e. lemma 3.1 and lemma 3.2) can be carried through with little modification. But estimate (17) is an immediate consequence of well-known results on linear partial differential equations (cf., e.g. [6], lemma 11.1, page 192) because equation (2) can be written Lu = c(x)u + F(x) + CO&) $,
(18)
1
wherefori=l,...,N, 04(x) = f(x, U, Vu) . /VU(X) /-‘g
if jVu(x) / f 0,
(19)
if jVu(x) / = 0.
(20)
‘ q(x) Since f(. , -) satisfies
= 0
(16) we have II4
* >IlLw2) s
IN *%Yn~~
i=l,...,N.
Therefore theorem 1 is applicable: If f(. , . , *) satisfies (16) and c(x) G &u< 0 for a.a. x E R for some number ?Xthen the BVP (2) is solvable for any F( .) E LP(Q). Actually, we can improve this result and prove 5. Suppose that f( . , . , .) satisfies (16) and c(x) G 0 for a.a. x E R. Then (2) has a solution in W’*p(Q) nw,‘.p(Q) for any F( .) E LP(S2). We first establish the following
THEOREM
LEMMA.
Suppose
the BVP
that d( .) E Lp(Q) (p > A’) and c(x) < 0 for a.a. x E R, then the BVP Lu = c(x)u + F(x) + d(x) IVul u=O
in R,
(21)
onan,
has a unique solution u E w*J’(Q) fl W,‘,p(Q) f or any F( *) E LP(Q). The conclusion if in the equation above h(x)]Vul is replaced by - d(x)/Vul. Proof of the lemma. To prove Letting w = u1 - uz we have
uniqueness
suppose
that there
are two solutions
still holds
ul and 4.
Lw = cw + b(x)[lVu*l - jVuzl] 3 cw - d(x)~Vwj or Lw 2 cw - f&(x) g I
where
oi(-),
i = 1,. . . , N,
are
defined
as in (19),
(20)
with
f(x, r, r]) = 8(x)/171. Since
706
N. P. CAC
wt( . ) E LP(Q). i = 1,. . A’. the maximum principle in the space W’+(Q) (cf., e.g. [4,7]) gives w 3 0, i.e., u1 a u:. on R. Interchanging ul and u2 we see that rll C U: on Q also. To prove solvability, for any aE [O, I] consider the BVP Lu = cu T F(x) A d(x)lT’n)
in R,
u=O
0naR.
(22)
Writing the equation in the form (18) we see (by [4], theorem 2; [7]. theorem 1.1) that there is a constant y depending only on the coefficients of L, on the LP(R)-norms of c( .) and d(e) such that every solution u of the BVP (22) satisfies the apriori estimate I]+~.P(*) c y.
(23)
On the other hand, for every u E C’(G) we know that the BVP Lu = cu f F(x)
+ d(x)lVu(x)/
u=O
in R,
ondR
has a unique solution (again by our theorem 0 above or [4,7]) which we denote by So and there is a constant y1 independent of u such that IlS~ll@P(R)=5 Y1(lll~ulllUG2, +
llav2))~
Therefore the mapping S of C,(Q) into itself is compact. It is not difficult to see that it is also continuous. From (23) we deduce, by using Leray-Schauder’s theorem, that S has a fixed point. This is the solution of the BVP (21). From the proof above it is not difficult to see that in (21) we replace d(x)]Vu( by -d(x)]Vu] then the existence and uniqueness of a solution still hold. n Proof of theorem W,‘J’(Q) such that
5. By the lemma Lp=q--
Furthermore,
just proved
F--t$x)iVc+ccp+
there
is (a unique)
v, EW’J’(R)
n
F+f(x,q,Vq).
since Lq-cy=F-+
&x)]V+=O
and c(x) s 0 a.a. x E 52, the maximum principle gives J$ b 0. Thus q is a nonnegative solution of the BVP (2). Similarly, there is (a unique) Q,E H’*J’(Q) rl w,lP(Q) such that
upper
Lrp = cq - F- - d(x)]Vg;/ G ccp + F + f (x, Q?,Vq).
Again, by the maximum principle, Q,G 0. Thus 97is a nonpositive (2). We then deduce that the BVP (2) is solvable. l
lower solution of the BVP
THEOREM 6. Suppose that f(. , . , -) satisfies (16) with b(v) E L”(R). F( .) E LP(Q) even if c( .) assumes positive values. To see this we prove
Suppose
also that
6. Suppose that f(. , . , .) satisfies (16) with b(e) E L”(R). Suppose also that c( *) E Lx(R) and there is a number & such that c(x) < &< Ai - ]]h( *)]]L-co,fl for a.a. x E R where Al is the first eigenvalue of -A subject to 0-Dirichlet boundary condition on THEOREM
Someremarks
on a quasihear
aQ. Then for every F(a) ELP(R),
elliptic
boundary
value problem
707
p ,Z 2, the BVP
-Au = cu f F(x) +f(x, u, Cu)
u=o
in 52,
0naQ,
has a soiution u E w’.p(Q). Proof. The operator u-+ Au = -Au - cu -f (x, K, vu) from 14’h,2(Q) into its dual H’-‘.?(Q) is pseudo-monotone (see, e.g. [13], Chapter 2, Section 2). Therefore ‘to prove the theorem it suffices to show that it is coercive. With (e, .) denoting the pairing between w;,‘(Q) and its dual, we have for any u E W$2(S2): (Au, u) = i, (1Vu/’ - cu* - f(x, u, Vu)u}dx,
(Au,ub+Vulza2 - II~(~)hdVul lull dx,
we deduce from the last inequality that (Au, u) 2
1 -t-!-j
jn~~u~‘dx
for Vu E W$‘(Q). Thus under the hypotheses of the theorem the operator A is coercive and therefore our BVP is solvable in W;.‘(Q) for every F( .) E LP(Q), p s 2. Using the familiar bootstrap argument we can deduce that its solution actually belongs to W’~“(Q). n Note. If the diameter of Q is sufficiently small, we shall have Ai - II@0) IIL=(oJV’& > 0 and under the hypotheses of theorem 6, c( .) can assume positive values. In view of theorem 6 it is natural to ask whether the BVP considered there is still solvable, at least for suitably chosen F( .), even if c is no longer smaller than A1- //a(. )]/L=(Q)d/;i; on R. We provide a partial answer to this question by proving the following theorem for the case f(x, r>II) = 1111. THEOREM 7. Let R = sup{lxl . x E 52). Suppose that c( .) is bounded from above away from At. Then the BVP Lu = cu + F(x) f [Vu I in R, is solvable if c G l/R and F =S2( N - R) on R. Before proving the theorem we make the following
u=O
ortaR,
705
N. P. c\c
,%‘ote. This theorem is of interest oniy if both of the following conditions are met: (i) N> R. In fact, if F s 0 on R then our BVP is solvable if c( .) is merely bounded above away from J.i: For in that case 0 is an upper solution of it and a nonpositive solution is provided by the solution of the BVP Lu = cu f F (ii) l/R 2 AI - fl,
otherwise
in 52,
it is already
u=O
contained
from lower
0nJQ. in theorem
6.
We give an example for which both of these conditions are satisfied. Take N = 2. The first eigenvalue of -A subject to 0-Dirichlet boundary condition on the disc of radius R is [ll] A, = (2.404825556)’ R’ Then
if we take R = 1.95 < N = 2 we certainly
’
shall have Ai > l/R 3 J.1 - G.
Proof of theorem 7. As pointed out in the discussion following theorem 4 and preceding theorem 5, it suffices to show that the BVP considered has an upper solution v, a lower solution QI with q c y on R. Since c is bounded from above away from Ai, we can find a nonpositive lower solution q by taking the solution of the BVP Lu = cu - FWe shall verify that under solution. Let 1x1 = r,
the conditions c-(x)
computations
u=O
of the theorem,
= max [c(x), 01.
E = Ilcf( .)ljY(cq, Elementary
in R,
F-(x)
on&?. q(x) = R’ - /xi’ is a positive = max [F(x), 01,
r: = llF_( .)iiL’(R,.
give:
cy+F+~Vt/,l~E(R’-?)+~+2r
forr
-AI& = 2N. Let p(r) = E(R’ - ;) value is
upper
+ p + 2r. The local maximum
of p is attained
when r = l/c : = ro and its
p(ro) = f + CR’ + i? On the other
hand, p(O) = ?R’ + F,
Therefore if e < l/R, the maximum solution is 2N 3 2R + l? n
p(R)
= 2R + I?
of p(r) for 0 < r =s R is 2R + F and by (24), y is an upper
Acknowledgements-I wish to thank the referee for raising thoughtful questions that led to improvements in the paper and to Professor Michael G. Crandall to whom this paper was submitted. H. F. Weinberger for I am grateful to Professor J. B. Serrin for sendin, 0 me a copy of [12] and to Professor informing me that in the case N = 2 a better estimate than theorem 3 (namely, theorem 4 with N = 2) can be deduced from [8, 91. iMy thanks also go to Professor Catherine Bandle for discussions about bounds for the eigenvalue 1,.
Some remarks
on a quastlinear
ellipnc
boundary
value problem
709
REFERENCES 1. A.LIANS H. & CRXND.+LL M. Cr.. On some existence theorems for semi-linear elliptic equations, Indiana L’nio. .Llarh. J. 27, 779-790 (1978). 2. BR~ZIS H. & NIRENBERG L.. Characterizarions of the ranges of some nonlinear operators and apphcations to boundary value problems. rlnnali Scu. norm. sup. Pisn Serie IV. V. 225-326 (1978). 3. C.X NGtiY’& P.. On an elliptic boundary value problem not in divergence form, Proc. rim. marh. Sot. 88. 17-52 (1983) 4. CHICCO M., Solvability of the Dirichiet problem in H’+‘(Q) for a class of linear second order elliptic partial differential equations. Boll. tin. mar. iral. 4. 371387 (19711. 5. QZDAN J. & KRAMER R.. Invariant criteria for ehstence of solutions to second order quasilinear elliptic equations, Commrcns pure uppl. Math. 31, 619-645 (1978). 6. LADYZHENSKAYA 0. A. & UR..XL’TSEVAiv.. Linear and Quasilinenr Elliptic Equafions, Academic Press. New York (1968). 7. LIONS P. L., Problemes elliptiques du Z&me ordre non sons forme divergence, Proc. R. Sot. Edinb. 84& 263271 (1979). 8. PAYNE L. E., New isoperimetric inequalities for eigenvalues and other physical quantities, Communs pure appl. Math. 9. 531-542 (1956). 9. PAYNE L. E., Bounds for the maximum stress in the St. Venant torsion problem, Indian J. .Mech. Jfarh. Special Issue 6. 51-59 (1968). 10. POHO~AEV S. f.. 6; equations of the form Au = f(x, u, Du),~,,$Marh. USSR Sbornik,,41, 269-280 (1982). Am. math. Sot. Transl. 11. OLIVER F. (ed.). Royal Society Mathemarical Tables. Vol. 7, Bessel Funcrions. Part III, Zeros and Associared V&es, Cambridge Univ. Press (1960). 12. WEINBERGER H. F. & SERRIN J. B., Optimal shapes for brittle beams under torsion, in Complex Analysis and Applications: VOL.dedicated to I. N. Vekua. Nauk, Moscow, pp. 88-91 (1978). 13. LIONS J. L., Quelques mPthodes de rholurion des problemes aux limires non lineaires, Dunod. Gauthier-Villars, Paris (1969).