We prove the existence of vectorvalued solutions for systems of the 

type −∂α(|∇u| p−2∂αu) = T  in a bounded open set Ω, u = 0 on ∂Ω, where T is a given distribution acting on the space C∞o (Ω,IRN ) of testfunctions

J. REULING

M. FUCHS

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Rendiconti di Matematica, Serie VIIVolume 15, Roma (1995), 311-319Non-linear elliptic systems involving measure dataM. FUCHS – J. REULINGRiassunto: Si dimostra l’esistenza di soluzioni a valori vettoriali per sistemi deltipo $"!(|%u|p!2"!u) = T in un insieme aperto e limitato " con valore nullo sullafrontiera ""; T è una distribuzione assegnata che agisce sullo spazio delle funzioni test.Abstract: We prove the existence of vectorvalued solutions for systems of thetype $"!(|%u|p!2"!u) = T in a bounded open set ", u = 0 on "", where T is a givendistribution acting on the space C%o (", IRN ) of testfunctions.– IntroductionFor a smooth bounded domain " 3 IRn, n 2 3, and a given distribu-tion T acting on the space C$o (", IRN) of vectorvalued testfunctions wewant to solve the nonlinear Dirichlet–problem(1)=$9!(|;u|p"29!u) = T on "u|%! = 0,where p > 2 $ 1ndenotes a fixed real number and equation (1) has to beunterstood in the sense that(2),!|;u|p"2;u · ;' = T (')Key Words and Phrases: Nonlinear elliptic systems – ExistenceA.M.S. Classification: 35R05 – 35 J 60312 M. FUCHS – J. REULING [2]holds for all ' " C$o (", IRN). For T su$ciently regular it is easy toconstruct a unique solution u of (1) in the space H1,p0 (", IRN) for exampleby applying variational methods. On the other hand the lefthand sideof (2) makes sense for functions u in the space H1,p"10 (", IRN) whichsuggests to relax the assumptions concerning the integrability of ;u. Infact, certain applications in physics (see [1]) lead to the study of equations(N = 1) of the type (1) where T =l3i=1.ai , ai " ", is a finite sum of Dirac–measures. In [7] and [8] Kichenassamy and Veron proved the existenceof solutions u in this case. Moreover, they showed|u(x)| C const |x $ ai|p!np!1 at least locally,that isu "S1+q< nn!1 (p"1)H1,q0 (", IR).The results of [7] are valid for any exponent p in the range (1,/).But if we consider the fundamental solution of the p-Laplacian #(x) =c(n, p)|x|p!np!1 , (N = 1, p < n) we see ;# " L1 near 0 if and only ifp > 2 $ 1n. This motivates the lower bound for p.Independently Boccardo and Gallouët [2] discussed equation (1)(N = 1 !) for signed Radon–measures T and obtained existence theoremsin H1,q0 (", IR), q <nn"1(p$1). In this note we concentrate on the vectorialcase N > 1 and proveTheorem 1. Let T denote a vectorial Radon–measure on " suchthat Ln(sptT ) = 0 and |T |(") < /. Then (1) admits a solution u in thespace S1+q< nn!1 (p"1)H1,q0 (", IRN) .Remarks.1. The theorem applies to measures of the formL3i=1Ei.ai , Ei " IRN , ai "", i " {1, . . . , L}, or HmDf, Hm = m-dimensional Hausdor!-measure,f : " $) IRN continuous with support contained in some m-dimensionalsubmanifold of ", m < n.[3] Non-linear elliptic systems involving measure data 3132. The condition Ln(sptT ) = 0 excludes measures like LnDf with f "L1(", IRN). The case of arbitrary Radon–measures causes some technicaldi$culties which can be overcome with the help of the “blow–up lemma”from [3]. Since the details are rather involved we will present them in asubsequent paper.3. In the linear case we can consider N equations and quote the results ofLittman, Stampacchia and Weinberger [9]. For p > n problem (1)is solvable with direct methods in H1,p0 (", IRN) for any vectorial Radon–measure of finite mass because T is in the dual space of H1,p0 (", IRN).1 – Proof of the TheoremFrom now on we assume that the hypothesis of the Theorem aresatisfied and that p " (2 $ 1n, n]. The first step in the proof is to ap-proximate the distribution T by a sequence of regular distributions Thk ,generated by functions hk " L1(", IRN) such that Thk $)T in the senseof distributions.Replacing T by Th, h " L1(", IRN) we can rewrite equation (2) as(1.1),!|;u|p"2;u · ;' =,!h · ' # ' " C$o (", IRN).Next we recall an apriori estimate for u satisfying (1.1).Lemma 1. Let u " H1,p0 (", IRN) denote the solution of (1.1). Forevery 1 . q < nn $ 1(p $ 1) there exists a positive constant C dependingonly on n, N, ", 1h11 such that1u11,q . C.The proof of Lemma 1 follows the lines in [2] where the scalar caseis treated. We just have to modify the testfunctions; define01(t) =IJJJJJKJJJJJL1 : t > k + 1t $ k : k . t . k + 10 : $k < t < kt + k : $k $ 1 . t . $k$1 : t < $k $ 1, k " IN314 M. FUCHS – J. REULING [4]and test (1.1) with'j := 01(uj) ej, j " {1, . . . , N},where ej is the j-th unitvector in IRN .We get,[k+|uj |+k+1]|;uj|p . 1h11 for every j " {1, . . . , N} .Choosing02(t) =IKLL : t > Lt : $L . t . L$L : t < $L, L " INand testing with 'j := 02(uj) ej gives,[|uj |+L]|;uj|p . L 1h11 for every j " {1, . . . , N} .Let q! denote the Sobolev exponent for q: q! :=nqn $ q . Hölder’sinequality and Sobolev’s imbedding Theorem imply (see [2] for details):1uj1qq& . Cand1;uj1qq . C(n, N, ", 1h11)for every j " {1, . . . , N}.Next we choose an approximating sequence (hk)k'IN 3 L1(", IRN)such that spt(hk) 3 Kk :=1x " ")) dist(x, K) . 1k2, K := spt(T ) andsupk1hk11 < /. Let uk " H1,p(", IRN) denote a solution of(1.2)=$9!(|;u|p"29!u) = hk in "u = 0 on 9",[5] Non-linear elliptic systems involving measure data 315and fix a ball BR in " such that BR 4 Kk = 7.Lemma 1 implies 1uk1H1,q . C for every 1 . q <nn $ 1(p $ 1).Consequently there exists u " Q1+q< nn!1 (p"1)H1,q0 (", IRN) such that uk : uweakly as k ) /. Since uk is p-harmonic on BR we have Caccioppoli’sestimate(1.3),BR/2|;uk|p .CRp,BR|uk|p .Using (1.2) for uk and uk# with the testfunction ' = &p (uk $ uk#), where& is the usual cuto! function& ==1 on BR/40 on " \ BR/2, 0 . & . 1, |;&| . CRwe get with an ellipticity argument (in case p 2 2)(1.4),BR/4|;uk $ ;uk# |p ..,BR/2&p|;uk $ ;uk# |p. c,BR/2&|;uk|p"2;uk $ |;uk# |p"2;uk#' ; (uk $ uk#) &p. c,BR/2&|;uk|p"2;uk $ |;uk# |p"2;uk#'(uk $ uk#) ;& &p"1. C(R)8 ,BR/2(|;uk|p + |;uk# |p)91" 1p 8 ,BR/2|uk $ uk# |p9 1p.Therefore (1.4) together with (1.3) gives(1.5),BR/4|;uk$;uk# |p . C(R)8 ,BR|uk$uk# |p9 1p8 ,BR|uk|p+|uk# |p91" 1p.316 M. FUCHS – J. REULING [6]For 2 $ 1n< p < 2 we argue as follows:,BR/2&p|;uk $ ;uk# |p .. c8 ,BR/2&2|;uk $ ;uk# |2 (|;uk| + |;uk# |)p"29 p2··8 ,BR/2(|;uk| + |;uk# |)p9 2!p2.. c8 ,BR/2&|;uk|p"2;uk $ |;uk# |p"2;uk#'&2 (;uk $ ;uk#)9 p2··8 ,BR/2(|;uk|p + |;uk# |p)9 2!p2.. C(R)8 ,BR/2|uk $ uk# |p9 128 ,BR/2|;uk|p + |;uk# |p9 12which proves the appropriate version of (1.4). Quoting (1.3) we alsoobtain an inequality of the form (1.5).We now use the compactness of the imbedding H1,q0 (", IRN) 7)Lp(", IRN)which is true if p < q!. Since q! <8nn $ 1(p $ 1)9!=n(p $ 1)n $ p , thisrestriction is fulfilled if n < p2. In this case (1.5) gives,BR/4|;uk $ ;uk# |p $) 0.We summarize;uk $);u in Lploc(" \ K, IRnN);uk(x) $);u(x) a.e. on " \ K;uk(x) $);u(x) a.e. on " at least for a subsequence.[7] Non-linear elliptic systems involving measure data 317Here our assumption Ln(K) = 0 enters. Since |;uk|p"1 is bounded inLqp!1 (", IRN), qp"1 <nn"1 we have |;uk|p"2;uk :: F in Lqp!1 (", IRN).With Egoro!s theorem we conclude |;uk|p"2;uk : |;u|p"2;u inLqp!1 (", IRN). Finally,!hk · ' =,!|;uk|p"2;uk · ;' $),!|;u|p"2;u · ;',i.e. u " Q1+q< nn!1 (p"1)H1,q0 (", IRN) is a solution of equation (1).To get rid of the restriction p2 > n we have to establish a maximumprinciple, i.e. we have to show(1.6) supBR/2|;uk|p . const .To prove (1.6) we show at first that uk is a subsolution of a certain ellipticoperator in BR.Lemma 2. Let us suppose that p 2 2 and v " H1,p(BR, IRN) isp-harmonic on BR. Then |;v|p solves,BRA!"(x)9" (|;v|p) 9!& . 0 # & " H1,20 (BR), & 2 0,with coe"cients A!" " L$(BR) satisfyingA!" %!%" 2 |%|2 # % " IRn.Proof. We follow the lines in [5] and compute A!" asA!" =p2.!" + p(p2$ 1)9!vi(x) 9"vi(x)|;v(x)|2 3[.v &=0].Now we can quote [4, p.86 “Remark”] (see also [10] for a completeproof) to getsupBR/2|;uk|p . c(R, !)8,B|;uk|p!91/!# ! > 0.318 M. FUCHS – J. REULING [8]For ! <nn $ 1.1 $ 1p/we arrive atsupBR/2|;uk|p . c(R, q)8,B|;uk|q9n"1npp"1for every q <nn $ 1(p$1) and we end up with (1.6). The case 2$1n< p <2 requires some minor technical modifications. The necessary changes canbe found in [6] or [11].It is now immediate that 1uk1H1,p(BR/2) . c(R) and therefore uk $)ustrongly in Lp(BR/2).The same calculation with & defined as above gives,BR/2&p|;uk $ ;uk# |p .. C(R)8 ,BR/2(|;uk|p + |;uk# |p)91" 1p# $! "+C(R)<$8 ,BR/2|uk $ uk# |p9 1p# $! ""*0.It follows;uk $);u in Lp(BR/2, IRnN);uk $);u in Lploc(" \ K, IRnN);uk(x) $);u(x) a.e. on " \ K;uk(x) $);u(x) a.e. on " at least for a subsequence,and the proof can be finished as before.[9] Non-linear elliptic systems involving measure data 319REFERENCES[1] S.L. Adler – T. Piran: Relaxation methods for gauge field equilibrium equations,Rev. Mod. Phys. 56 (1984), 1-40.[2] L. Boccardo – T. Gallouët: Non-linear Elliptic and Parabolic Equations In-volving Measure Data, J. Functional Anal. 87 (1989), 149-166.[3] M. Fuchs: The Blow-up of p-harmonic Maps, manuscripta math. 8 1 (1993),89-94.[4] M. Giaquinta: Introduction to Regularity Theory for Nonlinear Elliptic Systems,Lecture in Mathematics ETH Zürich, Birkhäuser, 1993.[5] M. Giaquinta – G. Modica: Remarks on the Regularity of Minimizers of certainDegenerate Functionals, manuscripta math. 57 (1986), 55-99.[6] C. Hamburger: Regularity of di!erential forms minimizing degenerate ellipticfunctionals, J. reine angew. Math. 431 (1992), 7-64.[7] S. Kichenassamy: Quasilinear Problems with Singularities, manuscripta math.57 (1987), 281-313.[8] S. Kichenassamy – L. Veron: Singular Solutions of the p-Laplace Equation,Math. Ann. 275 (1986), 599-615.[9] W. Littman – G. Stampacchia – H.F. Weinberger: Regular points for ellipticequations with discontinuous coe"cients, Ann. Scuola norm. sup. Pisa, Sci. fis.mat., III Ser. 17 (1963), 43-77.[10] L. Simon: Lecture notes on P.D.E., unpublished.[11] P. Tolksdorf: Everywhere-Regularity for Some Quasilinear Systems with a Lackof Ellipticity , Ann. Mat. Pura Appl. 134 (1983), 241-266.Lavoro pervenuto alla redazione il 26 ottobre 1994ed accettato per la pubblicazione il 15 febbraio 1995.Bozze licenziate il 10 aprile 1995INDIRIZZO DELL’AUTORE:Martin Fuchs – Jürgen Reuling – Fachbereich Mathematik – Universität des Saarlandes – D -66041 Saarbrücken – Germany

Non-linear elliptic systems involving measure data

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