Journal of Inequalities in Pure and Applied Mathematics

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1 Journal of Inequalities in Pure and Applied athematics Volume 4, Issue 5, Article 98, 2003 ASYPTOTIC BEHAVIOUR OF SOE EQUATIONS IN ORLICZ SPACES D. ESKINE AND A. ELAHI DÉPARTEENT DE ATHÉATIQUES ET INFORATIQUE FACULTÉ DES SCIENCES DHAR-AHRAZ B.P 1796 ATLAS-FÈS,FÈS AROC. C.P.R DE FÈS, B.P 49, FÈS, AROC. Received 26 arch, 2003; accepted 05 August, 2003 Communicated by A. Fiorenza ABSTRACT. In this paper, we prove an existence and uniqueness result for solutions of some bilateral problems of the form Au, v u f, v u, v K u K where A is a standard Leray-Lions operator defined on W0 1 L, with an N-function which satisfies the 2 -condition, and where K is a convex subset of W0 1 L with obstacles depending on some Carathéodory function gx, u. We consider first, the case f W 1 E and secondly where f L 1. Our method deals with the study of the limit of the sequence of solutions u n of some approximate problem with nonlinearity term of the form gx, u n n 1 gx, u n u n. Key words and phrases: Strongly nonlinear elliptic equations, Natural growth, Truncations, Variational inequalities, Bilateral problems athematics Subject Classification. 35J25, 35J INTRODUCTION Let be an open bounded subset of R N, N 2, with the segment property. Consider the following obstacle problem: Au, v u f, v u, v K, P u K, where Au = divax, u, u is a Leray-Lions operator defined on W 1 0 L, with being an N-function which satisfies the 2 -condition and where K is a convex subset of W 1 0 L. ISSN electronic: c 2003 Victoria University. All rights reserved

2 2 D. ESKINE AND A. ELAHI In the variational case i.e. where f W 1 E, it is well known that problem P has been already studied by Gossez and ustonen in [10]. In this paper, we consider a recent approach of penalization in order to prove an existence theorem for solutions of some bilateral problems of P type. We recall that L. Boccardo and F. urat, see [6], have approximated the model variational inequality: p u, v u f, v u, v K u K = {v W 1,p 0 : vx 1 a.e. in }, with f W 1,p and p u = div u p 2 u, by the sequence of problems: p u n + u n n 1 u n = f in D u n W 1,p 0 L n. In [7], A. Dall aglio and L. Orsina generalized this result by taking increasing powers depending also on some Carathéodory function g satisfying the sign condition and some hypothesis of integrability. Following this idea, we have studied in [5] the sequence of problems: p u n + gx, u n n 1 gx, u n u n p = f in D u n W 1,p 0, gx, u n n u n p L 1 Here, we introduce the general sequence of equations in the setting of Orlicz-Sobolev spaces Au n + gx, u n n 1 gx, u n u n = f in D u n W 1 0 L, gx, u n n u n L 1. We are interested throughout the paper in studying the limit of the sequence u n. We prove that this limit satisfies some bilateral problem of the P form under some conditions on g. In the first we take f W 1 E and next in L PRELIINARIES 2.1. N Functions. Let : R + R + be an N-function, i.e. is continuous, convex, with t > 0 for t > 0, t 0 as t 0 and t as t. t t Equivalently, admits the representation: t = t asds, where a : R+ R + is nondecreasing, right continuous, with a0 = 0, at > 0 for t > 0 and at tends to as t. The N-function conjugate to is defined by t = t 0 āsds, where a : R+ R + is given by āt = sup{s : as t} see [1]. The N-function is said to satisfy the 2 condition, denoted by 2, if for some k > 0: 2.1 2t kt t 0; when 2.1 holds only for t some t 0 > 0 then is said to satisfy the 2 condition near infinity. We will extend these N-functions into even functions on all R. Let P and Q be two N-functions. P Q means that P grows essentially less rapidly than Q, i.e. for each ɛ > 0, P t 0 as t. This is the case if and only if lim Qɛt t Q 1 t = 0. P 1 t J. Inequal. Pure and Appl. ath., 45 Art. 98,

3 ASYPTOTIC BEHAVIOUR OF SOE EQUATIONS IN ORLICZ SPACES Orlicz spaces. Let be an open subset of R N. The Orlicz class K resp. the Orlicz space L is defined as the set of equivalence classes of real valued measurable functions u on such that: uxdx < + resp. ux dx < + for some > 0. L is a Banach space under the norm { u, = inf > 0 : ux } dx 1 and K is a convex subset of L. The closure in L of the set of bounded measurable functions with compact support in is denoted by E. The equality E = L holds if only if satisfies the 2 condition, for all t or for t large according to whether has infinite measure or not. The dual of E can be identified with L by means of the pairing uvdx, and the dual norm of L is equivalent to,. The space L is reflexive if and only if and satisfy the 2 condition, for all t or for t large, according to whether has infinite measure or not Orlicz-Sobolev spaces. We now turn to the Orlicz-Sobolev space, W 1 L resp. W 1 E is the space of all functions u such that u and its distributional derivatives up to order 1 lie in L resp. E. It is a Banach space under the norm u 1, = D α u. α 1 Thus, W 1 L and W 1 E can be identified with subspaces of product of N + 1 copies of L. Denoting this product by L, we will use the weak topologies σ L, E and σ L, L. The space W0 1 E is defined as the norm closure of the Schwarz space D in W 1 E and the space W0 1 L as the σ L, E closure of D in W 1 L. We say that u n converges to u for the modular convergence in W 1 L if for some > 0 D α u n D α u dx 0 for all α 1. This implies convergence for σ L, L. If satisfies the 2 -condition on R +, then modular convergence coincides with norm convergence The spaces W 1 L and W 1 E. Let W 1 L resp. W 1 E denote the space of distributions on which can be written as sums of derivatives of order 1 of functions in L resp. E. It is a Banach space under the usual quotient norm. If the open set has the segment property then the space D is dense in W 1 0 L for the modular convergence and thus for the topology σ L, L cf. [8, 9]. Consequently, the action of a distribution in W 1 L on an element of W 1 0 L is well defined. J. Inequal. Pure and Appl. ath., 45 Art. 98,

4 4 D. ESKINE AND A. ELAHI 2.5. Lemmas related to the Nemytskii operators in Orlicz spaces. We recall some lemmas introduced in [3] which will be used in this paper. Lemma 2.1. Let F : R R be uniformly Lipschitzian, with F 0 = 0. Let be an N function and let u W 1 L resp. W 1 E. Then F u W 1 L resp. W 1 E. oreover, if the set D of discontinuity points of F is finite, then F u x i u a.e. in {x : ux / D}, F u = x i 0 a.e. in {x : ux / D}. Lemma 2.2. Let F : R R be uniformly Lipschitzian, with F 0 = 0. We suppose that the set of discontinuity points of F is finite. Let be an N-function, then the mapping F : W 1 L W 1 L is sequentially continuous with respect to the weak* topology σ L, E Abstract lemma applied to the truncation operators. We now give the following lemma which concerns operators of the Nemytskii type in Orlicz spaces see [3]. Lemma 2.3. Let be an open subset of R N with finite measure. Let, P and Q be N-functions such that Q P, and let f : R R be a Carathéodory function such that a.e. x and all s R: fx, s cx + k 1 P 1 k 2 s, where k 1, k 2 are real constants and cx E Q. Then the Nemytskii operator N f defined by N f ux = fx, ux is strongly continuous from } P E, 1k2 = {u L : du, E < 1k2 into E Q. 3. THE AIN RESULT Let be an open bounded subset of R N, N 2, with the segment property. Let be an N-function satisfying the 2 -condition near infinity. Let Au = divax, u be a Leray-Lions operator defined on W 1 0 L into W 1 L, where a : R N R N is a Carathéodory function satisfying for a.e. x and for all ζ, ζ R N, ζ ζ : 3.1 ax, ζ hx + 1 k 1 ζ 3.2 ax, ζ ax, ζ ζ ζ > ax, ζζ α ζ with α, > 0, k 1 0, h E. Furthermore, let g : R R be a Carathéodory function such that for a.e. x and for all s R: 3.4 gx, ss 0 J. Inequal. Pure and Appl. ath., 45 Art. 98,

5 ASYPTOTIC BEHAVIOUR OF SOE EQUATIONS IN ORLICZ SPACES gx, s b s 3.6 for almost x \ + there exists ɛ = ɛx > 0 such that: gx, s > 1, s ]q + x, q + x + ɛ[; for almost x \ gx, s < 1, s ]q x ɛ, q x[, there exists ɛ = ɛx > 0 such that: where b : R + R + is a continuous and nondecreasing function, with b0 = 0 and where q + x = inf{s > 0 : gx, s 1} q x = sup{s < 0 : gx, s 1} + = {x : q + x = + } = {x : q x = }. We define for s and k in R, k 0, T k s = max k, mink, s. Theorem 3.1. Let f W 1 E. Assume that hold true and that the function s gx, s is nondecreasing for a.e. x. Then, for any real number > 0, the problem Au n + gx, u n n 1 gx, u n un = f in D P n u n W0 1 L, gx, u n n L 1 admits at least one solution u n such that: un 3.7 k > 0 T k u n T k u for modular convergence in W 1 0 L where u is the unique solution of the following bilateral problem Au, v u f, v u, v K P u K = {v W0 1 L : q v q + a.e.}, Remark 3.2. If the function s gx, s is strictly nondecreasing for a.e. x then the assumption 3.6 holds true. Proof.Step 1: A priori estimates. The existence of u n is given by Theorem 3.1 of [3]. Choosing v = u n as a test function in P n, and using the sign condition 3.4, we get 3.8 Au n, u n f, u n. By Proposition 5 of [11] one has: un dx C, and ax, u n, u n u n dx C, 3.9 ax, u n, u n is bounded in L N, 3.10 gx, u n n 1 un gx, u n u n dx C. J. Inequal. Pure and Appl. ath., 45 Art. 98,

6 6 D. ESKINE AND A. ELAHI 3.11 We then deduce { u n >k} gx, u n n un dx C, for all k > 0. Since b is continuous and since b0 = 0 there exists δ > 0 such that b s 1 for all s δ. On the other hand, by the 2 condition there exist two positive constants K and K such that t t K + K for all t 0, which implies gx, u n n un dx K + K { u n δ} { u n δ} Consequently from 3.8 gx, u n n un dx C, for all n. un dx. Step 2: Almost everywhere convergence of the gradients. Since u n is a bounded sequence in W0 1 L there exist some u W0 1 L such that for a subsequence still denoted by u n 3.12 u n u weakly in W0 1 L for σ L, E, strongly in E, and a.e. in. Furthermore, if we have Au n = f gx, u n n 1 un gx, u n with gx, u n n 1 gx, u n un being bounded in L 1 then as in [2], one can show that 3.13 u n u a.e. in. Step 3: u K = {v W0 1 L : q v q + a.e. in }. Since s gx, s is nondecreasing, then in view of 3.6, we have: {s R : gx, s 1 a.e. in } = {s R : q s q + a.e. in }. It suffices to verify that gx, u 1 a.e. We have gx, u n n un which gives gx, u n n un and { gx,u n >k} { gx,u n >k} dx C, dx C un dx C k n J. Inequal. Pure and Appl. ath., 45 Art. 98,

7 ASYPTOTIC BEHAVIOUR OF SOE EQUATIONS IN ORLICZ SPACES 7 where k > 1. Letting n + for k fixed, we deduce by using Fatou s lemma un dx = 0 and so that, { gx,u >k} gx, u 1 a.e. in. Step 4: Strong convergence of the truncations. Let φs = s expγs 2, where γ is chosen such that γ 1 2 α. It is well known that φ s 2K φs 1, s R, where K is a constant which will be α 2 used later. The use of the test function v n = φz n in P n where z n = T k u n T k u gives Au n, φz n + gx, u n n 1 un gx, u n φz n dx = f, φz n which implies, by using the fact that gx, u n φz n 0 on {x : u n > k}, Au n, φz n + gx, u n n 1 un gx, u n φz n dx {0 u n T k u} { u n k} + gx, u n n 1 un gx, u n φz n dx f, φz n {T k u u n 0} { u n k} The second and the third terms of the last inequality will be denoted respectively by In,k 1 and I2 n,k and ɛ in denote various sequences of real numbers which tend to 0 as n +. On the one hand we have I 1 n,k {0 u n T k u} { u n k} {0 u n u} { u n k} gx, u n n un gx, u n n un φz n dx φz n dx, but since gx, u n 1 on {x : 0 u n u}, then we have I 1 n,k un φz n dx. { u n k} By using the fact that un K un + K we obtain I 1 n,k which gives K φz n dx + K α I 1 n,k ɛ1 n + K α ax, T k u n T k u n φz n dx, ax, T k u n T k u n φz n dx. J. Inequal. Pure and Appl. ath., 45 Art. 98,

8 8 D. ESKINE AND A. ELAHI Similarly, I 2 n,k un φz n dx { u n k} ɛ 1 n + K ax, T k u n T k u n φz n dx. α The first term on the left hand side of the last inequality can be written as: ax, u n [ T k u n T k u]φ z n dx = ax, u n [ T k u n T k u]φ z n dx { u n k} { u n >k} ax, u n T k uφ z n dx. For the second term on the right hand side of the last equality, we have ax, u n T k uφ z n dx C k ax, u n T k u χ { un >k}dx. { u n >k} 3.17 The right hand side of the last inequality tends to 0 as n tends to infinity. Indeed, the sequence ax, u n n is bounded in L N while T k uχ { un >k} tends to 0 strongly in E N. We define for every s > 0, s = {x : T k ux s} and we denote by χ s its characteristic function. For the first term of the right hand side of 3.16, we can write { u n k} = ax, u n [ T k u n T k u]φ z n dx [ax, T k u n ax, T k uχ s ][ T k u n T k uχ s ]φ z n dx + ax, T k uχ s [ T k u n T k uχ s ]φ z n dx ax, T k u n T k uχ \s φ z n dx. The second term of the right hand side of 3.17 tends to 0 since ax, T k u n χ s φ z n ax, T k uχ s strongly in E N by Lemma 2.3 and T k u n T k u weakly in L N for σ L, E. The third term of 3.17 tends to ax, T ku T k uχ \s dx as n since ax, T k u n ax, T k u weakly for σ E, L. J. Inequal. Pure and Appl. ath., 45 Art. 98,

9 ASYPTOTIC BEHAVIOUR OF SOE EQUATIONS IN ORLICZ SPACES Consequently, from 3.16 we have ax, u n [ T k u n T k u]φ z n dx = [ax, T k u n ax, T k uχ s ] [ T k u n T k uχ s ]φ z n dx + ɛ 2 n. We deduce that, in view of 3.17 and 3.18, [ax, T k u n ax, T k uχ s ] [ T k u n T k uχ s ] φ z n 2K α φz n dx ɛ 3 n + ax, T k u T k uχ \s dx, and so [ax, T k u n ax, T k uχ s ][ T k u n T k uχ s ]dx 2ɛ 3 n + 2 ax, T k u T k uχ \s dx. Hence ax, T k u n T k u n dx ax, T k u n T k uχ s dx + ax, T k uχ s [ T k u n T k uχ s ]dx + 2ɛ 3 n + 2 ax, T k u T k uχ \s dx. Now considering the limit sup over n, one has 3.19 lim sup ax, T k u n T k u n dx n + lim sup ax, T k u n T k uχ s dx + lim sup ax, T k uχ s n + n + [ T k u n T k uχ s ]dx + 2 ax, T k u T k uχ \s dx. The second term of the right hand side of the inequality 3.19 tends to 0, since ax, T k u n χ s ax, T k uχ s strongly in E, while T k u n tends weakly to T k u. The first term of the right hand side of 3.19 tends to ax, T ku T k uχ s dx since ax, T k u n ax, T k u weakly in L N J. Inequal. Pure and Appl. ath., 45 Art. 98,

10 10 D. ESKINE AND A. ELAHI for σ L, E while T k uχ s E. We deduce then lim sup ax, T k u n T k u n dx ax, T k u T k uχ s dx n ax, T k u T k uχ \s dx, by using the fact that ax, T k u T k u L 1 and letting s we get, since meas\ s 0 lim sup n + ax, T k u n T k u n dx ax, T k u T k udx which gives, by using Fatou s lemma, 3.20 lim ax, T k u n T k u n dx = n + On the other hand, we have Tk u n K + K α ax, T k u T k udx. ax, T k u n T k u n dx, then by using 3.20 and Vitali s theorem, one easily has Tk u n Tk u 3.21 strongly in L 1. By writing Tk u n T k u one has, by 3.21 and Vitali s theorem again, Tk u n 2 + Tk u n 3.23 T k u n T k u for modular convergence in W 1 0 L. Step 5: u is the solution of the variational inequality P. Choosing w = T k u n θt m v as a test function in P n, where v K and 0 < θ < 1, gives Au n, T k u n θt m v + gx, u n n 1 un gx, u n T k u n θt m vdx = f, T k u n θt m v, since gx, u n T k u n θt m v 0 on {x : u n 0 and u n θt m v} {x : u n 0 and u n θt m v} we have gx, u n n 1 un gx, u n T k u n θt m vdx gx, u n n 1 un gx, u n {0 u n θt mv} + gx, u n n 1 un gx, u n {θt mv u n 0} 2 T k u n θt m vdx T k u n θt m vdx. J. Inequal. Pure and Appl. ath., 45 Art. 98,

11 ASYPTOTIC BEHAVIOUR OF SOE EQUATIONS IN ORLICZ SPACES 11 The first and the second terms in the right hand side of the last inequality will be denoted respectively by Jn,m 1 and Jn,m. 2 Defining δ 1,m x = sup 0 s θt mv gx, s we get 0 δ 1,m x < 1 a.e. and Jn,m 1 k {0 u n θt mv} δ 1,m x n un dx. Since δ 1,mx n un χ { un m} Tm u n, we have then by using 3.23 and Lebesgue s theorem Similarly Jn,m 2 k where { u n m} J 1 n,m 0 as n +. δ 2,m x n Tm u n dx 0 as n +, δ 2,m x = inf gx, s. θt mv s 0 On the other hand, by using Fatou s lemma and the fact that ax, u n ax, u weakly in L N for σπl, ΠE, one easily has Consequently lim inf Au n, T k u n θt m v Au, T k u θt m v. n + Au, T k u θt m v f, T k u θt m v, this implies that by letting k +, since T k u θt m v u θt m v for modular convergence in W 1 0 L, Au, u θt m v f, u θt m v, in which we can easily pass to the limit as θ 1 and m + to obtain Au, u v f, u v. 4. THE L 1 CASE In this section, we study the same problems as before but we assume that q and q + are bounded. Theorem 4.1. Let f L 1. Assume that the hypotheses are as in Theorem 3.1, q and q + belong to L. Then the problem P n admits at least one solution u n such that: u n u for modular convergence in W 1 0 L, J. Inequal. Pure and Appl. ath., 45 Art. 98,

12 12 D. ESKINE AND A. ELAHI where u is the unique solution of the bilateral problem: Au, v u fv udx, v K Q u K = {v W0 1 L : q v q + a.e.}. Proof. We sketch the proof since the steps are similar to those in Section 3. The existence of u n is given by Theorem 1 of [4]. Indeed, it is easy to see that gx, s 1 on { s γ}, where γ = max {supess q +, infess q } and so that gx, s n ζ ζ for s γ. Step 1: A priori estimates. Choosing v = T γ u n, as a test function in P n, and using the sign condition 3.4, we obtain Tγ u n 4.1 α dx γ f and which gives and finally { u n >γ} gx, u n n un dx f 1, { u n >γ} un dx C un dx C. max{, } On the other hand, as in Section 3, we have gx, u n n un dx C. Step 2: Almost everywhere convergence of the gradients. Due to 4.2, there exists some u W 1 0 L such that for a subsequence u n u weakly in W 1 0 L for σπl, ΠE. Write Au n = f gx, u n n 1 un gx, u n and remark that, by 4.2, the second hand side is uniformly bounded in L 1. Then as in Section 3 u n u a.e. in. Step 3: u K = {v W0 1 L : q v q + a.e. in }. Similarly, as in the proof of Theorem 3.1, one can prove this step with the aid of property 4.3. Step 4: Strong convergence of the truncations. It is easy to see that the proof is the same as in Section 3. J. Inequal. Pure and Appl. ath., 45 Art. 98,

13 ASYPTOTIC BEHAVIOUR OF SOE EQUATIONS IN ORLICZ SPACES 13 Step 5: u is the solution of the bilateral problem Q. Let v K and 0 < θ < 1. Taking v n = T k u n θv, k > 0 as a test function in P n, one can see that the proof is the same by replacing T m v with v in Section 3. We remark that K L. Step 6: u n u for modular convergence in W0 1 L. We shall prove that u n u in L N for the modular convergence by using Vitali s theorem. Let E be a measurable subset of, we have for any k > 0 E un dx E { u n k} un dx + E { u n >k} un dx. Let ɛ > 0. By virtue of the modular convergence of the truncates, there exists some ηɛ, k such that for any E measurable un 4.4 E < ηɛ, k dx < ɛ 2, n. E { u n k} Choosing T 1 u n T k u n, with k > 0 a test function in P n we obtain: Au n, T 1 u n T k u n + gx, u n n 1 un gx, u n T 1 u n T k u n dx = ft 1 u n T k u n dx, 4.5 which implies { u n >k+1} gx, u n n un dx f dx. { u n >k} Note that meas{x : u n x > k} 0 uniformly on n when k. We deduce then that there exists k = kɛ such that f dx < ɛ 2, n, which gives { u n >k+1} { u n >k} gx, u n n un dx < ɛ 2, n. By setting tɛ = max {k + 1, γ} we obtain un dx < ɛ 2, n. { u n >tɛ} Combining 4.4 and 4.5 we deduce that there exists η > 0 such that un < ɛ, n when E < η, E measurable, E which shows the equi-integrability of un in L 1, and therefore we have un u strongly in L 1. J. Inequal. Pure and Appl. ath., 45 Art. 98,

14 14 D. ESKINE AND A. ELAHI By remarking that un u [ ] un u + one easily has, by using the Lebesgue theorem un u dx 0 as n +, 2 which completes the proof. Remark 4.2. The condition b0 = 0 is not necessary. Indeed, taking θ h u n, h > 0, as a test function in P n with hs if s 1 h θ h s = sgns if s 1, h we obtain gx, u n n 1 un gx, u n θ h u n dx fθ h u n dx. and then, by letting h +, gx, u n n un dx C. REFERENCES [1] R. ADAS, Sobolev Spaces, Academic Press, New York, [2] A. BENKIRANE AND A. ELAHI, Almost everywhere convergence of the gradients of solutions to elliptic equations in Orlicz spaces and application, Nonlinear Anal. T..A., , [3] A. BENKIRANE AND A. ELAHI, An existence theorem for a strongly nonlinear elliptic problem in Orlicz spaces, Nonlinear Anal. T..A., , [4] A. BENKIRANE AND A. ELAHI, A strongly nonlinear elliptic equation having natural growth terms and L 1 data, Nonlinear Anal. T..A., , [5] A. BENKIRANE, A. ELAHI AND D. ESKINE, On the limit of some nonlinear elliptic problems, Archives of Inequalities and Applications, , [6] L. BOCCARDO AND F. URAT, Increase of power leads to bilateral problems, in Composite edia and Homogenization Theory, G. Dal aso and G. F. Dell Antonio Eds., World Scientific, Singapore, 1995, pp [7] A. DALL AGLIO AND L. ORSINA, On the limit of some nonlinear elliptic equations involving increasing powers, Asympt. Anal., , [8] J.-P. GOSSEZ, Nonlinear elliptic boundary value problems for equations with rapidly or slowly increasing coefficients, Trans. Amer. ath. Soc., , [9] J.-P. GOSSEZ, Some approximation properties in Orlicz-Sobolev spaces, Studia ath., , [10] J.-P. GOSSEZ, A strongly nonlinear elliptic problem in Orlicz-Sobolev spaces, Proc. A..S. Symp. Pure ath., , J. Inequal. Pure and Appl. ath., 45 Art. 98,

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