On a class of singular quasilinear elliptic equations with general structure and distribution data

Pergamon

Nonlinear Analysis, Theory, Methods&Applications, Vol. 28, No. 11, pp. 1879-1902, 1997 © 1997ElsevierScienceLtd Printed in Great Britain. All fights reserved 0362-546X/97 $17.00+0.00 S0362-546X(96)00006-5

ON A CLASS OF SINGULAR QUASILINEAR ELLIPTIC EQUATIONS WITH GENERAL STRUCTURE AND DISTRIBUTION DATA LE D U N G t International Centre for Theoretical Physics, Trieste 34100, Italy

(Received 1 October 1993; received for publication 18 January 1996) Key words and phrases: Quasilinear elliptic equations, C a m p a n a t o - M o r r e y spaces, Sobolev-Hardy inequality, regularity. 1. I N T R O D U C T I O N

In this paper we study the regularity of weak solutions o f the quasilinear elliptic equation in an open set ~ C R n of the form

D i A i ( x , u, Du) + B(X, u, Du) = T

(1.1)

where A i, B are Borel measurable functions and T is a distribution. We will consider the following structure condition on (1.1): (E) There exist p > 1 and positive constants Vo, v~ and nonnegative measurable functions a 0 , a 1, b, g given on fl such that for any (x, u) e ~ x R and ( ~ R n we have a i ( x , U,

>-- V01(I" -- ao(X)

[Ai(x, u, OI ~ v~l([ p-1 + al(x) IB(x, u, O[ -< b(x)l(I p-1 + g(x). Here we follow the convention that summation over repeated indices from 1 to n will be taken. Our study is motivated by the papers of Rakotoson and Ziemer [1-3], who considered the case when T is in some Morrey-like spaces of distributions on Wol'P(fl) (roughly speaking, the set of all distributions whose restrictions onto any ball of radius R have their norms grow like R × for some 2 > 0, see Definition 2 of Section 2). An advantage of this setting is that one can at the same time investigate the regularity of solutions of variational inequalities involving the operators of the form (1.1) (say, the obstacle problems). On the other hand, the function spaces of the parameters in condition (E) have been enlarged in the works of [4-7]. There one can get a regularity theory in assuming that the parameters only belong to LS(g)) for some s sufficiently large (say, s > n/p). This condition was then relaxed by Campanato (see [8] and also [9]) for linear operator with coefficients in some Morrey spaces Lq'×(~)) which are larger than the traditional Lebesgue spaces LS(~)). The works [1-3]

t Current address: Department of Mathematics, Arizona State University, Tempe, A Z 85287-1804, U.S.A. 1879

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extended these results to quasilinear operators of the form (1.1). They also obtained an optimal result on the equivalence between the C °'~ regularity (for arbitrary p > 1), C 1'~ regularity (for p = 2 only) of solutions and the growth conditions on the right-hand side T of (1.1). Here we will consider weak solutions v of the Dirichlet problem associated to equation (1.1) which is supposed to satisfy (E). The nonhomogeneous case v = ~ on a ~ for some ~ ~ Wl'r(£)) can be reduced to the homogeneous case ~ - 0 by investigating u = v - c~ ~ WI'p(D). Therefore, we shall concentrate on the problem DiAi(x, u, Du) + B(x, u, Du) = T

in

u = 0

on 0~.

I

(1.2)

Thanks to the weighted Hardy-Sobolev inequalities one may consider (1.2) in a more singular situation. In fact, the parameters may not belong to any spaces like Lq'×(~) or LS(~) near the boundary 0ft. Roughly speaking, we can study the case O~ is smooth and consider the distance function dQ(x) -- dist(x, 0fl) for x e ~ . Then the parameter, say b, may behave like some negative power of de(x) near 0£) and even not belong to any Morrey space. It is then outside of any standard settings. We require that bPdg ~ Lr(~) for some fl > 0 and r > n/p. Our method based on [10] and [11] will show that a regularity theory continues to hold under this condition. Going back to the nonhomogeneous cases, of course, we cannot keep such a singular assumption. However, one can see that our argument still works, at least, in a fairly general Lq'x setting as in [2]. The other reason of this approach is that we want to study the local boundedness and Holder continuity o f the solutions up to the boundary. It also completes the results of [1,2] where only the interior treatment and nonsingular case have been investigated. This will be done in Section 3. In addition, in connection with the equivalence between C °'~ Holder continuity and the growth of T, we do not require the operator to be strongly monotone as in [1, 2]. Actually, this is necessary only when one studies the Holder continuity of the first derivatives of solutions (see Section 4). Also, we will complete the two results o f [2] where only the case p = 2 is considered: one on the decomposition theorem of the space Mx,- 1lo, pc ' introduced by Rakotoson; and the other on the equivalence between the C 1'" property and the growth of T (see Theorem 4.1 and Theorem 4.2 of Section 4). Finally, in Section 5, we study the Holder c o n t i n u i t y o f the first derivatives near O~ for the case p = 2 but under singular assumptions. After this work was complete, we also learned that equations with measure data have been investigated in [12, 13]. They obtained the weak Harnack inequalities for equations having coefficients lie in suitable Lp spaces which do not cover our case. Moreover, our method in getting C o, ~ regularity based on a modified device of logarithmic function by Moser rather than on weak Harnack inequalities. Also, our C 1'~ results are not included in their works. 2. PRELIMINARIES AND NOTATIONS Let £) be an open subset o f R n with smooth boundary 0fl and x 0 ~ ~. We denote by BR (Xo) the open ball of radius R > 0 centered at Xo. Set ~R (Xo) = £) t~ B~ (Xo). If the center x 0 is understood, we will simply write BR = BR(xo) and ~R = £)R(Xo). We also write ISI for the Lebesgue measure of a measurable subset S of ~ . For a point x e £), we denote by do(x) the distance function from x to 0~.

A class of singular quasilinear elliptic equations

The notions of the

Definition 1.

Morrey spaces Lq'×(f~)

and the

Campanato spaces £q,h(~-~) are

1881

given by:

For q _> 1 and 2 _> 0, we define the Morrey space

Lq'×(~) = IU ~ Lq(~) : llU"Lq'x(n) = sup p-X/q'JUl'Lq(~P(x))< °° p~l and the Campanato space

~l~q'X(~')) = [u E Lq(~')) : [u]*~q'x(fl) = sup p-X/qIIu - (U)pIILq(°P(x)) < °° p~l where I = (0, diam(f~)) and (U)p = l/[f~al ~ap u(x) dx, the mean value of u over Dp. We note that (Lq'h(~), I[" I1 o x) is a Banach space and so is (£q.x(O), ]j. i[zq.x(m) with

This definition can be applied for vector-valued functions as well. The following basic properties of these spaces can be found in [141. PROPOSITION 2.1. For ~ with smooth boundary 0O we have ( = means isomorphic to): (i) Lq'°(O) = Lq(~) and Lq'n(~) = L®(~). (ii) Lq'x(~) = £q,x(~) for 0 _ 2 < n. (iii) £q'×(fl) = C°'~(~) for n < 2 _< n + q and o~ = (2 - n)/q. In [1, 2] we have the notion of Morrey like space for the distributions in W-~'P'(D), the dual space of wd'P(f~).

Definition 2. Forp >_ 1,

2 _> 0, we define the space

M;~'P'(f~)

by

Mxl'P'(~) = I T E W-I'P'(~) : sup p-X/P'IITIIw-"F'(°P(x)) < °° p~I w h e r e p ' is the conjugate of p, 1/p + 1/p' = 1 and I]Tllw-~,p'(np)is the norm of the restriction of T t o W~'P(~a). We can introduce the notions of the local spaces Ll°dx, £I°ox and Mx,lo -l.u' c in a similar way by replacing f~ in the definitons by any subdomain f~' C f2. One important result of the papers [1-3] is given in the following proposition. PROPOSITION 2.2. If we set

cxl'P'(~) = I T~ W-I"p'(D): T= f° + i=l~Difi'fi~ ~P"x(f~)l then (i) CxZ'P'(f2) C Mx-I'P'(D) for e v e r y p , 2. (ii) Cx-l'P'(f2) = Mx--l'P'(~) for p > 1, 2 ~ (n - p , n). Off) If T ~ Mx~o'2(f~) and n < 2 _< n + 2 then there exist functions f / ~ C°'~(f~) for some ct > 0 such that T = - ~ = 1 Difi.

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The proof of (i) can be found in [1]. For (ii) and (iii) see Theorem 1 and Lemma 5 of [2]. We will given an extension of (iii) in Section 4 for p ~ 2. We need the following result of Adams (see [15, p. 54] or [16, p. 213]). PROPOSITION 2.3. Let p be a positive Radon measure supported in t2. There exist positive constants M, 2, 1 < p < q = p k / ( n - p) such that

P(g)R) <- MR×

(2.1)

if and only if there is a constant C such that for every u e Wo~'P(f/)

(f fl lul q dp 11/0 <_ CM1/q[IDuIILp~n>.

(2.2)

A direct consequence of (2.2), the definition of Ll'X(f~) and a simple use of Holder's inequality is the following lemma. LEMMA 2.1. Let 0 be a nonnegative function in Ll'X(f/) with )~ > n - p then for any u ~ W01'P(f/)

l°lu'Pdx<-cf~

o IDulPdx

where C depends only on )~, p and is bounded by diam(f]),

IIollv. ¢a) and II01lv< ).

In this paper, we also deal with the singularity of the coefficients of the equations near the boundary. There, these functions may not define a density of some bounded Radon measure near Or2. In this connection, a general Hardy-Sobolev inequality will be needed. The following one is a special case of the results in [10]. LEMMA 2.2. Let p e (1, n), r e (1, oo] and ,8 be such that - p + n/r <_ fl <_ O. Let 0 be a nonnegative measurable function satisfying Od~ ~ ~ Lr(D). There exists constant C = C(n, p, r, ~,) such that for any u e wd'P(D)

f. fl IOulPax. lfl lulPOdx<_CllOdffall,: Finally, the following technical lemma extends Lemma 6 in [10]. L p u ~ 2.3. Let 0 be as in Lemma 2.1 or Lemma 2.2. For every t / e CI(R n) and u e WI'p(~'~) such that r/u e WI'p(t2) and for any e > 0, we can find positive constants C, r depending only on p, n and [1011v,
IlOd~llLrta)or

A class of singular quasilinear elliptic equations

1883

Proof. Suppose first that Odff ~ • Lr(Q), we can choose a real m < p which is near to p such 1/r < p / m r < p / n = 1/r < m / n and fl > n / r - m. Putting U = (qu) p/m, we then have IDUI m <_ C(m)llrtlv-mlDrtlmlul p + IrllPlulp-mlDul 'n} Using the simple inequality ap-mb m <_ a t' + b v, we get

[ulPrllp-mlortl " <_ lul"(l~l p 1,71"lul"-mloul " <_

+

IOr/Ip)

I l°(elDul" + *-TluI')

with r = m / ( p - m). From L e m m a 2.2 we have

I~ O[rlu[P dx = fa O[U[m dx <- C Io 'DU[m dX" Taking into account the above estimates, we get the lemma. The case 0 • LI'X(Q) is proved similarly by choosing m such that ~. > n - m and using L e m m a 2.1. • 3. LOCAL BOUNDEDNESS AND HOLDER CONTINUITY In this section we suppose that the parameters in the structure condition (E) satisfy the following: (H) There exist 2 • (n function ao + af' + For the function 0 satisfying - p + n / r

p , n] and a nonnegative measureable function 0 on f~ such that the gP'O -1 • LI'X(D) and b p <_ O. we suppose either 0 • LI'x(Q) or there exist real r • (1, oo], and // < fl ~ 0 such that Odff ~ • Lr(Q).

We will prove that weak solutions of (1.2) is locally bounded and Holder continuous if and only if the following condition on the right-hand side T is fulfilled -1,p' (T) T • Mx,loc for some 2 • (n - p , hi. Thanks to a result of [2] mentioned in Proposition 2.2, we can find functions f~ in Ll°o~× such that T = D i f i .

We have the following theorem. THEOREM 3.1 (local estimate). Let u be a weak subsolution (supersolution) of (1.2). Suppose that (T) holds. Then for every R _< 1, there exists constant C depending on the global quantities

n,p, vo/Vl,2

II01lv, .)

and

or

r, A IlOdff llLr a).

but not on R such that

sup,,,u ( - u )

<_ C (

1

f

lu+(u-)l " dp)\ t / p + k(R)

(3.1)

where P = (1 + R"-X(h/llhllt,,xtm))dr, h = ao + a~' + gP'O -l + Z f P ' , is a Radon measure supported in QzR and k(R) = R~(1 + [[hllt,,xtm) with a = (p - n + ;O/P > O.

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From this we have the following corollary. COROLLARY 3.1. Let u be a solution of (1.2) then Theorem 3.1 holds with u and u + replaced by lul. We then prove the following optimal results on the Holder continuity of weak solutions and the growth o f the right-hand side T of equation (1.2). THEOREM 3.2. Under the conditions (E), (H), if (T) holds them any weak solution u of (1.2) is Holder continuous up to the boundary 0£). More precisely, there is a positive constant C, c~ not depending on R such that osc(u, ~ R ) <-- CRY, (3.2) where osc(u, ~R) = super u - infoR u. In addition, we have the following growth estimate for D u l rite

IOulPdx _< CRn-p+p

c~"

(3.3)

THEOREM 3.3 (converse of Theorem 3.2). If (E), ( H ) hold any u ~ Cl~ ¢ for some a > 0 then -l,p' T ~ Mx,,~oc for some ~.l > n - p. Let us give some remarks before proving these theorems. R e m a r k 2.1. In literature, the coefficients and data were usually required to belong to some L s spaces with s sufficiently large (say, s > n / p ) . This condition is stronger than ours since we know that if a(x) ~ LS(f]) then a(x) ~ Lllox with ). = n - n / s > n - p . However, there is a

function in Lidx which needs not be in L~oc for any q > 1. One can see that our hypothesis cover those considered in classical works (e.g. [4, 7]). R e m a r k 2.2. In recent works of Rakotoson and Ziemer, the operator is less general than ours

for the same reason as above. They treated the nonsingular cases where the standard methods of truncation of De Giorgi (see [4]) were available for proving that the weak solutions satisfy the Hartmann-Stampachia maximum principle and then they can get the local boundedness of the solutions. Let us consider the simple limit case r -- oo and f l e ( - p , 0]. In this singular situation, 0 and then gr', b p _ dg near the boundary and therefore will not define a density for some Radon measure there. This fact does not allow the truncation method to be applicable near the boundary (actually, only the interior estimates were considered in [1, 2]). In addition, we could use the full form of the Hardy-Sobolev inequalities as in [10] to consider the case v 0, vl are measurable functions in x and they behave like d~ (ct > 0) near 0ft. Again, the truncation methods cease to work in this case. R e m a r k 2.3. By using Moser's methods as in [10, 11, 17], we do not require A ( x , u, D u ) to be u n i f o r m l y m o n o t o n e . This condition is essential in the works o f Rakotoson and Ziemer to

obtain the results as in Theorem 3.2 and Theorem 3.3 (see [1-3]). We will see that it is needed only when we deal with the Holder continuity of the first derivatives of weak solutions.

A class of singular quasilinear elliptic equations

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In the sequel, we will use C for various positive constants which m a y change f r o m line to line but depend only on the global parameters of the structure conditions (E) and (H).

Proof of Theorem 3.1.

Suppose u is a subsolution of (1.2). For any k > 0, l _> 1 and N >

k,

we introduce the function

~t t - k t J(t) = (.lNl-l(t _ iV) + N t - k t Then J ' is a bounded increasing function on [k, w=u+

for t e [k, N]

(3.4)

for t > N.

oo), J"

exists for t ~ N. We also set

+k.

At first, we prove the following energy estimate.

LEMMA 3.1. For every k > 0 and r / e quantities but not on k, r/such that

CI(R'),

f lD(rIJ(w))lPdx<_ CITf

(IDr/I p +

wJ(w), h = ao + af' + gP' O-1 + fP'

where W =

Proof.

there exist constants C, r depending on the global

k-Phlrllp +

I~IP)WPOx

(3.5)

w i t h f = (ETfiz) 1/2.

Define

4, ---

i wIJ'(t)l p at,

( = I,rl'0.

k

Because 0 < 6 < u + IJ'(w)[ p, DO = DwlJ'(w)l p = Du+lJ'(w)l p, it is easy to see that ( e WI'p(f~) and then ( is a valid test function. Therefore

la Ai(x' u'Du)Di( dx + t4~ B(x,u, Du)(dx <_ (T, () = Ifl fiDi(dx implies

f A,(x, u, Du)lr/lPD,¢dx <_p f Iai(x, u, Du)l~lr/lP-'lDrl[ dx [~

12

+t'~ ]B(x'u'Du)lu+HP(w)dx+la flD(ldx

(3.6)

where we denoted H = IJ'[ for brief notation. F r o m the structure condition (E) and the fact that u ÷, k ___ w, we have

I

fl

A~(x, u, Du)l,Tl~n,~ dx >_ Vo

I

fl

I~l~"~(w)lDwl~ dx -

i °°

~ I,Tl~a~(w)w~ dx.

fl

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Let e > 0 be given, the standard uses of Young's inequality give

IflIAAx,u,Du)Ic/,IrllP-llD'll dx <-e I lrIIPHP(w)IDwlP dx fl + C(e,Vo/VOIn (wH(w)Y'lDrllPdx ÷ fo ~-~p(wH(w)Y'lrtlPdx Ill] B ( x ,

!1,

Du)IHP(w)u+ dx ~ Ill(b(x)IDu[p-I["IPHP(I)BI+"}-g(x)l"IPHP(w)I1+)dx

i°b(:IDui'-11":'(w)u+ dx<_.I°i,t'.'(,~)iZ'wi" dx ÷ c(c) f. b'(x)(u+/-/(w)~'l~l" dx

Xg(x)lrtlPHP(w)u÷dx <_f ---k"-1T- (wH(w))p dx + f ol,71,(u+Nw)y"dx. 12

0

fl

Similarly,

l f[D~ldx<-fn o flrllp-llDrtlwH~'(w)+flrllPlDwlPHP(w)dx <__e

l

l"

IrtlPHP(w)IDwl p dx + C(.)

fl

--p-1,71"(wH(w))" dx

fl

+ In ]DrllP(wH(w): dx. Choosing e small in these estimates, (3.6) then gives

X [rllPHP(w)'Dw[Pdx <

o ['Drl'P + k-Ph'rllP]WPdx + fa OlrllPUPdxl

where C depends on VO/Vl, and U = u+H(w) = u +IJ'(w)l ~ Wo~'P(~). Since ]uDH(w)I <- 2IH(w)tDwl, applying Lemma 2.3 we derive

IDUI <-IH(Iw)Dul ÷

Again, with a suitable choice of e, we obtain from (3.7) and (3.8) that

S lrtlPlJ'(w)lPlOwlp dx <_el ~S (Io~/l" + k-Phlrtlp ÷ Ir/l')W p dx [1

since l _ 1. This immediately gives (3.5).

fl

•

(3.7)

A class of singular quasilinear elliptic equations

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P r o o f o f Theorem 3.1 continued. F o r m 1, m 2 such that 1 _ m~ < m 2 _< 2, in L e m m a 3.1 we choose a c u t - o f f function r / e C~(R n) such that i/ = 1 in BmtR, r/ = 0 outside Bm2R and IDol - 1/(m2 - ml)e. W e also take k = k(R) described in the theorem. T h e n obviously, (Io~l p + k-Phlrl[ p + 1,71p) dx _< and for e v e r y x ~ z R a n d 0 <

r
1 ((m2 - ml)R) p

d/u

1

/u(Br(X)) < C(rn + Rn_X

h B,.> Ilhllv

dx) <__CR~-Xr x.

Therefore, f r o m P r o p o s i t i o n 2.3, L e m m a 3.1 and the fact that rlJ(w) ~ Wd'P(t2) we have

(fftll~J(w)lqd/u) l/q<_ CR'~-X)/q(flD(rIJ(w))lVdx)l/" < C

---- -m 2 -- m 1

f~m2R W vd/u

(3.9)

where q = p2/(n - p) > p. Letting N ~ e¢, this becomes

",,R

< c

d/u/

m2 _ ml

am2R

w',

where f = r/p + 1. Since/z(f~m~R) ->/U(D s) -> R ~ (recall that Of~ is s m o o t h ) and/U(D.m2R) <-/U(f22R) <_ R ~ + R~-×R × <_ CR ~, we also have

(

1 /u(ffm~R)

f

,1/q< C

(W I - kl) q d/uJ

n.,g

-

- -l ÷ ( 1 S w l P d / u ) l / p " m2 ml U(dm2R) am~R

Using the M i n k o w s k y inequality and recalling that w _ k, we can replace the integrand o n the left by w tq. The standard iteration scheme with l = (q/p)i, mi = 1 + 2 -i, i = 0, 1 . . . . yields 1 supw<_ C ( fiR ~

l

\l/p . w p d/u~ •2R

This and the definition o f w, k immediately give o u r theorem.

•

To prove T h e o r e m 3.2 we follow the m e t h o d used in [10]. W e put

Mi = sup u, fliR

rni = inf u, OiR

o~(~ R) = sup u - inf u. fir fir

Applying an elementary result in [7], we see that (3.2) holds if we can prove that og(~R) _< zco(~4R ) + Ck(R)

(3.10)

where r ~ (0, 1) and C are positive constants not depending on R ; k(R) is defined as in T h e o r e m 3.1.

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If B4R CC o , (3.10) comes from

O)(~'~4R) ~ C[O.)(~'~4R) -- (.O(~'~R) -f" k(R)}.

(3.11)

For this, we consider the following functions

Wl = l°g((2~-----m-~ k [ ) 4 - \ u, +

and

w2 = log(2--~- - km4 ) m+4+)

where k = k(R). It is clear that (3.11) is proved if there exists a constant C such that either super wl -< C or sup~ Rw2 -< C. The idea is to prove that wl, w2 are subsolutions of certain equations having the same structure as before. One then uses Theorem 3.1 to estimate the sup-norm of wi by the Lv norm of w~. Obviously, Wl >- 0 ¢~ w2 -< 0. Therefore at least one of w/+ vanishes on a subset S of f~2R with IsI ---½1f 2RI. This allows us to use the Poincar6 inequality to estimate the Lp norm of such wi by the quantity Ja2R Iow*l p dx which in turn can be bounded independently from R. This process can be adapted either from the lines of [10] or from the p r o o f below for the case BR is not contained completely in ~ . The latter is, of course, more delicate. Therefore, from now on we will be concerned with the case BR ~ fL (3.10) is then a result of adding the following two inequalities M 4 <_ "cM1 +

Ck(R)

(3.12)

- m 4 <_ - z m I + Ck(R).

(3.13)

Because s u p ( - u ) = - i n f u, (3.13) comes from (3.12). Consequently, we need only consider (3.12). For this, we study the following function w = l ° g ( (p-N(u)I)M4+ k )

where N ( u ) = ( p -

1)(M4 - u ) + k

which is defined in t)4R (we can suppose that M4 -> 0, otherwise we can consider - u ) . Again, we find a bound for w from above by a constant that does not depend on R. The p r o o f o f Theorem 3.2 will be divided into four steps.

Step 1. We prove that w is a subsolution of a certain equation. Let r / e CI(t)4R) and r/_> 0, we set ~ = rl/Np-l(u). Then, Dd~ = (Drl + (p - 1)rIDw)/NP-l(u) and N(u)Dw = (p - 1)Du. Since N(u) >_ k in f~4R, (P is a legal test function in Wo~'V(f~). Therefore, i n A i ( x , u , Du)(Dirl + (P--1)rlDiw_'~ N p- 1 j dx + l n(x, u, Ou)

rl

fl

3n

fi Dirl + (p - 1)rIDiW dx NV-1 "

(3.14)

From the structure hypothesis and the Young inequality we see that A i ( x , u, D u ) r I ~

d x >_ v o

IDwl",t e_,c -

~

rl

A class of singular quasilinear elliptic equations

1889

and

3elfrlOWlp-I N

dx <.

f

(P --21)Vo ~ [DwlPrldx

+ C , ~rldx.

f:

Then (3.14) yields

(3.15) where

fi, i(x, Dw)

Ai(x, u, N(u)Dw/(p - 1)) =

N p - 1( u )

B(x, Dw) = NI-p(u) B

'

, u, P _ 1 /

ao

- NP(u ) ,

a n d f = f/NP-~(u). For any ( • R ~, using the structure conditions of the original equation (1.2) it is clear that : t i ( X, ()(i

=

A i ( x , u, (p - 1 ) - I N ~ ) N ( i > VolCl" - a0 g p

1

IA~(x, OI ~-~-~(v~lN~l IB(x,

p-~ + al) =

villi p-1 + a 1

OI -< b(x)l(l ~-' + g

where d o = ao/N p, al = a l / N p-l, g = g / N p-1 + ao/N p-1 + f P ' / N p. Since N = N(u) >_ k > O, it is clear that (3.15) satisfies the same structure conditions (E), (H).

Step 2. Thus, we can apply Theorem 3.1 for the subsolution w of (3.15) to get the following estimate super w <

C(1

f

~

x~X/P + /¢(R)

~2R (w+)Vd/i)

(3.16)

where/i, h are defined as in that theorem. We have,/~P(R) = Rp-n÷xlI~IILI,~ with Llhllv,~(~ --IlhllL~,~ta~/kP(R) <_ R ~-p-x. Hence, we also have/~(R) is bounded by a constant not depending on R.

Step 3. We now estimate the integral term of (3.16). Since

for every ball Br(x), we can apply the Holder inequality and then the result of Adams to get

1 #(fl2n)

l

( w + F d/i _< ~(t~2R)_p/q .~

cl

(w+)q d/i

',oo2R (since /i(f~Eg) >- Rn)

;q

_

lOw+ ip dx \#(ta2R),/

<- RP-n I J fl2R

a2,

[Dw + Ip dr.

(3.17)

1890

LE DUNG

To estimate the last term, we take a cut-off function r/ in B4R such that r/ = 1 in B2R, r/ = 0 outside B4n and IDa/[ _< I/2R. We denote N÷ = (p - 1)(M4 - u +) + k(R), N÷(0) = (p - 1)M4 + k(R) and set 4~ = rlPu÷/Nr+-1. Obviously, ~ is a valid test function and one can check that

prlP-lu+Drl + N+(O)rlUDw+ D~p =

Np+_ 1

Putting ~ into the equation of u, using the hypothesis and dividing both sides by N÷(0) we obtain

v°t

Np+

f

dx <_p

~

+ i J

IAi(x, u, Du)l rf-lu+ IDr/[ N+(O)NP+_l dx [B(x,u, Du)[ N+ ~tPu+ (0)N+p- 1 dx

I f':-'u+lO'rl

+ p j~ N+(O)NP+_1 dx +

l"frlPlDw+l l doq p dx. dx + NP+-1

fl

Taking into account the facts that N+(0) _ (p - 1)u +, iV+ >_ k(R) and using the Young inequality, we easily get

v°.fu rlPlDw+[P dx <- e f , rlPlDw+[P dx + C(e)(

J fl

(IDrtl p + k-PhPrl p) dx

+ I Off Updx

(3.18)

J [1 where U = u+/N+(O), h = ao + a p' + gP'O-1 + fP" and t is an arbitrary positive number. Since 0 <_ U<_ 1, N+ <_ N+(O), [DU[ = [Du+I/N+(O) < CIDw+[, we can apply Lemma 2.3 to estimate

Altogether, by choosing e small, we have

fl2R

f~

~4R

<-- C(R n-p + Rn-P-XR x + Rn) <- CR n-p. We get the result immediately f r o m this and (3.17).

A class o f singular quasilinear elliptic equations

1891

Step 4 (estimating the growth of J [Du[ p dx). We first consider the boundary case BR q~ ~ . Let l = 1 in the energy estimate o f L e m m a 3.1 and then let N tend to infinity, we get dx. 3 fl

,J f

Take r/ to be the cut-off function: r / = 1 in BR and r/ = 0 outside BzR and IDFtl -< 1/R. Since u + is Holder continuous, we can find a positive real o~ such that u ÷ + k(R) <_ CR ~. We then have

i ~R IDul"<- I fl2R ( e -p

+ k-Ph + 1)((u+) p + k P ( R ) ) d x

CRP~'(R n - r + R n - p - X R x + R n)

CR" -P +P¢~ which gives the claim. I f BR C f~, this estimate can be proved similarly. We can use 4J = ~lP(u - (U)R) in the p r o o f of L e m m a 3.1 and a simple use of Poincar6's inequality will also lead to the same result. The p r o o f o f Theorem 3.2 is thus complete. •

P r o o f o f Theorem 3.3. We follow the same lines of [1] with a little modification. Let e W01'P(DR) and IIO~llLptf) --- 1. Take ~ as a test function and use the structure condition (E), we have

I
<_ v~ 1 [IOulp-llo4~l + b(x)lOulp-llckl + g(x)14,1 + al(x)lOckl] dx. J fir

Applying Holder's and Young's inequality in a standard way and L e m m a 2.2 for ¢ we derive easily II <- C

IDulPdx fir

+

(gP'O -l + a~') fir

< C[R<,-p+o/p ' + R x/p'] because of the growth of Du (3.3) and (H). This implies that T ~ M£~dff'(O) for some 2' = m i n [ 2 , n - p + e } > n - p . •

R e m a r k 2.4. In (E), concerning the assumption on B(x, u, Du) we could allow the term IDul p to appear on the right side. In this case, we can modify our power type test function by multiplying it with a suitable exponential of u to get the energy estimate (3.5) (see also [3]). The p r o o f can then go on as before with some minor modifications. 4. C 1'~ R E G U L A R I T Y

In this section we study the Holder continuity of the first derivatives o f the weak solutions o f the equation (1.2). We shall assume that a solution u o f (1.2) is Holder continuous, say u ~ C~°~~ for some ot > 0, and then investigate the relation between the right-hand side T and the fact that Du ~ C1°'~~' for some c~' > 0. For this purpose, of course, we must first strengthen our hypothesis on the structure of (1.2).

1892

LE DUNG

Let us assume that (M) (Monotonicity) For every (x, u) • f~ × R, Ai(x, u, h) is differentiable in h and there exist positive constants Vo, v I and x _> 0 such that for every (, h • R n

OAi Vo(~ + IhlZ)p/2-1l(I 2 -< -5-~j (x, u, h)(i(j _< vl(x + IhlZ)p/2-1l(I 2. (C) (Continuity) There exist positive constants C, o such that for every (x, u), (x', u') • f~ x R and h • R n

IAAx, u,h) - Ai(x', u',h)l <-C(Ix- x'l + lu - u'l)~(1 + Ihl) p-1. (L) (Lower order terms) There exist nonnegative measurable functions b(x), g(x) and O(x) on f~ and 2 > n such that

In(x, u, 01 --- cl~l ~ + b(x)l~l ~-1 +

g(x).

Moreover, b p and gP' belong to Llo~' for some ).' > n - 1. At first, we notice that if in (L) there was not the term I~lp then these conditions would ensure the hypothesis (H) of Section 3, so that weak solutions of (1.2) are locally Holder continuous in EL See also Remark 3.4 of the previous section. Regarding the functions b, g if ~ • Wol'P(BR) one has the following estimates

BR

bPq~ dx

and

BR

g~ dx <_ CR ×''/p'

\ JBR

D4~p

(4.1)

where C depending on the norms in LI'X'(BR) of b p, gP' and some 2" > n. Indeed, using Holder's inequality and the result of Adams we have

f BR

bP4Jdx <

II BR

b p dx~l-1/q(1

/

\ ,JBR

bPOqdx)N~l/q <_ cgX"/,"ll~ll~d.~t~R>

with q = p A ' / ( n - p ) and 2" = A ' ( 1 - 1/q)p' = ( A ' p - n + p ) / ( p applying the Poincar6 inequality we get

l

BR

ggJ dx

1)>n.

Similarly,

<- [[glIL"'tBR)II*IIL"
with2" =2'+p'>n. Our main result in this section is: THEOREM 4.1. Let u be a solution i n Cl°c¢~of the equation (1.2). Assume that (M), (C), (L) hold. Then (a) I f Du • C~°'¢~ for some a > 0 then T • M~1o~' for some ;t > n. (b) Conversely, if T • Mx-~1oc p' with ~. > n then Du • C~°o'~¢ for some a > 0.

P r o o f o f (a). Let f~' CC ~, BR(X o) C ~ ' and ~b • WoLP(BR(x)) with lt*ll ~ 1. Since u • C]ol,ot c , we see that [Du] is bounded in ~ ' and the functions F/(x) = Ai(x, u(x), Du(x)) belong to C°'~'(BR(Xo)) for some or' > 0 (by condition (C)). Therefore, using the Holder inequality, (4. l) and the Poincar6 inequality for ~b we easily get

A class of singular quasilinear elliptic equations

t

loCI dx + BR(XO)(IDul p

<- CR'~' aR(Xo)

1893

+ b p + g)dp dx

<_ ce°'+°/"'ll~llwd,.
< CRx/P ' where it = minln + p'o~', n + p ' , it"} > n.

•

Together with (b), we also have the following representation theorem of Mx,- 1lo, pc' which completes the result p = 2 of Rakotoson (see [2, L e m m a 5 or (iii) of Proposition 2.1]). However, we should point out that this fact will not be used in our p r o o f of (b) as in [2].

THEOREM 4.2. I f T e Mx,lo c-l'p' for 2 e (n, n + p] then there exist functions f / e Ci°o~~ for some o~ > 0 such that T = Difi.

Before giving the proofs of this theorem and (b) of the main theorem, let us briefly recall some known facts about the equation

I Di ai((Du)) = 0

in BR

(4.2)

on OBR

u = g

where the a : s do not depend on x, u but satisfy the Monotonicity condition (M); and g e C°'V(BR) for some y > 0. It is well known (see [7 or 181) that (4.2) has a unique solution v e CI(BR) fq C°'~'(BR). Moreover, by the m a x i m u m principle, we have osc(v, BR) _< osc(g, BR). In addition, by a result of Liebermann (see L e m m a 5.1 of [19 or 20]), we can find positive constants C, o~ depending on p, n, Vo/V 1 such that

sup0¢ +

f IDvlF' < CR-" |

BR/2

l

Bp

IDv -

(x + IDvl:d~

(4.3)

,J B R

(Dv).lp dx _<

(p)"°l

[Dv - (Dv)R] p dx

(4.4)

BR

for 0 < p < R. These estimates will be crucial for our proofs later. The following fact related to the functions A i is standard but we state it separately for easy reference.

1894

LEDUNG

Lm~MA 4.1. Let A~ satisfy (M). Then for every weakly differentiable functions u, v and w=uvwehave I

(Ai(x ,

u, Du) - A i ( x , u, D v ) ) D i w d x

>Icf, IDwl~dr

ifp > 2

_

(4.5)

/

kdO

(x + [Du[ 2 + [Dv[2y '/2 dr)(p-2)/p

i f p < 2.

Proof. Using the mean theorem and easy inequalities the left-hand side can be written as -~-~j (x, u, tDu + (1 - t ) D v ) D i w D j w d t dr >_

i flf o1(K + [tDu + (1 -

t)Dvl2)p/2-1lDwl2dt dx

>- i (x + IOu[ z + IovlZy/2-11Owl z dr. d fl

I f p _ 2, we immediately have the result. Otherwise, we can use the reverse Holder inequality for the exponent p / 2 < 1 and p / ( 2 - p) to get (4.5). • Now let u ~ Wd'P(f~) with a ~ smooth and x o e ~ . If BR(xo) CC f~ we define

• (u, Xo, R) = i IOu - (Du)RIP dr. J BR(xo)

(4.6)

I f xo ~ aft, since 0f~ is smooth, up to a diffeomorphism we m a y assume that x0 lies in a part of af2, say F, which is given by the equation xn = 0 and £2 C {xn > 0l. We then define

,V(U, Xo,R) =

f

ID, u - (D,u)RlPdx +

~R(xo)

f

~ ID, ulP dr.

(4.7)

flR(Xo) i = 1

We have the following lemma which is useful in proving the Holder continuity of derivatives of solutions in various cases. LE~MA 4.2. Suppose that for some 0 < R 0 _< 1 and for every 0 < p < R < Ro and Xo e D such that B R (Xo) C f~, we have the following decay estimate

• (u, Xo, P) -< C /P/n+~ ~ ( u , x0, R) + R ~ l

(1 + [Dul") dx

(4.8)

~ R(Xo)

where C, o~, z are positive numbers not depending on R. Then D u ~ C~c for some fl > 0. Moreover, if (4.8) also holds for Xo e 0f~ and • given by (4.7) then D u ~ C ~ up to the boundary.

A class of singular quasilinear elliptic equations

1895

Proof. Thanks to the characterization of the Campanato space, we need only to prove that OO(u, x o , R) <_ CR × for some A > n and R small. The case B R CC g) can be proved by the same method and more easy to handle. We will give here only the p r o o f when BR is not contained completely in ~ . Let R > 0 be given such that BR f3 0 ~ = BR f3 F. For any y • ~R, Y = (X, . . . . . X,), put Y0 = (X, . . . . . Xn_l, 0) • F. Then, x, < R and ~)R(Y) C ~2R(Yo) = B;R(Yo) = BzR(Yo) f') {Xn > 0}. Suppose that for some integer k _> 0 and some positive constant K which may depend on R o and IIDullLp(t~) we have the estimate

f

B~(yo)

IDul p dx _< KRk~"

WR _< R o .

(4.9)

Notice that this holds trivially for k = 0. By the assumption (4.8), we get

OO(u, Yo, P) <- C ~

OO(u,Yo, R) + K R (k÷l)T

f o r 0 < p < R _< R o. I f ( k + 1)r < n + c~ we can use Lemma 2.1 in [141 to conclude that there exists C such that

/'p'~(k+ 1)~" oo(U, Yo,p) <- C~-~)

(OO(U,Yo,R) +

KR(k+')").

Because JB, IDu - (DU)R[P dx < C JnR IDu - zl p for any z • R", we see that OO(u,y, R) <_ COo(u, Yo, 2R). Therefore, if R _ R o then we can take p = 2R, R = R o to get (k+l)7

oo(u,y, R) < COO(u,y o, 2R) < _

_

\R0/

(OO(U,Yo, Ro) + KR(ok+l),) < K (k+')"

(because it is easy to see that oo(u,y, Ro) < CIIDulILp(o)). The above estimate holds trivially when Ro/2 <_ R <_ R o . Thus Du • ~3P'(t'+l)~(B~o(Yo)). If (k + 1)z < n, this space is isomorphic to the Morrey space LP'(k+l)*(B~o(yo)). Consequently, we obtain again (4.9) with k increased by 1. Going on in this way, after finite of steps, we get

OO(u,y, R) <_ K R (k+l)7 for 0 < R _< Ro and some k such that kr < n < (k + l)z < n + o~(we may assume that r < cx). K depends only on oL, r, [lDul[Lp(o) and Ro. This gives the result. •

We are now ready to prove Theorem 4.2 and (b) of Theorem 4.1.

P r o o f o f Theorem 4.2. From the theory of monotone operator (e.g. see [21]), there exists a solution u • W0~,P(F~) of the following problem Diai (Du) = T

(4.10)

where ai(Du) = (1 + IDulZ)P/2-1Diu. The result o f Section 3 shows that u • C'~(BR) for some o~ > 0 and B,~ C ~. Now we need only to prove that Du is Holder continuous. This will give our conclusion.

1896

LE DUNG

Because

Oa_j~= ahj

(1 +

[hJZ)P/z-l((p - 2)hihj(1 +

[hi2) -1 + JO)

max{p

we see that the (4.10) satisfies condition (M) with v o = m i n [ p - 1, 1} and vl = The following problem

Diai(Dv)

I

= 0

- 1, 1].

in BR

v = u

in

(4.11)

OBg

thus has a solution v e CI(BR) (q C~(/~R). Therefore (4.4) gives

i so [Dv - (Dv)p]P dx < C (P)n+a I BR [Dv-(Dv)RIu~, -~

0
It is easy to see that

f

Be

[Du-(DU)PIP~<-Cf

Bp

[Dv-(Do)p'v~+ C I

BR

,Du-Dv,P~.

Hence

[Du - (DU)p]PClx <_ C

BR

[Du - (Du)RlV dx + C

S [Du - Dv[P dx.

(4.12)

BR

We n o w estimate the last term. Substracting (4.10) by (4.11) and then multiplying by

W = U -- V 6 WI'p(BR), we derive

If

BR

{ai(Du) - ai(Dv)lDiwdxl < l[T[lw-l,p'nRllwl[wo~.pWR)<--CRX/P'llDwl[ze(sR).

I f p ___ 2, L e m m a 4.1 and (4.13) give at once

l

BR

[Du

l

By[ p ~ <

CR x.

Since 2 > n, this and (4.12) and L e m m a 4.2 show that I f p < 2, we also have

Du

is H o l d e r continuous. (2-p)/p

(fBR 'Dw[p ~)2/P < CRX/P'[[Dw[[~(B~)(IBR (1 + [Dulv + IDvlp) dx Multiplying (4.11) by u - v we easily prove that

l

BR

1 + [Dv[P~
BR

1 + [DulPdx.

So that

f

BR

[DwI p dx < CR (p-1)x

\ JBR

1 + IDul p

V"

(4.13)

A classof singularquasilinearellipticequations

1897

Now let p > 0 be such that 2' = 2 - p > n. Putting r = / z ( p - 1)/(2 - p) > 0 and using the Young inequality we obtain

l [Dwl pdX <_ CRX'+ CR" I (1 + IDulP)dx < CRY'( (1 + [ D u f ) d x ,JB~ JB R J BR

(4.14)

for some r' > 0. Again, (4.12), (4.14) and lemma 4.2 can be combined as before. Our p r o o f of Theorem 4.2 is then complete. •

Proof of (b) of Theorem 4.1. Let B R = B R(x0) C ~. From the discussion after Theorem 4.2, we can consider the solution v of the problem I D i A i ( X o , U(Xo) , Dr) = 0

in BR(Xo) on OBR(Xo).

v = u As before, we can find o~ > 0 such that l B, lay - (Dr)p[ p dx <_ C (P)n+~ f B~ -~

IDv -

(Dv)RIP dx

0 < p < R.

Again, it yields

l Bp [Du

- (Du)pl" dx <_ C (R)n+~

f BR [Du

- (Du)R[ p dx + C

f BR IOu -

Dvl p dx. (4.15)

We consider the last term. Because

Di{Ai(x, u, Du) - Ai(xo, U(Xo), Do)} + B(x, u, DU) = T. Multiplying it by w = u - v, we get

f

BR

{Ai(x, u, Du) - Ai(x, u, Dv)lDiw dx <-

~ MAx,u, Dv)

J BR

+ t"

J BR

- Ai(xo, U(Xo), Dv) l IDwl dx

IB(x, u, Du)llwl dx +

Il.

(4.16)

Denote the terms on the right side b y / 1 , 12 and 13. We estimate them as follows: Let v be a positive real which will be determined later. Since u is Holder continuous and thanks to (C), there is oq > 0 such that I1 "< CRWl t" (1 J BR

_
<_ C(e)R'~. I

+

IDvI)P-IIDwldx

1+ 'Dv[Pdx + ~,R'XlI ]DwlPdx BR

l + [Du,P dx + eR~'l I Bt¢

,Dw[P dx BR

1898

LE DUNG

where e is an arbitrary positive number. By (L) and then Y o u n g ' s and H o l d e r ' s inequalities, we have

I2 < C t~ IDulPlwl dx + I BR

BR

blDu'p-l'w'dx + f glwldx BR

<-C f BR IDul'lwldx + i BR (bp +g)lwldx. CR~z

Note that Iw I _< in BR for some a 2 > 0 and using (4.1) to b o u n d the term involving b, g, we can majorize 12 by

CR ~'2

f

(l

IDul p dx + CR x"

BR

IDwl p

\ ,)BR

<_C(p)R'~3I

(1+ BR

V"

IDulV)dx + eRVl IDwlPdx BR

where a3 = min{a2, :t" - v - n} > 0. This choice o f v is possible because 2" > n. Again, we can take v to be positive and small e n o u g h such that g - v = n + oL4 with 0~4 > 0. T h e n

13<-llTllw-,,,'
f (Ai(x,u,Du)-A,(x,u, Dv))D,wdx<_C(e)R'~f (1 +,Du,V)dx+eR" f ,Dw,Vdx BR

BR

BR

if z = minlai} and R < 1. N o w if p > 2. using L e m m a 4.1 and choosing e small, we get

f [DwlVdx<_CR"I BR

(1 + BR

IOulV)dx.

(4.17)

I f p < 2, we also have

IDwIPdx)v/2

I IDwlPdx<-(C(e)R¢I (l+lDulV)dx+eR" f BR

×

BR

(l

(1

+

BR

,(2-p)/2

IDulp + DvlP)dx)

BR

<-R-"(C(e)R~IBR(I + [Duf)dx + eR"lss lDwf dx) + R" l

(1 + IDulp + IDwl') dx. BR

Again, if e, R are small and v is chosen such that r > v then we obtain (4.17). Finally, a c o m b i n a t i o n o f (4.17), (4.15) and L e m m a 4.2 shows that is H o l d e r continuous. •

Du

A class of singular quasilinear elliptic equations 5.

C l'ct

1899

R E G U L A R I T Y NEAR THE B O U N D A R Y

We are going to study the Holderness near 0D o f the first derivatives o f a solution u o f the problem

I D i (aij(x, u)Dj u) + B(x, u, Ou) = T u = 0

in t) on #fL

(5.1)

We assume the following singular situation: (A.1) Suppose as in (E) with p = 2. We also assume that u are in C°'"(D) for some o~ > 0. Also, there is tr > 0 such that for (x, u), (x', u') in D x R

]aij(X, U) -- aij(x',

u ' ) [ ----.C(Ix

- x'[ + [u -

u'lf'.

(A.2) For some nonnegative measurable functions b, c, g

IB(x,u, Du)l < ClOul 2 + b(x)lVul + c(x)lul + g(x). For b, c, g, we assume that there are reals/~, and r e (1, co] and a measurable function 0 such that - 2 + n/r < fl <_ 0 and Odff~ e If(D). Then c _< 0 and g20-1, b40 -1 e Lll~,Xc for some 2 > n. -1,2 We also assume that T e Mx.loc • We then have: THEOREM 5.1. Under the assumptions (A.1), (A.2), the first derivatives of weak solutions of (5.1) are Holder continuous up to the boundary. The interior case has been treated in Section 4 where there is no singularity of the parameters. We will be concerned here with the case near OD. As before, we can straighten the boundary of D by a local diffeomorphism. Consequently, from now on, we will assume that D is the unit upper half-ball in R n D =B

+ = [xeR":lxl

~ 1,xn>O} =BNR~

and u = 0 on the flat part o f the boundary r = [x~R~:

Ixl ~ 1,xn

=

0}.

Proof. Let us consider the upper half ball B~ = B~(xo) = BR(xo) fq D with the center Xo e F and R < dist(x 0, 0B+\F). We consider the following Dirichlet problem I Di(aij(Xo, U(Xo))Div) = 0 v = u

in B~(xo) on aBe.

(5.2)

F r o m the standard theory of linear equation with constant coefficients (see [4 or 14, Chap. 3]), we k n o w that D e v exist in B ~ and for k - I ..... n - I, the function w = D k v is a solution of the following equation (recall that u = 0 on F)

Dj(aiflXo, U(Xo))Diw) = 0 w=O

I

in B~(xo) onF.

(5.3)

1900

LE DUNG

In addition, (see [14, Chap. 3]) we also have [Dwl 2 dx < C

f

[Dwl2dx<_CR-2f

BR/2 +

]Dwl 2 dx

(5.4)

w2dx

(5.5)

B~

for 0 < p < R and C is a constant depending only on Ilai[l®. Now from the uniform ellipticity of (5.3), we can express the derivative Dnnv in terms of Dijv for i = 1 . . . . . n and j = 1. . . . . n - 1. Therefore, from (5.4) we have

l

Iz:vl 2 dx <_C/R/n

E iI

IDwkl~dx.

(5.6)

Using the Poincar6 inequality and the Cacciopoli inequality (5.5), we obtain

B;

ID,,v - (Dnv)alEdx + E

k = 1 B;

IDkvl 2 dx < C --

-e

k = 1 B~

[Dkv[z dx

(5.7)

for 0 < p < R/2. Then by (5.7) and because IB;~lu - (u)RI 2dx <- fn;~lu - zl2dx for every z e R, as in the proof of (b) of theorem 4.1 we easily see that (5.7) implies

Cb(u, Xo, p) < C ~ where ~ is defined by (4.7) and p < ep • H~(B~), we have

¢(u, Xo, R) + C

ID(u - v)l z dx

(5.8)

R/2. We now estimate the term I [D(u - v)l 2 dx. For

t"B~aij(Xo' U(X°))(DJU - Dju)Di~ dx <- IB~ I(aij(X°' g(X°)) - gij(X' u))DjuDi~)I dx

÷i" IB(x,u,Du)[16l ÷ J B~

Ildx.

In the sequel, we use ai to denote various positive constants which depend on the parameters of the hypothesis but not on R. Taking cb = u - v • H~(B~) and using the hypothesis, we get

l lD(u-v)[2dx<-CR'~°f IDullD(u-v)ldx+ f IDul2lu-vldx + l

÷ t'

B~

J B~

b[Dullu- vl dx + f clul lu- vl dx B~

glu - v[ dx + [(T. u - v>l.

(5.9)

For any e > 0. the first integral on the right side can be treated by the Young inequality and then majorized by

eI

B~

[D(u-v),2dx+C(e)R'~l

I

B~

,Dul2dx.

(5.10)

A class of singular quasilinear elliptic equations

1901

For the other terms, we first note that from the maximum principle and the Holder continuity of u there exists ~3 > 0 such that sup lu

- vl < CR '~3.

Therefore, the second integral on right of (5.9) can be bounded as in (5.10). Next, using the Young inequality and condition on b, g, we see that

t" blDu"u - v' dx <- t' el,

e~

'Du'Z'u - v' dx + f

e~

bZ'u - v' dx

< CR'~' t" IOulzdx + CRX + l" O[u - v{Zdx Oe~ J e~ l

J n/~

glu - V[ dx <_ CR x + i

3e~

Olu -

u[2dx

(5.11)

(5.12)

and

l

e~

Because u - v = 0 on

n~

c[ullu-vldx<-I OB~, by

e~

Ou2dx+ 18~ Olu - vl 2 ~.

Lemma 2.3 we have

O[u - vl2 dx <_ e

B~

ID(u - v)lZdx + C(e) f B~ lu

vl 2

<- e ln~ ID(u - v)12dx + C(e)R "*~3. This is also applied for the last term in (5.11) and (5.12). Finally, Lemma 2.2 and u = 0 on F give

i Ou2dx< CRC~4llOd~#l'Lr(a)f ]Oul2dx n~

where o~4 = fl as follows

n/r

el,

+ 2 > 0. The last term o f (5.9) can be treated by the Young inequality

f l - < IITIl.,-.~,e~)llD(u - v)ll..g,~(..~ ~ C(e)R ~ + e i

J s~

In(u

U)[2 dx.

Altogether, for R small (say, R < Ro), we deduce from (5.9) and the above estimates

ID(u -

v)[2 dx

< R"

IDulZdx + CR "+~

with z = min[eti, 2 - n} > 0. This and (5.8) yield O(u, Xo, P) < C

O(u, Xo, R) + R ~

l

el

(1 +

]Du[z) dx

(5.13)

1902

LE DUNG

for any p < R/2 and x0 ~ F. It is trivial when p >_ R/2. If xo ~ F and B R (Xo) C ~ , by the similar and easier arguments we can prove that (5.13) also holds with ¢~(U,Xo,p) given by (4.6). Consequently, lemma 4.2 can be applied here to give our theorem. • Acknowledgements--It is a pleasure to thank my advisor, Professor Alberto Verjovsky, and my teacher, Professor Duong Minh Due, for their warm support and continual advice. The author would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for the hospitality at the International Centre for Theoretical Physics, Trieste. REFERENCES 1. RAKOTOSON J. M., Equivalence between the growth of . . . . J. diff. Eqns 86, 102-122 (1990). 2. RAKOTOSON J. M., Quasilinear equations and spaces of Campanato-Morrey, Communs partial diff. Eqns 16(6,7), 1155-1182 (1991). 3. RAKOTOSON J. M. & ZIEMER W. P., Local behavior of solutions of quasilinear elliptic equations with general structure, Trans. Am. math. Soc. 319(2) (1990). 4. LADYZENSKAYA O. A. & URAL'TSEVA N. N., Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968). 5. TRUDINGER N. S., Linear elliptic operators with measurable coefficients. Annali Scu. norm. sup. Pisa 27(3), 265-308 (1973). 6. TRUDINGER N. S., On Harnack type inequalities and their applications to quasilinear elliptic equations. Communs pure appl. Math. 20, 721-747 (1967). 7. GILBARG D. & TRUDINGER N. S., Elliptic Partial Differential Equations o f Second Order. Springer, Berlin (1983). 8. CAMPANATO S., Equazioni ellitiche del II ordine e spazi L 2"1. Annali Mat. pura. appL 69(4), 321-381 (1965). 9. KADLEC & NECAS J., Sulla regolarita delle soluzioni di equazioni ellitiche negli spazi H k't. Annuli Scu. norm. sup. Pisa 3, 21 (1967). 10. DUC D. M. & LE DUNG, On the regularity of solutions of singular linear elliptic equations. ICTP preprint 1992, IC/92/401. (Submitted for publication.) 11, LE DUNG, Boundary C 1'~ regularity of singular semilinear elliptic equations. ICTP preprint 1993, IC/93/35. 12. LIEBERMAN G. M., Sharp forms of estimates for subsolutions and supersolutions of quasilinear elliptic equations. (Preprint.) 13, KILPELAINEN & MALY, The Wiener test and potential estimates for quasilinear elliptic equations. (Preprint.) 14. GIAQUINTA M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, New Jersey (1983). 15. MAZ'DA V., Sobolev Spaces. Springer, Berlin (1985). 16. ZIEMER W. P., Weakly Differentiable Functions. Springer, Berlin (1989). 17. MOSER J. K., On Harnack's theorem for elliptic differential equations. Communspure appl. Math. 14, 577-591 (1961). 18. DIBENEDETTO E., C I÷'~ local behavior of solutions of quaslinear equations. Nonlinear Analysis 7, 827-850 (1983). 19. LIEBERMANN G. M., The natural generalization of the natural conditions of Ladyzenskaya and Ural'tseva for elliptic equations. Communs partial d~ff. Eqns 16(2, 3), 311-361 (1991). 20. MANFREDI J. J., Regularity for minima of functionals with p-growth. J. diff. Eqns 76, 203-212 (1988). 21. ZEIDLER E., Nonlinear Functional Analysis and its Applications. Nonlinear Monotone Operator. II/B. Springer, Berlin (1990). 22. CAMPANATO S., Sistemi ellitici in forma divergenza. Regolarita alrinterno, Quaderni Scuola Norm. Sup. Pisa (1980).