In this paper we prove a Harnack inequality for some quasilinear elliptic equations, extending the results in [6] and [5]

P. ZAMBONI

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Rendiconti di Matematica, Serie VIIVolume 15, Roma (1995), 251-262Local behavior of solutions of quasilinear ellipticequations with coe"cients in Morrey spacesP. ZAMBONIRiassunto: In questa nota proviamo una disuguaglianza di Harnack per equazioniquasilineari di tipo ellittico, estendendo i risultati contenuti in [6] e [5].Abstract: In this paper we prove a Harnack inequality for some quasilinear el-liptic equations, extending the results in [6] and [5].1 – IntroductionIn his work [6] J. Serrin extended the Harnack inequality to quasi-linear equations of the form(1.1) divA&x, u(x),;u(x)' + B&x, u(x),;u(x)' = 0 .In his paper Serrin assumes that the coe$cients in the structure condi-tions belong to appropriate Lp spaces. Precisely his assumptions are(1.2)IJJJJKJJJJLa = constantb(x), e(x) " Ln/(p"1)c(x) " Ln/(1"-)d(x), f(x), g(x) " Ln/(p"-)Key Words and Phrases: Quasilinear equations – Morrey spaces - Harnack inequaltyA.M.S. Classification: 35J60 – 35B65252 P. ZAMBONI [2]where a, b, c, d, e, f and g are the coe$cients in the structure conditionsgiven in (2.1) of the following section 2 and * is a positive small number.Recently Rakotoson and Ziemer in [5] somewhat improved theabove mentioned result by Serrin. Indeed in Theorem 3.12 of [5] a Har-nack inequality is proved for equationdivA&x, u(x),;u(x)' + B&x, u(x),;u(x)' = µunder the following assumptions(1.3)IJJJJJJJKJJJJJJJLa '= constant , c(x) " L$locb(x) = d(x) = 0e(x) " Ln/(p"1+-)f(x) " Ln/(p+-) + L1,,g(x) " Ln/(p+-) .We wish to point out that, because of the form of the L$ estimatethey get in Theorem 3.4, their constant is blowing up as the radius ofthe ball, on which the inequality is considered, approaches zero. This inparticular makes it impossible to deduce Hölder continuity of the localsolution from Harnack inequality.Our aim in this note is to extend the validity of the Harnack inequal-ity in both papers [6] and [5] weakening the assumptions in [6] and [5] andobtaining the very same conclusion as in Serrin, i.e. an inequality withconstant independent of the radius of the ball. This is done in Theorem 2in the following section 4.As an obvious consequence we obtain the local Hölder continuity ofsolutions improving Theorem 3.7 in [5], Corollary 1 in [4] and our ownTheorem 3 in [7].Essentially our hypotheses are done using everywhere the Morreyspace scale instead of the Lp spaces and it can be shown by examplesthat in this scale they are optimal in order to have the boundedness ofthe local solutions.A comparison between our assumptions and the above recalled onescan be found in Remark 1 at the end of this paper.About the technique we tried to follow as closely as possible the bynow classical work [6], i.e. we will use the Moser’s iteration technique.[3] Local behavior of solutions of quasilinear elliptic etc. 253Clearly, because of the lack of high integrability of our coe$cients,many modifications are needed, which are pointed in the proof of thetheorems below. In particular we have to use on a regular basis Adamsinequality (see Lemma 1 below) instead of Hölder plus Sobolev to estimateproducts of coe$cients times test functions. This we learnt from thepaper [5] and the later [4].We did not give the details of the proof whenever it is essentially thesame as in Serrin only stressing the di!erences in treatment which leadto the basic inequalities.After this work was completed we heard by Professor Ziemer, whosekindness we gratefully aknowledge, that G. Lieberman has recently ob-tained a result similar to ours. In his work [3] G. Lieberman provesL$ estimate and Harnack principle for positive solutions of a quasilinearsecond order equation. While the structure assumptions seem to be moregeneral, in some cases, than ours, the class of solutions he considers isdi!erent. This because he is forced by the structure to assume a priorithe boundedness of the solutions. This, on the contrary, we prove inTheorem 1 in this paper.2 – Structure hypotheses and preliminary resultsLet " be a bounded open subset of IRn. LetA(x, u, %) : " & IR & IRn $) IRnandB(x, u, %) : " & IR & IRn $) IRtwo continuous functions satisfying inequalities of the form(2.1)IJKJL))A(x, u, %))) . a|%|p"1 + b|u|p"1 + e))B(x, u, %))) . c|%|p"1 + d|u|p"1 + f% · A(x, u, %) 2 |%|p $ d|u|p $ gfor a.e. x " ", #u " IR, # % " IRn. Here p is a fixed number in ]1, n[, ais a positive constant and the functions b(x), c(x), d(x), e(x), f(x) and254 P. ZAMBONI [4]g(x) are such that(2.2)IJJKJJLb(x), e(x) " Lpp"1 ,,c(x) " Lp,,d(x), f(x), g(x) " L1,,where ( = n $ p + !, ! > 0.We recall that for q " [1,+/[, & "]0, n[, Lq,& = Lq,&(") denotes theclassical Morrey space of the functions f " Lq(") such thatsupx$!">015&,B(x,.))!))f(y)))qdy = 1f1qq,&;! < +/where B(x, 5) is the ball centered at x with radius 5 (whenever x is notrelevant we will write B.).In the following two lemmas are vitalLemma 1 (Adams [1]). Let µ a nonnegative measure in IRn suchthat for all x " IRn and 0 < r < +/, there is a constant M with theproperty that µ4B(x, r)5 . Mr,, where ( = sp(n $ p), 1 < p < s < +/and p < n. If u " C$0 (IRn) then8 ,IRn|u|sdµ9 1s . C M1s8 ,IRn|;u|pdx9 1pwhere C = C(p, (, n).Lemma 2 (Poincaré, John-Nirenberg [2]). Let Br̃ 3 Rn, u(x) "H1,p(Br̃) and suppose that for all Br 3 Br̃ there exists a constant K suchthat8 ,Br|;u|pdx91/p. Kr(n"p)/p .Then there exist two positive constants p0 and C depending on K, p, nsuch that8 ,Br̃ep0udx9 8 ,Br̃e"p0udx9. C|Br̃|2 .[5] Local behavior of solutions of quasilinear elliptic etc. 255A function u(x) is said to be a local solution of (1.1) in " if u "H1,ploc (") and,!6A&x, u(x),;u(x)';'(x)+B&x, u(x),;u(x)''(x)7dx = 0(2.3)#' " C$0 (") .Using Young’s inequality and Lemma 1 it is immediately seen that(2.3) is meaningful.3 – Local boundedness of solutionsThe purpose of this section is to show that weak solutions of equation(1.1) are locally bounded.Theorem 1. There exist a positive number r0, independent of u,such that, if we assume that u(x) is a local solution of equation (1.1) in", that conditions (2.1) and (2.2) hold and that Br and B2r are balls withthe same center with B2r 33 ", then for r . r0 we have1u1L%(Br) . Cr"n/p=8 ,B2r|u|pdx91/p+ hrn/p@whereh = (rp"1p!1e1 pp"1 ,,;B2r+ r!1f11,,;B2r)1p"1 + (r!1g11,,;B2r)1p .Proof. First we prove the theorem for the special case r = 1.Set v = |u|+h, where h is a positive constant. From (2.1) we deduce(3.1)IJKJL))A(x, u, %))) . a|%|p"1 + b1|v|p"1))B(x, u, %))) . c|%|p"1 + d1|v|p"1% · A(x, u, %) 2 |%|p $ d1|v|pwhere b1 = b + h1"pe and d1 = d + h1"pf + h"pg.256 P. ZAMBONI [6]Thus, settingh =&1e1 pp"1 ,,;B2+ 1f11,,;B2' 1p"1 +&1g11,,;B2' 1pwe have that the norms of b1 and d1 are bounded,(3.2)= 1b11 pp"1 ,,;B2. 1b1 pp"1 ,,;B2+ 11d111,,;B2 . 1d11,,;B2 + 2 .For fixed numbers q 2 1 and 6 > h we consider the functionsF (v) ==vq if h . v . 6q6q"1v $ (q $ 1)6q if 6 . vandG(u) = signu4F (v)F %(v)p"1 $ qp"1h"5 , $/ < u < +/where " is such that pq = p + " $ 1.As a test function in (2.3) we take'(x) = &p(x)G(u)where &(x) is a non negative smooth function with support in B2.Substituting '(x) in (2.3) and using the assumptions (3.1) we obtain,as in [6],(3.3),B2&p|;w|pdx . ap,B2))(;&)w))))&(;w)))p"1dx++ qp"1p,B2b1))(;&)w))|&w|p"1dx++,B2c|&w|))&(;w)))p"1dx + (1 + p)qp"1,B2d1|&w|pdx ,where w = w(x) = F (v).[7] Local behavior of solutions of quasilinear elliptic etc. 257We simplify (3.3) using Young’s inequalityabp"1 . 1p*(1"p)ap +.1 $ 1p/*bp , # * > 0obtaining(3.5),B2&p|;w|pdx . C1(p + 1)qpA ,B2))w(;&)))pdx +,B2|&w|pdµBwhere C1 is a positive constant depending only on p and a and µ denotesthe measure defined bydµ = (bpp"11 + cp + d1)dx .Noting that for 0 < 5 < +/ we haveµ[B.] . M5,whereM = 1b11p/(p"1)p/(p"1),,;B2 + 1c1pp,,,B2+ 1d111,,;B2using Lemma 1 we obtain(3.6),B2|&w|pdµ .8 ,B2|&w|sdµ9p/sµ(B2)1"p/s .. C2(p, (, n)MA ,B2))(;&)w))pdx +,B2))(;w)&))pdxBwhere s is defined by (n $ p)s/p = (.Substituting (3.6) in (3.5) and assuming that(3.7) C1(p + 1)C2M < 1we obtain **&(;w)**Lp(B2). C3(p, a)q**w(;&)**Lp(B2)258 P. ZAMBONI [8]and by Sobolev inequality we have(3.8) 1&w1Lp&(B2). C4q**w(;&)**Lp(B2)where C4 is a constant depending only on p, a and n.Now proceeding exactly as in [6] pp. 258-259, by the familiar iterationprocedure, we obtain1u1L%(B1) . C11u1Lp(B2) + h2.This proves the theorem in the special case r = 1. The general caser %= 1 is obtained by dilation.We note that, in this case, condition (3.7) yields the number r0 inthe statement of the theorem, this is easily seen having in mind the wayin which the Lp,, norms in the “constant” M vary under dilation.4 – Harnack’s inequality and the Hölder continuity of solutionsTheorem 2. There exists a number r0 > 0, independent of u, suchthat if assume that u(x) is a non negative local solution of equation (1.1)and that conditions (2.1) and (2.2) hold, then for r . r0 and B3r 33 "we havemaxBru(x) . C1 minBru(x) + h2whereC = C&p, n, a, (, r0, 1b1p/(p"1),,;B3r , 1c1p,,;B3r , 1d11,,;B3r'h =&rp"1p!1e1 pp"1 ,,;B3r+ r!1f11,,;B3r' 1p"1 +&r!1g11,,;B3r' 1p .Proof. Also in this case we prove the theorem assuming r = 1.Proceeding as in Theorem 1, setting v = |u| + h, we deduce the newconditions (3.1) and (3.2) and then, taking as a test function in (2.3)'(x) = &pv", where &(x) is a non negative smooth function with support[9] Local behavior of solutions of quasilinear elliptic etc. 259in B3 and " " IR, we obtain(4.1),B3|;v|p&pv""1dx . C1(p, a)&1 + |"|"1'A ,B3|;&|pvp+""1dx++,B3&pvp+""1dµB.Here we may assume that v(x) 2 * > 0. Otherwise we may replacev by v + * and let * ) 0 in the final result.Settingw(x) ==vq(x) where pq = p + " $ 1 if " %= 1 $ plog v(x) if " = 1 $ p(4.1) yields(4.2),B3&p|;w|pdx . C1|q|p&1 + |"|"1'pA ,B3|;&|pwpdx++,B3&pwpdµBif " %= 1 $ p(4.2)%,B3&p|;w|pdx . C1A ,B3|;&|pdx +,B3&pdµBif " = 1 $ p .We consider first the (4.2)%. Using Lemma 1 we have(4.3),B3&p|;w|pdx . C2,B3|;&|pdxwhere C2 is a positive constant depending on p, n, a, (, 1b1p/(p"1),,;B3 ,1c1p,,;B3 and 1d11,,;B3 .Let Bh be an arbitrary open ball contained in B2 with the samecenter. We choose &(x) so that &(x) = 1 in Bh, 0 . & . 1 in B3 \ Bh and|;&| . 3/h, h > 1.From (4.3) we have1;w1Lp(Bh) . C3h(n"p)/p260 P. ZAMBONI [10]where C3 is a positive constant depending on the same arguments of C2.Thus, from Lemma 2, we have that there exist two positive constantsp0 and k depending on p, n and C3 such that8 ,B2ep0wdx9 8 ,B2e"p0wdx9. K .Since w = log v we have(4.4) /(p0, 2) . K/($p0, 2)where, for any real number p %= 0, we have/(p, h) =8 ,Bh|v|pdx91/p.We consider now the (4.2). Using Lemma 1 and proceeding as inTheorem 1 we have(4.5),B3&p|;w|pdx . C1|q|p&1 + |"|"1'p,B3|;&|pwpdxprovidedC1|q|p&1 + |"|"1'pC(p, (, n)11b11p/(p"1)p/(p"1),,;B3 + 1c1pp,,,B3+ 1d111,,;B32< 1and using Sobolev inequality, (4.5) yields(4.6) 1&w1Lp&(B3). C4|q|&1 + |"|'**(;&)w**Lp(B3)where C4 is a positive constant depending only on p, a and n.Now following the pattern of [6] pp. 267-268 we have/(/, 1) . C %/(p%0, 2)and/($/, 1) 2 C %%/(p0, 2)[11] Local behavior of solutions of quasilinear elliptic etc. 261where C % and C %% are two positive constants depending only on p, a andn, and p%0 . p0. Recalling (4.4) we obtain/(/, 1) . C/($/, 1)where C = C&p, n, a, (, 1b1p/(p"1),,;B3 , 1c1p,,;B3 , 1d11,,;B3'that ismaxB1u(x) . C1 minB1u(x) + h2.The general case r %= 1 follows again by dilation.As in [6], using Harnack’s inequality, we deduce the following resultCorollary 1. Let u(x) be a weak solution of equation (1.1). Ifwe assume that (2.1) and (2.2) hold, then u is locally Hölder continuouson ".Remark 1 We wish to point out that it can be seen by simple ex-amples that our assumptions (2.2) on the coe$cients of the structureconditions are sharp in the scale of Morrey spaces (for more details seeRemark 2 in [7]).Furthermore, we wish to show that our hypotheses are more generalthan those in [6] and, at least in some instances, than those in [5] and [4].Indeed comparing (2.2) with (1.2) we haveLn/(p"-) ! L1,, ; Ln/(1"-) ! Lp,, ; Ln/(p"1) ! Lp/(p"1),, .While, with respect to [5] where it is assumed b = d = 0 we haveb " Lp/(p"1),, d " L1,, and about e, f and g the comparison goes similarlyto the above with [6].Similar considerations can be seen to hold with respect to [4].262 P. ZAMBONI [12]REFERENCES[1] D. Adams: Traces of potentials arising from translation invariant operators, Ann.Scuola Norm. Sup. Pisa 25 (1971), 203-217.[2] F. John – L. Nirenberg: On function of bounded mean oscillation, Comm. PureAppl. Math., 14 (1961), 415-426.[3] G.M. Lieberman: Sharp forms of estimates for subsolutions and supersolutionsof quasilinear elliptic equations involving measures, Comm. P.D.E., 18 (1993),1191-1212.[4] J.M. Rakotoson: Quasilinear equations and spaces of Campanato Morrey type,Comm. P.D.E. 16 (1991), 1155-1182.[5] J.M. Rakotoson – W.P. Ziemer: Local behavior of solutions of quasilinearelliptic equations with general structure, Trans. Amer. Math. Soc., 319 (1990),747-764.[6] J. Serrin: Local behavior of solutions of quasilinear equations, Acta Math., 111(1964) 247-302.[7] P. Zamboni: Local boundedness of solutions of quasilinear elliptic equations withcoe"cients in Morrey spaces, Boll. U.M.I. 8-B (1994), 985-997.Lavoro pervenuto alla redazione il 16 febbraio 1994ed accettato per la pubblicazione il 26 ottobre 1994.Bozze licenziate il 23 febbraio 1995INDIRIZZO DELL’AUTORE:P. Zamboni – Università di Messina – Dipartimento di Matematica – Contrada Papardo –Salita Sperone 31 – 98166 Sant’Agata – MESSINA – ITALIA.

Local behavior of solutions of quasilinear elliptic equations with coefficients in Morrey spaces

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Local behavior of solutions of quasilinear elliptic equations with coefficients in Morrey spaces

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