ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES

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1 Electronic Journal of Differential Equations, Vol. 2008(2008), No. 76, pp ISSN: URL: or ftp ejde.math.txstate.edu (login: ftp) ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES GE DONG Abstract. This article shows the existence of solutions to the nonlinear elliptic problem A(u) f in Orlicz-Sobolev spaces with a measure valued righthand side, where A(u) div a(x, u, u) is a Leray-Lions operator defined on a subset of W 1 0 L M (), with general M. 1. Introduction Let M : R R be an N-function; i.e. M is continuous, convex, with M(u) > 0 for u > 0, M(t)/t 0 as t 0, and M(t)/t as t. Equivalently, M admits the representation M(u) u φ(t)dt, where φ is the derivative of M, with 0 φ non-decreasing, right continuous, φ(0) 0, φ(t) > 0 for t > 0, and φ(t) as t. The N-function M conjugate to M is defined by M(v) t ψ(s)ds, where ψ is 0 given by ψ(s) sup{t : φ(t) s}. The N-function M is said to satisfy the 2 condition, if for some k > 0 and u 0 > 0, M(2u) km(u), u u 0. Let P, Q be two N-functions, P Q means that P grows essentially less rapidly than Q; i.e. for each ε > 0, P (t)/q(εt) 0 as t. This is the case if and only if lim t Q 1 (t)/p 1 (t) 0. Let R N be a bounded domain with the segment property. The class W 1 L M () (resp., W 1 E M ()) consists of all functions u such that u and its distributional derivatives up to order 1 lie in L M () (resp., E M ()). Orlicz spaces L M () are endowed with the Luxemburg norm u (M) inf { λ > 0 : M ( u(x) ) } dx 1. λ The classes W 1 L M () and W 1 E M () of such functions may be given the norm u,m D α u (M). α Mathematics Subject Classification. 35J15, 35J20, 35J60. Key words and phrases. Orlicz-Sobolev spaces; elliptic equation; nonlinear; measure data. c 2008 Texas State University - San Marcos. Submitted October 8, Published May 27,

2 2 G. DONG EJDE-2008/76 These classes will be Banach spaces under this norm. I refer to spaces of the forms W 1 L M () and W 1 E M () as Orlicz-Sobolev spaces. Thus W 1 L M () and W 1 E M () can be identified with subspaces of the product of N 1 copies of L M (). Denoting this product by ΠL M, we will use the weak topologies σ(πl M, ΠE M ) and σ(πl M, ΠL M ). If M satisfies 2 condition, then L M () E M () and W 1 L M () W 1 E M (). The space W0 1 E M () is defined as the (norm) closure of C0 () in W 1 E M () and the space W0 1 L M () as the σ(πl M, ΠE M ) closure of C0 () in W 1 L M (). Let W 1 L M () (resp. W 1 E M ()) denote the space of distributions on which can be written as sums of derivatives of order 1 of functions in L M () (resp. E M ()). It is a Banach space under the usual quotient norm (see 12]). If the open set has the segment property, then the space C0 () is dense in W0 1 L M () for the modular convergence and thus for the topology σ(πl M, ΠL M ) (cf. 12, 13]). Let A(u) div a(x, u, u) be a Leray-Lions operator defined on W 1,p (), 1 < p <. Boccardo-Gallouet 7] proved the existence of solutions for the Dirichlet problem for equations of the form A(u) f in, (1.1) u 0 on, (1.2) where the right hand f is a bounded Radon measure on (i.e. f M b ()). The function a is supposed to satisfy a polynomial growth condition with respect to u and u. Benkirane 4, 5] proved the existence of solutions to in Orlicz-Sobolev spaces where A(u) g(x, u, u) f, (1.3) A(u) div(a(x, u, u)) (1.4) is a Leray-Lions operator defined on D(A) W 1 0 L M (), g is supposed to satisfy a natural growth condition with f W 1 E M () and f L 1 (), respectively, but the result is restricted to N-functions M satisfying a 2 condition. Elmahi extend the results of 4, 5] to general N-functions (i.e. without assuming a 2 -condition on M) in 8, 9], respectively. The purpose of this paper is to solve (1.1) when f is a bounded Radon measure, and the Leray-Lions operator A in (1.4) is defined on D(A) W 1 0 L M (), with general M. We show that the solutions to (1.1) belong to the Orlicz-Sobolev space W 1 0 L B () for any B P M, where P M is a special class of N-function (see below). Specific examples to which our results apply include the following: div( u p 2 u) µ in, div( u p 2 u log β (1 u )) µ div M( u ) u u 2 µ in in where p > 1 and µ is a given Radon measure on. For some classical and recent results on elliptic and parabolic problems in Orlicz spaces, I refer the reader to 2, 3, 6, 10, 11, 12, 14, 16, 18].

3 EJDE-2008/76 ELLIPTIC EQUATIONS WITH MEASURE DATA 3 2. Preliminaries We define a subset of N-functions as follows. { P M B : R R is an N-function, B /B M /M and 1 where H(r) M(r)/r. Assume that 0 } B H 1 (1/t 1 1/N )dt < P M (2.1) Let R N be a bounded domain with the segment property, M, P be two N-functions such that P M, M, P be the complementary functions of M, P, respectively, A : D(A) W0 1 L M () W 1 L M () be a mapping given by A(u) div a(x, u, u) where a : R R N R N be a Caratheodory function satisfying for a.e. x and all s R, ξ, η R N with ξ η: a(x, s, ξ) βm( ξ )/ ξ (2.2) a(x, s, ξ) a(x, s, η)]ξ η] > 0 (2.3) a(x, s, ξ)ξ αm( ξ ) (2.4) where α, β, γ > 0. Furthermore, assume that there exists D P M such that D H 1 is an N-function. (2.5) Set T k (s) max( k, min(k, s)), s R, for all k 0. Define by M b () as the set of all bounded Radon measure defined on and by T 1,M 0 () as the set of measurable functions R such that T k (u) W0 1 L M () D(A). Assume that f M b (), and consider the following nonlinear elliptic problem with Dirichlet boundary A(u) f in. (2.6) The following lemmas can be found in 4]. Lemma 2.1. Let F : R R be uniformly Lipschitzian, withf (0) 0. Let M be an N-function, u W 1 L M () (resp. W 1 E M ()). Then F (u) W 1 L M () (resp. W 1 E M ()). Moreover, if the set D of discontinuity points of F is finite, then (F u) x i { F (u) u x i a.e. in {x : u(x) D} 0 a.e. in {x : u(x) D}. Lemma 2.2. Let F : R R be uniformly Lipschitzian, with F (0) 0. I suppose that the set of discontinuity points of F is finite. Let M be an N-function, then the mapping F : W 1 L M () W 1 L M () is sequentially continuous with respect to the weak topology σ(πl M, ΠE M ). 3. Existence theorem Theorem 3.1. Assume that (2.1)-(2.5) hold and f M b (). Then there exists at least one weak solution of the problem u T 1,M 0 () W 1 0 L B (), B P M a(x, u, u) φdx f, φ, φ D()

4 4 G. DONG EJDE-2008/76 Proof. Denote V W 1 0 L M (). (1) Consider the approximate equations u n V div a(x, u n, u n ) f n (3.1) where f n is a smooth function which converges to f in the distributional sense that such that f n L1 () f Mb (). By 4, Theorem 3.1] or 8], there exists at least one solution {u n } to (3.1). For k > 0, by taking T k (u n ) as test function in (3.1), one has a(x, T k (u n ), T k (u n )) T k (u n )dx Ck. In view of (2.4), we get M( T k (u n ) )dx Ck. (3.2) Hence T k (u n ) is bounded in (L M ()) N. By 9] there exists u such that u n u almost everywhere in and T k (u n ) T k (u) weakly in W 1 0 L M () for σ (ΠL M, ΠE M ). (3.3) For t > 0, by taking T h (u n T t (u n )) as test function, we deduce that a(x, u n, u n ) u n dx h f Mb () which gives and by letting h 0, t< u n th 1 M( u n )dx f Mb () h t< u n th d dt u n >t M( u n )dx f Mb (). Let now B P M. Following the lines of 17], it is easy to deduce that B( u n )dx C, n. (3.4) We shall show that a(x, T k (u n ), T k (u n )) is bounded in (L M ()) N. Let ϕ (E M ()) N with ϕ (M) 1. By (2.2) and Young inequality, one has ( M( Tk (u n ) ) ) a(x, T k (u n ), T k (u n ))ϕdx β M dx β M( ϕ )dx T k (u n ) β M( T k (u n ) )dx β This last inequality is deduced from M(M(u)/u) M(u), for all u > 0, and M( ϕ )dx 1. In view of (3.2), a(x, T k (u n ), T k (u n ))ϕdx Ck β, which implies {a(x, T k (u n ), T k (u n ))} n being a bounded sequence in (L M ()) N. (2) For the rest of this article, χ r, χ s and χ j,s will denoted respectively the characteristic functions of the sets r {x ; T k (u(x)) r}, s {x ; T k (u(x)) s} and j,s {x ; T k (v j (x)) s}. For the sake of

5 EJDE-2008/76 ELLIPTIC EQUATIONS WITH MEASURE DATA 5 simplicity, I will write only ε(n, j, s) to mean all quantities (possibly different) such that lim lim lim ε(n, j, s) 0. s j n Take a sequence (v j ) D() which converges to u for the modular convergence in V (cf. 13]). Taking T η (u n T k (v j )) as test function in (3.1), we obtain a(x, u n, u n ) T η (u n T k (v j ))dx Cη (3.5) On the other hand, a(x, u n, u n ) T η (u n T k (v j ))dx a(x, T k (u n ), T k (u n ))( T k (u n ) T k (v j ))dx { u n T k (v j) η} { u n k} { T k u n T k (v j) η} a(x, u n, u n )( u n T k (v j ))dx a(x, T k (u n ), T k (u n ))( T k (u n ) T k (v j ))dx a(x, u n, u n ) u n dx a(x, u n, u n ) T k (v j )dx By (2.4) the second term of the right-hand side satisfies a(x, u n, u n ) u n dx 0. Since a(x, T kη (u n ), T kη (u n )) is bounded in (L M ()) N, there exists some h kη (L M ()) N such that a(x, T kη (u n ), T kη (u n )) h kη weakly in (L M ()) N for σ (ΠL M, ΠE M ). Consequently the third term of the righthand side satisfies a(x, u n, u n ) T k (v j )dx since { u T k (v j) η} { u >k} a(x, T kη (u n ), T kη (u n )) T k (v j )dx h kη T k (v j )dx ε(n) T k (v j )χ { un T k (v j) η} { u n >k} T k (v j )χ { u Tk (v j) η} { u >k} strongly in (E M ()) N as n. Hence a(x, T k (u n ), T k (u n )) T k (u n ) T k (v j )]dx { T k u n T k (v j) η}

6 6 G. DONG EJDE-2008/76 Cη ε(n) h kη T k (v j )dx { u T k (v j) η} { u >k} Let 0 < θ < 1. Define Φ n,k a(x, T k (u n ), T k (u n )) a(x, T k (u n ), T k (u))] T k (u n ) T k (u)]. For r > 0, I have 0 {a(x, T k (u n ), T k (u n )) a(x, T k (u n ), T k (u))] T k (u n ) T k (u)]} θ dx r Φ θ n,kχ { Tk (u n) T k (v j) >η}dx r r Φ θ n,kχ { Tk (u n) T k (v j) η}dx Using the Hölder Inequality (with exponents 1/θ and 1/(1 θ)), the first term of the right-side hand is less than ( ) θ ( ) 1 θ. Φ n,k dx χ { Tk (u n) T k (v j) >η}dx r r Noting that Φ n,k dx r a(x, T k (u n ), T k (u n )) T k (u n )dx a(x, T k (u n ), T k (u)) T k (u n )dx r r a(x, T k (u n ), T k (u n )) T k (u)dx a(x, T k (u n ), T k (u)) T k (u)dx r r ( M( Tk (u) ) ) Ck β M dx β M( T k (u n ) )dx r T k (u) r ( M( Tk (u n ) ) ) β M dx β M( T k (u) )dx r T k (u n ) r β M( T k (u) )dx r Ck β M( T k (u) )dx β M( T k (u n ) )dx r β M( T k (u n ) )dx β M( T k (u) )dx β M( T k (u) )dx r r (2β 1)Ck 3M(r) meas it follows that r Φ θ n,kχ { Tk (u n) T k (v j) >η}dx C(meas{ T k (u n ) T k (v j ) > η}) 1 θ, where C (2β 1)Ck 3M(r) meas ] θ. Using the Hölder Inequality (with exponents 1/θ and 1/(1 θ)), r Φ θ n,kχ { Tk (u n) T k (v j) η}dx ( ) θ ( Φ n,k χ { Tk (u n) T k (v j) η}dx r dx r ) 1 θ

7 EJDE-2008/76 ELLIPTIC EQUATIONS WITH MEASURE DATA 7 ( ) θ( ) 1 θ Φ n,k χ { Tk (u n) T k (v j) η}dx meas r Hence 0 {a(x, T k (u n ), T k (u n )) a(x, T k (u n ), T k (u))] T k (u n ) T k (u)]} θ dx r C ( meas{ T k (u n ) T k (v j ) > η} ) 1 θ ( ) θ( ) 1 θ Φ n,k χ { Tk (u n) T k (v j) η}dx meas r C ( meas{ T k (u n ) T k (v j ) > η} ) 1 θ ( a(x, Tk (u n ), T k (u n )) a(x, T k (u n ), T k (u)) ] r T k (u n ) T k (u) ] ) θ( ) 1 θ dx meas For each s r one has 0 a(x, Tk (u n ), T k (u n )) a(x, T k (u n ), T k (u)) ] r T k (u n ) T k (u) ] dx a(x, Tk (u n ), T k (u n )) a(x, T k (u n ), T k (u)) ] s T k (u n ) T k (u)]dx a(x, Tk (u n ), T k (u n )) a(x, T k (u n ), T k (u)χ s ) ] s ] T k (u n ) T k (u)χ s dx a(x, Tk (u n ), T k (u n )) a(x, T k (u n ), T k (u)χ s ) ] T k (u n ) T k (u)χ s ]dx a(x, Tk (u n ), T k (u n )) a(x, T k (u n ), T k (v j )χ j,s ) ] ] T k (u n ) T k (v j )χ j,s dx a(x, T k (u n ), T k (u n )) ] T k (v j )χ j,s T k (u)χ s dx a(x, Tk (u n ), T k (v j )χ j,s ) a(x, T k (u n ), T k (u)χ s ) ] T k (u n )dx a(x, T k (u n ), T k (v j )χ j,s ) T k (v j )χ j,s dx a(x, T k (u n ), T k (u)χ s ) T k (u)χ s dx I 1 (n, j, s) I 2 (n, j, s) I 3 (n, j, s) I 4 (n, j, s) I 5 (n, j, s)

8 8 G. DONG EJDE-2008/76 On the other hand, a(x, T k (u n ), T k (u n )) T k (u n ) T k (v j )]dx a(x, Tk (u n ), T k (u n )) a(x, T k (u n ), T k (v j )χ j,s ) ] ] T k (u n ) T k (v j )χ j,s dx a(x, T k (u n ), T k (v j )χ j,s ) ] T k (u n ) T k (v j )χ j,s dx a(x, T k (u n ), T k (u n )) T k (v j )χ { Tk (v j) >s}dx The second term of the right-hand side tends to a(x, T k (u), T k (u)χ s ) T k (u) T k (v j )χ s ]dx { T k (u) T k (v j) η} since a(x, T k (u n ), T k (u)χ s )χ { Tk (u n) T k (v j) η} tends to a(x, T k (u), T k (u)χ s )χ { Tk (u) T k (v j) η} in (E M ()) N while T k (u n ) T k (v j )χ s tends weakly to T k (u) T k (v j )χ s in (L M ()) N for σ(πl M, ΠE M ). Since a(x, T k (u n ), T k (u n )) is bounded in (L M ()) N there exists some h k (L M ()) N such that (for a subsequence still denoted by u n ) a(x, T k (u n ), T k (u n )) h k weakly in (L M ()) N for σ(πl M, ΠE M ). In view of the fact that T k (v j )χ { Tk (u n) T k (v j) η} T k (v j )χ { Tk (u) T k (v j) η} strongly in (E M ()) N as n the third term of the right-hand side tends to h k T k (v j )χ { Tk (v j) >s}dx. { T k (u) T k (v j) η} Hence in view of the modular convergence of (v j ) in V, one has I 1 (n, j, s) Cη ε(n) h kη T k (v j )dx Therefore, { T k (u) T k (v j) η} { T k (u) T k (v j) η} { u T k (v j) η} { u >k} h k T k (v j )χ { Tk (v j) >s}dx a(x, T k (u), T k (u)χ s ) T k (u) T k (v j )χ s ]dx Cη ε(n) ε(j) h k T k (u)χ { Tk (u) >s}dx a(x, T k (u), 0)χ { Tk (u) >s}dx I 1 (n, j, s) Cη ε(n, j, s) (3.6) For what concerns I 2, by letting n, one has I 2 (n, j, s) h k T k (v j )χ j,s T k (u)χ s ]dx ε(n) { T k (u) T k (v j) η}

9 EJDE-2008/76 ELLIPTIC EQUATIONS WITH MEASURE DATA 9 since a(x, T k (u n ), T k (u n )) h k weakly in (L M ) N for σ(πl M, ΠE M ) while χ { Tk (u n) T k (v j) η} T k (v j )χ j,s T k (u)χ s ] approaches χ { Tk (u) T k (v j) η} T k (v j )χ j,s T k (u)χ s ] strongly in (E M ) N. By letting j, and using Lebesgue theorem, then I 2 (n, j, s) ε(n, j). (3.7) Similar tools as above, give I 3 (n, j, s) a(x, T k (u), T k (u)χ s ) T k (u)χ s dx ε(n, j) (3.8) Combining (3.6), (3.7), and (3.8), we have a(x, Tk (u n ), T k (u n )) a(x, T k (u n ), T k (u)) ] r T k (u n ) T k (u) ] dx ε(n, j, s). Therefore, 0 {a(x, T k (u n ), T k (u n )) a(x, T k (u n ), T k (u))] T k (u n ) T k (u)]} θ dx r C(meas{ T k (u n ) T k (v j ) > η}) 1 θ (meas ) 1 θ (ε(n, j, s)) θ Which yields, by passing to the limit superior over n, j, s and η, { lim a(x, Tk (u n ), T k (u n )) a(x, T k (u n ), T k (u)) ] n r T k (u n ) T k (u) ]} θ dx 0. Thus, passing to a subsequence if necessary, u n u a.e. in r, and since r is arbitrary, u n u a.e. in. By (2.2) and (2.5), D H 1( a(x, u n, u n ) ) dx β Hence D( u n )dx C a(x, u n, u n ) a(x, u, u) weakly for σ(πl D H 1ΠE D H 1). Going back to approximate equations (3.1), and using φ D() as the test function, one has a(x, u n, u n ) φdx f n, φ in which I can pass to the limit. This completes the proof.

10 10 G. DONG EJDE-2008/76 References 1] R. Adams; Sobolev spaces, Academic Press, New York, ] L. Aharouch, E. Azroul and M. Rhoudaf; Nonlinear Unilateral Problems in Orlicz spaces, Appl. Math., 200 (2006), ] A. Benkirane, A. Emahi; Almost everywhere convergence of the gradients of solutions to elliptic equations in Orlicz spaces and application, Nonlinear Analysis, 28 (11), 1997, ] A. Benkirane, A. Elmahi; An existence for a strongly nonlinear elliptic problem in Orlicz spaces, Nonlinear Analysis, 36 (1999) ] A. Benkirane, A. Emahi; A strongly nonlinear elliptic equation having natural growth terms and L1 data, Nonlinear Analysis, 39 (2000) ] L. Boccardo, T. Gallouet; Non-linear elliptic and parabolic equations involving measure as data, J. Funct. Anal. 87 (1989) ] L. Boccardo, T. Gallouet; Nonlinear elliptic equations with right hand side measures, Comm. Partial Differential Equations, 17 (3/4), 1992, ] A. Elmahi, D. Meskine; Existence of solutions for elliptic equations having natural growth terms in orlicz spaces, Abstr. Appl. Anal. 12 (2004) ] A. Elmahi, D. Meskine; Non-linear elliptic problems having natural growth and L1 data in Orlicz spaces, Annali di Matematica (2004) ] A. Elmahi, D. Meskine; Strongly nonlinear parabolic equations with natural growth terms in Orlicz spaces, Nonlinear Analysis 60 (2005) ] A.Fiorenza, A. Prignet; Orlicz capacities and applications to some existence questions for elliptic PDEs having measure data, ESAIM: Control, Optimisation and Calculus of Variations, 9 (2003), ] J. Gossez; Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc. 190 (1974) ] J. Gossez; Some approximation properties in Orlicz-Sobolev spaces, Studia Math. 74 (1982), 17C24. 14] J. Gossez, V. Mustonen; Variational inequalities in Orlicz-Sobolev spaces, Nonlinear Analysis TMA, 11 (3), 1987, ] M. Krasnoselski, Y. Rutickii; Convex functions and Orlicz space, Noordhoff, Groningen, ] D. Meskine; Parabolic equations with measure data in Orlicz spaces, J. Evol. Equ. 5 (2005) ] G. Talenti; Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura. Appl. IV, 120 (1979) ] T. Vecchio; Nonlinear elliptic equations with measure data, Potential Anal. 4 (1995) Ge Dong 1. Department of Mathematics, Shanghai University, No. 99, Shangda Rd., Shanghai, China 2. Department of Basic, Jianqiao College, No. 1500, Kangqiao Rd., Shanghai, China address: dongge97@sina.com