EXISTENCE OF WEAK SOLUTIONS FOR ELLIPTIC SYSTEMS WITH p,q-growth

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1 Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 40, 015, EXISTENCE OF WEAK SOLUTIONS FOR ELLIPTIC SYSTEMS WITH p,q-growth Giovanni Cupini, Francesco Leonetti and Elvira Mascolo Università di Bologna, Dipartimento di Matematica Piazza di Porta S. Donato 5, 4016 Bologna, Italy; Università di L Aquila, Dipartimento di Ingegneria e Scienze dell Informazione, Matematica L Aquila, Italy; leonetti@univaq.it Università di Firenze, Dipartimento di Matematica e Informatica U. Dini Viale Morgagni 67/A, Firenze, Italy; mascolo@math.unifi.it Abstract. We consider a non-linear system of m equations in divergence form and a boundary condition: A i x,dux))) = 0, 1 m, in x i=1 i u = ũ on. The functions A i x,z) are Hölder continuous with respect to x and m z p c 1 A i x,z)zi c 1+ z ) q, p q. We prove the existence of a weak solution u in ũ+w 1,p 0 ;R m )) W 1,q loc ;Rm ), provided p and q are close enough and under suitable summability assumptions on the boundary datum ũ. 1. Introduction In this paper we are concerned with the existence and regularity of solutions to the Dirichlet problem associated to a non-linear system of m equations in divergence form 1.1) i=1 u = ũ A i x,dux))) = 0, 1 m, in x i on, where is a bounded open set in R n, n, and ũ is a vector-valued function in a suitable Sobolev space. We assume that A i : R nm R, 1 i n, 1 m, are Carathéodory functions satisfying the following properties: there exist p q, 0 < γ 1 and ν,m,c > 0, such that for every x,y and for every z, z R nm : doi: /aasfm Mathematics Subject Classification: Primary 35J5; Secondary 35J47, 49N60. Key words: Existence, regularity, weak, solution, elliptic, system, growth. Acknowledgement: We would like to thank Dominic Breit for raising a question that led us to the present study. The authors have been supported by the Gruppo Nazionale per l Analisi Matematica, la Probabilità e le loro Applicazioni GNAMPA) of the Istituto Nazionale di Alta Matematica INdAM).

2 646 Giovanni Cupini, Francesco Leonetti and Elvira Mascolo H1) ν z + z ) p z z m [A i x,z) A i x, z)][z i z i ], H) A i x,z) M1+ z ) q 1, m H3) A i x,z) A i y,z) c x y γ 1+ z ) q 1. Notice that we are in the framework of thep,q-growth, since H1) and H) imply that there exist c 1,c > 0 such that ν m 1.) z p c 1 A i x,z)z i c 1+ z ) q. For ũ W 1,q ;R nm ), a weak solution u to problem 1.1) is a function u such that 1.3) u ũ W 1,p 0 ;R m ) W 1,q loc ;Rm ) and 1.4) m A i x,du)ϕ x i x)dx = 0 ϕ W 1,q ;R m ), suppϕ. If p = q, the existence of weak solutions to 1.1) can be established using the theory of coercive, monotone operators, see Leray Lions [4], Browder [4] and Hartman Stampacchia [0]. Moreover, by H1) the solution is unique. Also the regularity issue has been extensively studied, see the monographs [18], [19] and the surveys [8] and [9]. If p < q the above classical existence results cannot be applied due to the lack of coercivity in W 1,q. Moreover, the request u W 1,q loc ;Rm ) in the definition of weak solution, needed to have a well defined integral in 1.4), is an additional difficulty. Notice that such a request is a priori assumed in some regularity results under the p, q-growth, see for example Leonetti [], Bildhauer Fuchs [1] and Cupini Marcellini Mascolo [10]. In this paper we prove the existence of a weak solution u W 1,q loc ;Rm ) to 1.1). Moreover, under additional assumptions, we show that u is locally bounded. The existence of weak solutions is proved assuming that the integrability of the boundary datum is high enough. Precisely, our result is the following. Theorem 1.1. Assume that H1) H3) hold, with 1.5) p q < p n+γ n. For every ũ W 1,pq 1 p 1 ;R m ) there exists a weak solution u to the Dirichlet problem 1.1), that is u satisfies 1.3) and 1.4). Moreover, u W 1,s loc ;Rm ) for all 1 s < p n. n γ As far as the regularity of solutions is concerned, the obstructions are essentially two: we are dealing with systems if m ) and under non-standard growth p < q). Indeed, in the vectorial case, even under the standard growth, the everywhere regularity of solutions for systems, or of minimizers of integrals, cannot be expected unless some structure conditions are assigned, and this holds also for the local boundedness, see e.g. the counterexamples by De Giorgi [13] and Sverák Yan [30].

3 Existence of weak solutions for elliptic systems with p,q-growth 647 Since the pioneering paper by Marcellini [6], the theory of regularity in the framework of non-standard growth has been deeply investigated. The results and the contributions to regularity are so many, that it is a hard task to provide a comprehensive overview of the issue. For this, we refer to the survey of Mingione [8] for an accurate and interesting account on this subject. A common feature is that to get regularity results p and q must be not too far apart, as examples of irregular solutions by Giaquinta [17], Marcellini [5], [7] and Hong [1] show. Notice that the rate of Hölder continuity ofa,z) interacts with the ratioq/p precisely as in 1.5), see Esposito Leonetti Mingione [15] and Colombo Mingione [8] where the same bound appears. This is suggested by an example in [15], where the minimizer of a functional with q/p > 1+γ/n fails to be locally W 1,q -regular; see also Fonseca Malý Mingione [16]. Moreover, the condition on the distance between the exponents p and q can usually be relaxed if the solutions/minimizers are assumed locally bounded, see e.g. Choe [7], D Ottavio [14], Bildhauer Fuchs [], Canale et al. [5], Carozza Kristensen Passarelli [6], Breit [3], Colombo Mingione [9] and Cupini Marcellini Mascolo [1]. A particular case of systems such that its weak solutions in W 1,q loc ;Rm ) are locally bounded is given by ) 1.6) a ij x,du)u x x j = 0, = 1,...,m. i i=1 j=1 This result is a consequence of Theorem 4.1 in [10], see Theorem 5.1 below. See also Leonetti Mascolo [3] and Cupini Marcellini Mascolo [11]. Collecting the existence result Theorem 1.1 and Theorem 5.1, we obtain the existence of a locally bounded weak solution to 1.1) under the structure condition 1.6), see Theorem 5.. We emphasize that no boundedness condition is imposed on the boundary datum ũ. Theorem 5. can be applied to the Euler equation of the functionals Dux) +ax) Dux) q) dx and 1+ Dux) ) px) dx, as described at the end of paper. The plan of the paper is briefly described. In Section we give the strategy of the proof of the existence result Theorem 1.1, which relies on an approximation argument, and state useful preliminary results. The a priori estimates needed to let the approximation work are proved in Section 3. The conclusion of the proof of Theorem 1.1 is in Section 4. In the last section, we will apply the existence result to a particular class of nonlinear systems, see 5.3), and we will prove that for this class there exists a locally bounded weak solution of the corresponding Dirichlet problem, see Theorem 5.. Applications of our results are given.. Preliminary results We consider a bounded open set in R n, n, and Carathéodory functions A i : Rnm R, with i = 1,...,n and = 1,...,m, satisfying H1), H), H3), with p q, 0 < γ 1 and ν,m,c > 0.

4 648 Giovanni Cupini, Francesco Leonetti and Elvira Mascolo Fixed ũ W 1,q loc ;Rm ) we want to study the existence and the summability properties of the solutions to the elliptic system 1.1) in a weak sense; precisely, u is a weak solution to 1.1) if u ũ W 1,p 0 ;R m ) W 1,q loc ;Rm ) satisfies 1.4). For the sake of clarity, we report the scheme of the proof of Theorem 1.1. We approximate the system in 1.1) with the systems.1) m where A ǫ,i : Rnm R is defined as x i A ǫ,i x,du) ) = 0, 0 < ǫ < 1,.) A ǫ,i z) := A i z)+ǫ z q z i. By 1.), A ǫ,ix,z) satisfies m Moreover, by H1) and H), we have that A ǫ,ix,z)zi ǫ z q λ..3) A ǫ,i x,z) M +1)1+ z )q 1 and, since z z q z is a monotone operator,.4) ν z + z ) p z z m [ A ǫ,i x,z) A ǫ,i x, z)] [zi z i ]. The classical theory of monotone operators, see e.g. [4], [4] and [0], permits to conclude that there exists and, by H1), unique) u ǫ ũ+w 1,q 0 ) such that.5) A ǫ,ix,du ǫ )ϕ x i dx = 0 ϕ W 1,q 0 ;R m ).,i Then, we prove estimates in Lebesgue spaces for the gradient of the functions u ǫ, uniform with respect to ǫ: the norms of Du ǫ are bounded in L p ) and in L q ),, see Propositions 3.1 and 3., respectively. We conclude by proving that there existsu 0 ũ+w 1,p 0 ) such thatu 0 W 1,q loc ) and, up to subsequences, Du ǫ Du 0 strongly in L p loc ). This will imply that A ǫ,ix,du ǫ ) converges, up to subsequences, to A i x,du 0 ) in L 1 loc ). Passing to the limit in the equation.5), with ϕ W 1,q ;R m ) with compact support in, we conclude. We here recall some known results. To prove that the L p norms of Du ǫ are bounded with respect to ǫ, see Lemma 3.1, we need the following inequality, see also [7]. Lemma.1. Assume that H1) and H) hold. Let A ǫ,i be defined as in.). Then there exists c > 0 such that for all 0 < ǫ < 1, for allx and for allξ,η R nm { }.6) ξ p c A ǫ,i x,ξ)ξ i ηi )+1+ η )pq 1) p 1.,i

5 Existence of weak solutions for elliptic systems with p,q-growth 649 Proof. By.4), ) ξ p p 1 ξ η p ξ η + η p ) c ξ + η ) p ξ η + η p ) c A ǫ,i x,ξ) A ǫ,i x,η)) ξi ηi )+ η p, ξ,η R nm.,i By H) and the Young inequality, c A ǫ,i x,η)ξ i ηi ) cm +1)1+ η ) q 1 ξ η 1 ξ p +c1+ η ) pq 1) p 1.,i Thus,.6) follows. To get that the L q norms of Du ǫ are locally bounded with respect to ǫ, see Proposition 3., we need technical results. For the reader s convenience we list them below. The first one is based on an iteration argument, see e.g. [19]. Lemma.. Let Zt) be a bounded and non-negative function in the interval [ρ,r]. Assume that for ρ r < d R A Zr) θzd)+ d r) +B γ with A,B 0 and 0 θ < 1. Then [ Zρ) cγ) A R ρ) γ +B The remaining results are related to fine properties of the difference quotient; they can be found e.g. in [15]. We recall that, given a function f: R n R k, then τ s,h fx) = fx+he s ) fx), where e s is the s-th vector of the canonical basis in R n. Lemma.3. If 0 < ρ < R, h < R ρ, 1 t < +, s {1,...,n}, and G,D s G L t B R ), theṋ τ s,h Gx) t dx h t D s Gx) t dx. B ρ B R Lemma.4. If G: R n R k, G L B R ), 0 < R 1 and for some ρ 0,R), d 0,1), M > 0, η: R n R with η Cc 1 ), 0 η 1 in R n, Dη 4 in Bρ+R R ρ R n, η = 1 on B ρ, τ s,h Gx) η x)dx M h d B R s=1 for every h with h R ρ, theng 4 Wb, B ρ ;R k ) L n n b Bρ ;R k ) for every b 0,d) and G n c L n bb ρ) R ρ) b+d+m + G L B R )), where c = cn,k,b,d). Lemma.5. For every p > 1 and G: B R R k, we have τ s,h µ + Gx) ) p 4 Gx)) ck,p)µ + Gx) + Gx+he s ) ) p τs,h Gx) for every x B ρ, with h R ρ and every s = 1,...,n, where 0 µ 1 and the constant c = ck, p) is independent of µ. ].

6 650 Giovanni Cupini, Francesco Leonetti and Elvira Mascolo 3. Uniform estimates In this section we study the approximating systems.1). The properties of A ǫ,i, i = 1,...,n, = 1,...,m, ensure the existence of a unique function u ǫ ũ+w 1,q 0 ) such that A ǫ,i x,du ǫ)ϕ x i dx = 0 ϕ W 1,q 0 ;R m ).,i In the following, c is a positive constant that may vary from line to line. First we claim that the L p norms of Du ǫ are bounded with respect to ǫ. Proposition 3.1. Assume that H1) and H) hold and considerũ W 1,pq 1) p 1 ). For all ǫ, let u ǫ ũ+w 1,q 0 ) satisfy.5). Then there exists a positive constant c 0, independent of ǫ, such that 3.1) Du ǫ p dx c 0 1+ Dũ ) pq 1) p 1 dx. Moreover, u ǫ ũ W 1,p 0 ) c with c independent of ǫ. Proof. By.5) with test function ϕ = u ǫ ũ, and using Lemma.1, with ξ = Du ǫ x), η = Dũx), we get 3.1). Moreover, since p 1 Du ǫ Dũ p dx Du ǫ p + Dũ p )dx, then 3.1) implies the boundedness of the W 1,p 0 -norm of u ǫ ũ with respect to ǫ. In the following proposition we claim that the L q norms of Du ǫ are locally bounded with respect to ǫ. Precisely, we will prove the stronger result that the L s -norms of Du ǫ are locally bounded, with q < s < p n n γ. Proposition 3.. Assume that H1) H3) hold with q < p n+γ, and consider n ũ W 1,pq 1) p 1 ). Let uǫ ũ + W 1,q 0 ) satisfy.5), with 0 < ǫ < 1. For all s [q,p n ) there exist σ,τ positive and independent of ǫ, such that for all B n γ Rx 0 ), with R 1, and for all ρ < R, we have that 3.) B ρx 0 ) Du ǫ x) s dx c 1 R ρ) τ with positive constants c 1,c independent of ǫ. σ 1+ Du ǫ x) ) dx) p +c R n, B R x 0 ) Proof. From now on, we write u in place of u ǫ. Consider B R x 0 ), ρ r < d R 1, and define η Cc ), suppη Bd+rx 0 ), η 1 in B r and Dη 4. d r Let ϕ = τ s, h η τ s,h u) with h < d r. Then.5) implies 3.3),i η τ s,h A ǫ,i x,du) ) τ s,h u x i dx = τ s,h A ǫ,i x,du) ) ηη xi τ s,h u dx.,i We now proceed by steps. Step 1. We prove that +r η τs,h Du )p 4 Du) dx c d r where the constant c is independent of ǫ. 1+ Du ) q dx h γ

7 Existence of weak solutions for elliptic systems with p,q-growth 651 Indeed, since η τ s,h A ǫ,i x,du)) ) τ s,h u x i dx,i =,i +,i η A ǫ,i x+he s,dux+he s ) A ǫ,i x,dux+he s)) ) τ s,h u x i dx η A ǫ,i x,dux+he s)) A ǫ,i x,dux))) τ s,h u x i dx, then 3.3) implies η A ǫ,i x,dux+he s)) A ǫ,i x,dux))) τ s,h u x i dx 3.4),i =,i,i By Lemma.5, τ s,h A ǫ,i x,du) ) ηη xi τ s,h u dx η A ǫ,i x+he s,dux+he s )) A ǫ,i x,dux+he s)) ) τ s,h u x i dx. Du ) τ )p 4 s,h Du cn,m,p) Dux) + Dux+he s ) )p τ s,h Dux). Therefore, using.4), there exists a positive constant c, depending on n,m,p and ν, such that Du τ ) p ) 4 [ s,h Du c A ǫ,i x,dux+he s )) A ǫ,i x,dux))] τ s,h u x i.,i This inequality, together with 3.4), implies 1 { η τs,h Du ) p } 4 Du dx τ s,h A c ǫ,i x,du) ) ηη xi τ s,h u dx,i η A ǫ,ix+he s,dux+he s )) A ǫ,ix,dux+he s )) ) τ s,h u x i dx,i 3.5) =: I 1 +I. Let us estimate I 1. By.3) I 1 c η Dη 1+ Dux) + Dux+he s ) ) q 1 τ s,h u dx c d r B r+d 1+ Dux) + Dux+he s ) ) q dx q 1 q B r+d 1 q τ s,h u q dx.

8 65 Giovanni Cupini, Francesco Leonetti and Elvira Mascolo Then, using Lemma.3 and taking into account that h < 1, we get I 1 c { }q 1 { 1+ Dux) ) q q dx u xs q dx d r c 3.6) 1+ Dux) ) q dx h γ. d r }1 q h Let us estimate I. By Hölder continuity of A,z) in H3), I c η 1+ Dux+he s ) ) q 1 τ s,h Du dx h γ 3.7) c 1+ Dux) + Dux+he s ) ) q dx h γ c 1+ Du ) q dx h γ. B r+d Thus, collecting 3.5), 3.6) and 3.7), the claim follows. Step. Since q < n+γ, then we can choose 0 < θ < γ such that q p n ) < n+θ. Such p n q a choice is obtained when 0 n 1 < θ < γ. We claim that Du L qδ p loc ), where δ = p n n. Notice that p < qδ < p. q n θ n γ Indeed, fixed a ball B R x 0 ), with ρ r < d R 1, by Step 1 and by Lemma.4, applied with b = θ, we have Du p L n n θ Br ). Moreover, Du p L n n θ B r) c d r) θ+γ+ { ) Dux) ) q dx + Dux) p dx d r } )1 with c independent of ǫ. Therefore, if we set = θ+γ+ 5) n the above inequality n θ gives )qδ 3.8) Dux) qδ c dx 1+ Dux) ) q p dx. B r d r) The arbitrariness of r, d and x 0 allows to conclude. If q = p then we go to Step 5. If q > p we need Steps 3 and 4. Step 3. Since q < n+θ then δq < δq p. In this step we prove that there exist p n p q p positive constants t,σ,c, all independent of ǫ, satisfying δq < t δq p, 0 < σ < 1, p q p such thaṱ 3.9) Indeed, t B r Dux) qδ dx c d r) 1 σ ] δq, δq p p q p 1+ Dux) ) p dx implies 3.10) 1 < qδ p )t 1)σ 1 σ + 1 < t, qt δ t 1 p. Dux) qδ dx+cr n.

9 By Hölder inequality Existence of weak solutions for elliptic systems with p,q-growth Dux) ) q dx 3.11) 1+ Dux) ) qδ dx )qδ )qδ p = 1+ Dux) ) Bd qδ t 1+ Dux) ) q1 δ p t) dx )qδ pt Bd Thus, 3.8) and 3.11) imply Dux) qδ dx B r ) σ 1+ Dux) ) qδ c dx d r) where c is the constant in 3.8) and By Young inequality, we get B r Dux) qδ dx c d r) 1 σ σ := qδ t δ, b := pt t Dux) ) qb dx 1+ Dux) ) q t δ t 1 dx )qδ pt t 1). ) t 1)σ 1+ Dux) ) qb dx, )t 1)σ 1 σ Dux) ) qδ dx. qδ Notice that by Step and 3.10) the right-hand side is finite. Since qb p and x + y ) qδ qδ 1 x qδ + y qδ ), we get 3.9). Step 4. By using Lemma. with Zr) := B r Dux) qδ dx, the inequality 3.9) implies )t 1)σ Dux) qδ c dx 1+ Dux) ) p 1 σ dx +cr n. B ρ R ρ) 1 σ B R ) ) q Step 5. Notice that since θ is any number in n 1,γ, then s := qδ can be p ) p any number in,p n. Note that q p. The estimate 3.) follows when p q n γ p q [ ] s belongs to such an interval. For s p, we use Hölder inequality and we get p p q 3.) also in this case. 4. Proof of Theorem 1.1 We are now ready to prove the existence of a weak solution to 1.1). Proof of Theorem 1.1. For all 0 < ǫ < 1, let us define the operators A ǫ,i as in.) and let u ǫ ũ+w 1,q 0 ) satisfy 4.1) A ǫ,ix,du ǫ )ϕ x i dx = 0 ϕ W 1,q 0 ), as in Section 3.,i

10 654 Giovanni Cupini, Francesco Leonetti and Elvira Mascolo Step 1. We claim that there exists u 0 ũ + W 1,p 0 ;R m ) such that, up to subsequences, Du ǫ Du 0 strongly in L p loc ). By Proposition 3.1, the W 1,p 0 ) norm of u ǫ ũ is bounded with respect to ǫ. Moreover, the estimate 3.1) and Proposition 3. imply that, fixed, Du ǫ is bounded in L q ;R nm ) uniformly with respect to ǫ. Thus, there exists u 0 ũ+w 1,p 0 ;R m ) such that Du 0 L q loc ;Rnm ) and, up to subsequences, u ǫ ũ u 0 ũ weakly in W 1,p 0 ;R m ) and Du ǫ Du 0 weakly in L q loc ;Rnm ). Since q < p n+γ, by Rellich Theorem we get n u ǫ u 0 in L q loc ;Rm ). To conclude, consider η Cc ) and define the test function ϕ = u ǫ u 0 )η. Notice that ϕ W 1,q ;R nm ), with suppϕ. By 4.1) A ǫ,i x,du ǫ)u ǫ,x i u 0,x i )ηdx = A ǫ,i x,du ǫ)u ǫ u 0 )η x i dx 4.),i M +1) Notice that.4) implies 1+ Du ǫ ) q 1 u ǫ u 0 Dη dx. Du ǫ Du 0 p = Du ǫ Du 0 ) p Duǫ Du 0,i p Duǫ + Du 0 ) p Duǫ Du 0 p ν A ǫ,i x,du ǫ) A ǫ,i x,du 0))u ǫ,x i u 0,x i ).,i Thus, 4.) and the definition of A ǫ,i imply p ν Du ǫ Du 0 p ηdx A ǫ,i x,du ǫ) A ǫ,i x,du 0))u ǫ,x i u 0,x i )ηdx M +1) 1+ Du ǫ ) q 1 u ǫ u 0 Dη dx ǫ,i u 0,x i q u 0,x i u ǫ,x i u 0,x i )ηdx.,i A i x,du 0 )u ǫ,x i u 0,x i )ηdx Thus, passing to the limit, as ǫ goes to 0 we get that the right hand side goes to 0. We have so proved that 4.3) Du ǫ Du 0 in L p loc ). Step. We claim that, up to subsequences, A ǫ,i x,du ǫ) converges to A i x,du 0) in L 1 loc ). Indeed, for all, adding and subtracting A i x,du ǫ), we get A ǫ,ix,du ǫ ) A i x,du 0 ) dx ǫ Du ǫ q 1 dx+ A i x,du ǫ) A i x,du 0) dx.,i

11 Existence of weak solutions for elliptic systems with p,q-growth 655 By the uniform boundedness of the L q )-norms of Du ǫ the first term at the right hand side goes to zero. As far as the last term is concerned, define the functions f ǫ x) := A i x,du ǫ ) A i x,du 0 ), g ǫ x) := M { 1+ Du ǫ x) ) q Du 0 x) ) q 1} and g 0 x) := M1+ Du 0 x) ) q 1. By H) we get that f ǫ x) g ǫ x) for a.e. x. Moreover, by what previously proved, g ǫ g 0 almost everywhere. Let us now prove that g ǫ dx g 0 dx as ǫ goes to 0. This easily follows by 4.3) and recalling that q p+1. A generalization of the Lebesgue Convergence Theorem allows to conclude. Step 3. By 4.1) we get that A ǫ,ix,du ǫ )ϕ x i dx = 0 ϕ W 1,q ;R m ), suppϕ.,i By Step, up to subsequences, A i x,du 0 )ϕ x i dx = lim A ǫ,ix,du ǫ )ϕ x i dx = 0 ǫ 0,i for all ϕ W 1,q ;R m ), suppϕ. Therefore, u 0 is a weak solution to 1.1). 5. Local boundedness of the weak solutions Let us assume that the functions A i in 1.1) are such that 5.1) A i x,z) := a ij x,z)zj, where a ij : R nm R are Carathéodory functions such that 5.) a ij x,z)λ i λ j 0 λ R n, z R nm and a.e. x. i,j=1 Let us consider the corresponding system, that is ) 5.3) a ij x,dux))u x x j x) = 0, = 1,...,m. i i=1 j=1 The following boundedness result is consequence of Theorem 4.1 in [10]. Theorem 5.1. Assume 5.1), 5.) and m 5.4) [A i x,z) A i x, z)][z i z i ] 0 z, z Rnm and a.e. x. Assume also ν z p c 1 m j=1,i A i x,z)z i c 1+ z ) q z, z R nm and a.e. x, for some ν,c 1,c > 0, with 1 < p q, and also q < p n 1 n p if p < n.

12 656 Giovanni Cupini, Francesco Leonetti and Elvira Mascolo Let u W 1,q loc ) be a weak solution to 5.3), i.e., m A i x,du)ϕ x i x)dx = 0, ϕ W 1,q ;R m ), suppϕ. Then u L loc ;Rm ). Proof. It is easy to check that the proof of Theorem 4.1 in [10] works using 5.) and 5.4) instead of the following condition assumed there: )p m a ij x,z)λ i λ j M 1 zi, i,j=1 for all z R mn, λ R n and with M 1 > 0. By this result and Theorem 1.1, we obtain the following existence and regularity theorem. Theorem 5.. Assume that H1) H3) hold, with p q < p n+γ. Let n A i be as in 5.1), with a ij ) satisfying 5.). For every ũ W 1,pq 1 p 1 ;R m ) there exists a weak solution u to the Dirichlet problem 1.1), such that u is locally bounded. Proof. The existence follows by Theorem 1.1 and the local boundedness follows by Theorem 5.1. i=1 λ i =1 We conclude with two applications of the above result. Example 5.3. Let us consider 5.5) A i x,z) := z i +ax)q z q z i, where q, a C 0,γ ), ax) 0, 0 < γ 1. The assumptions of Theorems 1.1 and 5. are satisfied with p =. Therefore, if q < n+γ and ũ W 1,q 1) ;R m ) n then there exists a locally bounded weak solution to the Dirichlet problem 1.1). Notice that 5.5) comes from the Euler equation of the functional Dux) +ax) Dux) q) dx. Example 5.4. Let us consider 5.6) A i x,z) := px)1+ z ) px) z i, with p C 0,γ ), 0 < γ 1. Assume that p 1 and p < p 1 n+γ n, where p 1 := min{px)} and p := max{px)}. The assumptions of Theorems 1.1 and 5. are satisfied for p = p 1 and any q with n+γ p < q < p 1. n Therefore, if ũ W 1,p 1 q 1) p 1 1 ;R m ) then there exists a locally bounded weak solution to the Dirichlet problem 1.1). Notice that 5.6) comes from the Euler equation of the functional 1+ Dux) ) px) dx.

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