Electronic Journal of Differential Equations, Vol 2007(2007, o 54, pp 1 13 ISS: 1072-6691 URL: http://ejdemathtxstateedu or http://ejdemathuntedu ftp ejdemathtxstateedu (login: ftp EXISTECE OF BOUDED SOLUTIOS FOR OLIEAR DEGEERATE ELLIPTIC EQUATIOS I ORLICZ SPACES AHMED YOUSSFI Abstract We prove the existence of bounded solutions for the nonlinear elliptic problem div a(x, u, u = f in, with u W 1 0 L M ( L (, where a(x, s, ξ ξ M 1 M(h( s M( ξ, and h : R + ]0, 1] is a continuous monotone decreasing function with unbounded primitive As regards the -function M, no 2 -condition is needed 1 Introduction Let be a bounded open set of R, 2 We consider the equation div(a(x, um 1 (M( u u = f in, u u = 0 on, (11 where M 1 1 (M( a(x, s β, (12 (1 + s θ with 0 θ 1, and β is a positive constant For M(t = t 2, existence of bounded solutions of (11 was proved under (12 in [4] and in [5] when f L m ( with m > 2 This result was then extended in [3], to the study of the problem div a(x, u, u = f in, u = 0 on, (13 in the Sobolev space W 1,p 0 (, under the condition α a(x, s, ξ ξ (1 + s θ(p 1 ξ p, (14 when f L m ( with m > p In this paper, we prove the existence of bounded solutions of (13 in the setting of Orlicz spaces under a more general condition than (14 adapted to this situation 2000 Mathematics Subject Classification 46E30, 35J70, 35J60 Key words and phrases Orlicz-Sobolev spaces; degenerate coercivity; L -estimates; rearrangements c 2007 Texas State University - San Marcos Submitted December 11, 2006 Published April 10, 2007 1
2 A YOUSSFI EJDE-2007/54 The main tools used to get a priori estimates in our proof are symmetrization techniques such techniques are widely used in the literature for linear and nonlinear equations (see [3] and the references quoted therein We remark that our result is in some sense complementary to one contained in [13] As examples of equations to which our result can be applied, we list: α e u p 1 div( (e + u γ log(e + u u 2 u = f in, u = 0 on, where α > 0, γ < 1 and M(t = e tp 1 with 1 < p < ; and α div( (1 + u γ u p 2 ulog q (e + u = f in, u = 0 on, where α > 0 and 0 γ 1, here M(t = t p log q (e + t with 1 < p < and q R 2 Preliminaries Let M : R + R + be an -function; ie, M is continuous, convex, with M(t > 0 for t > 0, M(t t 0 as t 0 and M(t t as t The -function conjugate to M is defined as M(t = sup{st M(t, s 0} We will extend these -functions into even functions on all R We recall that (see [1] M(t tm 1 (M(t 2M(t for all t 0 (21 and the Young s inequality: for all s, t 0, st M(s + M(t If for some k > 0, M(2t km(t for all t 0, (22 we said that M satisfies the 2 -condition, and if (22 holds only for t greater than or equal to t 0, then M is said to satisfy the 2 -condition near infinity Let P and Q be two -functions the notation P Q means that P grows essentially less rapidly than Q, ie ɛ > 0, that is the case if and only if P (t 0 as t, Q(ɛt Q 1 (t P 1 0 as t (t Let be an open subset of R The Orlicz class K M ( (resp the Orlicz space L M ( is defined as the set of (equivalence class of real-valued measurable functions u on such that: M(u(xdx < (resp Endowed with the norm u M = inf{λ > 0 : M( u(x dx < for some λ > 0 λ M( u(x dx < }, λ L M ( is a Banach space and K M ( is a convex subset of L M ( the closure in L M ( of the set of bounded measurable functions with compact support in is denoted by E M (
EJDE-2007/54 EXISTECE OF BOUDED SOLUTIOS 3 The Orlicz-Sobolev space W 1 L M ( (resp W 1 E M ( is the space of functions u such that u and its distributional derivatives up to order 1 lie in L M ( (resp E M ( This is a Banach space under the norm u 1,M = D α u M α 1 Thus, W 1 L M ( and W 1 E M ( can be identified with subspaces of the product of ( + 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 The space W0 1 E M ( is defined as the norm closure of the Schwartz space D( in W 1 E M ( and the space W0 1 L M ( as the σ(πl M, ΠE M closure of D( in W 1 L M ( We say that a sequence {u n } converges to u for the modular convergence in W 1 L M ( if, for some λ > 0, M( Dα u n D α u dx 0 for all α 1; λ this implies convergence for σ(πl M, ΠL M If M satisfies the 2 -condition on R + (near infinity only if has finite measure, then the modular convergence coincides with norm convergence Recall that the norm Du M defined on W0 1 L M ( is equivalent to u 1,M (see [10] Let W 1 L M ( (resp W 1 E M ( denotes 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 If the open has the segment property then the space D( is dense in W0 1 L M ( for the topology σ(πl M, ΠL M (see [10] Consequently, the action of a distribution in W 1 L M ( on an element of W0 1 L M ( is well defined For an exhaustive treatments one can see for example [1] or [12] We will use the following lemma, (see[6], which concerns operators of emytskii Type in Orlicz spaces It is slightly different from the analogous one given in [12] Lemma 21 Let be an open subset of R with finite measure let M, P and Q be -functions such that Q P, and let f : R R be a Carathéodory function such that, for ae x and for all s R, f(x, s c(x + k 1 P 1 M(k 2 s, where k 1, k 2 are real constants and c(x E Q ( Then the emytskii operator f, defined by f (u(x = f(x, u(x, is strongly continuous from P (E M, 1 k 2 = {u L M ( : d(u, E M ( < 1 k 2 } into E Q ( We recall the definition of decreasing rearrangement of a measurable function w : R If one denotes by E the Lebesgue measure of a set E, one can define the distribution function µ w (t of w as: µ w (t = {x : w(x > t}, t 0 The decreasing rearrangement w of w is defined as the generalized inverse function of µ w : w (σ = inf{t R : µ w (t σ}, σ (0,
4 A YOUSSFI EJDE-2007/54 It is shown in [15] that w is everywhere continuous and w (µ w (t = t (23 for every t between 0 and ess sup w More details can be found for example in [2, 13, 14] 3 Assumptions and main result Let be an open bounded subset of R, 2, satisfying the segment property and M is an -function twice continuously differentiable and strictly increasing, and P is an -function such that P M Let a : R R R be a Carathéodory function satisfying, for ae x, and for all s R and all ξ, η R, ξ η, a(x, s, ξ ξ M 1 M(h( s M( ξ (31 where h : R + R + is a continuous monotone decreasing function such that h(0 1 and its primitive H(s = s h(tdt is unbounded, 0 a(x, s, ξ a 0 (x + k 1 P 1 M(k 2 s + k 3 M 1 M(k 4 ξ (32 where a 0 (x belongs to E M ( and k 1, k 2, k 3, k 4 to R +, (a(x, s, ξ a(x, s, η (ξ η > 0 (33 Let A: D(A W 1 0 L M ( W 1 L M ( be a mapping (non-everywhere defined given by A(u := div a(x, u, u, We are interested, in proving the existence of bounded solutions to the nonlinear problem A(u := div(a(x, u, u = f in, (34 u = 0 on, As regards the data f, we assume one of the following two conditions: Either or f L (, (35 f L m ( with m = r/(r + 1 for some r > 0, + t and ( M(t r dt < + We will use the following concept of solutions: (36 Definition 31 Let f L 1 (, a function u W0 1 L M ( is said to be a weak solution of (34, if a(, u, u (L M ( and a(x, u, u v dx = fv dx holds for all v D( Our main result is the following Theorem 32 Under the assumptions (31, (32, (33 and either (35 or (36, there exists at least one weak solution of (34 in W 1 0 L M ( L (
EJDE-2007/54 EXISTECE OF BOUDED SOLUTIOS 5 Remark 33 In the case where M(t = t p, with p > 1, assumptions (35 and (36 imply that m > p Our result extends those in [5] and [4] where M(t = t2 and [3] where M(t = t p, with p > 1 Remark 34 ote that the result of theorem (31 is independent of the function h which eliminates the coercivity of the operator A The result is not surprising, since if we look for bounded solutions then the operator A becomes coercive Remark 35 The principal difficulty in dealing with the problem (34 is the non coerciveness of the operator A, this is due to the hypothesis (31, so the classical methods used to prove the existence of a solution for (34 can not be applied (see [11] and also [9] To get rid of this difficulty, we will consider an approximation method in which we introduce a truncation The main tool of the proof will be L a priori estimates, obtained by mean of a comparison result, which then imply the W 1 0 L M ( estimate, since if u is bounded the operator A becomes uniformly coercive 4 Proof of theorem 32 For s R and k > 0 set: T k (s = max( k, min(k, s and G k (s = s T k (s Let {f n } W 1 E M ( be a sequence of smooth functions such that and f n f strongly in L m ( f n m f m, where m denotes either or m, according as we assume (35 or (36, and consider the operator: A n (u = div a(x, T n (u, u By assumption (31, we have A n (u, u = a(x, T n (u, u u dx M 1 (M(h(n M( u dx Thus, A n satisfies the classical conditions from which derives, thanks to the fact that f n W 1 E M (, the existence of a solution u n W0 1 L M (, (see [11] and also [9], such that a(x, T n (u n, u n vdx = f n vdx (41 holds for all v W 1 0 L M ( To prove the L a priori estimates, we will need the following comparison lemma, whose proof will be given in the appendix Lemma 41 Let B(t = M(t t and µ n (t = {x : u n (x > t}, for all t > 0 We have for almost every t > 0: µ n(t { u f n >t} n dx h(t µ n (t 1 1 where C is the measure of the unit ball in R µ n(t 1 1 (42
6 A YOUSSFI EJDE-2007/54 step 1: L -bound If we assume (35, using the inequality f n dx f µ n (t 1 1, (42 becomes ( µ h(t n(t µ n(t 1 1 Then we integrate between 0 and s, we get H(s hence, a change of variables yields H(s By (23 we get So that H(u n(σ H(u n(0 f f f f f 0 s µ n(t dt; µ n (t 1 1 dt µ n(s t 1 1 σ dt t 1 1 1/ Since u n(0 = u n, the assumption made on H (ie, lim s + H(s = + shows that the sequence {u n } is uniformly bounded in L ( Moreover if we denote by H 1 the inverse function of H, one has: u n H 1( f 1/ (43 ow, we assume that (36 is filled Then, using the inequality f n dx f m µ n (t 1 1 m in (42, we obtain H(s A change of variables gives H(s As above, (23 gives H(u n(τ Then, we have H( u n s 0 µ n(t µ n (t 1 1 µ n(s τ 0 f m µ n(t 1 m 1 f m σ 1 m 1 f m σ 1 m 1 f m σ 1 m 1 dσ σ 1 1 dσ dt σ 1 1 dσ σ 1 1
EJDE-2007/54 EXISTECE OF BOUDED SOLUTIOS 7 A change of variables gives where c 0 = H( u n H( u n f r m (M 1 (M(1 r+1 r C r+1 + c 0 rt r 1 B 1 (tdt, f m Then, an integration by parts yields r 1 f r ( m B 1 (c 0 (M 1 (M(1 r+1 r C r+1 c r + 0 + B 1 (c 0 ( s rds M(s The assumption made on H guarantees that the sequence {u n } is uniformly bounded in L ( Indeed, denoting by H 1 the inverse function of H, one has u n H 1( f r m (M 1 (M(1 r+1 r C r+1 ( B 1 (c 0 + c r 0 + B 1 (c 0 ( s rds M(s (44 Consequently, in both cases the sequence {u n } is uniformly bounded in L (, so that in the sequel, we will denote by c the constant appearing either in (43 or in (44, that is u n c (45 Step 2: Estimation in W0 1 L M ( It is now easy to obtain an estimate in W0 1 L M ( under either (35 or (36 Let m denotes either or m according as we assume (35 or (36 Taking u n as test function in (41, one has a(x, T n (u n, u n u n dx = f n u n dx Then by (31 and (45, we obtain M( u n dx c f m 1 1 m M 1 (M(h(c (46 Hence, the sequence {u n } is bounded in W0 1 L M ( Therefore, there exists a subsequence of {u n }, still denoted by {u n }, and a function u in W0 1 L M ( such that and u n u in W 1 0 L M ( for σ(πl M, ΠE M (47 u n u in E M ( strongly and ae in (48 Step 3: Almost everywhere convergence of the gradients Let us begin with the following lemma which we will use later Lemma 42 The sequence {a(x, T n (u n, u n } is bounded in (L M ( Proof We will use the dual norm of (L M ( Let ϕ (E M ( such that ϕ M = 1 By (33 we have ( a(x, T n (u n, u n a(x, T n (u n, ϕ ( u n ϕ 0 k 4 k 4
8 A YOUSSFI EJDE-2007/54 Let λ = 1 + k 1 + k 3, by using (32, (45, (46 and Young s inequality we get a(x, T n (u n, u n ϕdx k 4 a(x, T n (u n, u n u n dx k 4 a(x, T n (u n, ϕ u n dx k 4 + a(x, T n (u n, ϕ ϕdx k 4 k 4 c f m 1 1 m + k 4 λ c f m 1 1 m M 1 M(h(c ( + (1 + k 4 M(a 0 (xdx + k 1 MP 1 M(k 2 c which completes the proof + k 3 (1 + k 4 + λ, From (45 and (48 we obtain that u W0 1 L M ( L (, so that by [8, Theorem 4] there exists a sequence {v j } in D( such that v j u in W0 1 L M ( as j for the modular convergence and almost everywhere in, moreover v j ( + 1 u For s > 0, we denote by χ s j the characteristic function of the set s j = {x : v j (x s} and by χ s the characteristic function of the set s = {x : u(x s} Testing by u n v j in (41, we obtain a(x, T n (u n, u n ( u n v j dx = f n (u n v j dx (49 Denote by ɛ i (n, j, (i = 0, 1,, various sequences of real numbers which tend to 0 when n and j, ie lim lim ɛ i(n, j = 0 j n For the right-hand side of (49, we have f n (u n v j dx = ɛ 0 (n, j (410 The left-hand side of (49 is written as a(x, T n (u n, u n ( u n v j dx ( = a(x, Tn (u n, u n a(x, T n (u n, v j χ s j ( u n v j χ s j dx + a(x, T n (u n, v j χ s j ( u n v j χ s jdx a(x, T n (u n, u n v j dx \ s j (411 We will pass to the limit over n and j, for s fixed, in the second and the third terms of the right-hand side of (411 By Lemma 42, we deduce that there exists
EJDE-2007/54 EXISTECE OF BOUDED SOLUTIOS 9 l (L M ( and up to a subsequence a(x, T n (u n, u n l weakly in (L M ( for σ( L M, E M Since v j χ \ s j (E M (, we have by letting n, a(x, T n (u n, u n v j dx l v j dx \ s j \ s j Using the modular convergence of v j, we get as j l v j dx l u dx \ s \ s j Hence, we have proved that the third term a(x, T n (u n, u n v j dx = l udx + ɛ 1 (n, j (412 \ s \ s j For the second term, as n, we have a(x, T n (u n, v j χ s j ( u n v j χ s j dx a(x, u, v j χ s j ( u v j χ s j dx, since a(x, T n (u n, v j χ s j a(x, u, v jχ s j strongly in (E M ( as n by lemma 21 and (48, while u n u weakly in (L M ( by (47 And since v j χ s j uχs strongly in (E M ( as j, we obtain a(x, u, v j χ s j ( u v j χ s j dx 0 as j So that a(x, T n (u n, v j χ s j ( u n v j χ s j dx = ɛ2 (n, j (413 Consequently, combining (410, (412 and (413, we obtain ( a(x, Tn (u n, u n a(x, T n (u n, v j χ s j ( u n v j χ s j dx = l u dx + ɛ 3 (n, j \ s On the other hand (a(x, T n (u n, u n a(x, T n (u n, uχ s ( u n uχ s dx ( = a(x, Tn (u n, u n a(x, T n (u n, v j χ s j ( u n v j χ s j dx + a(x, T n (u n, u n ( v j χ s j uχ s dx a(x, T n (u n, uχ s ( u n uχ s dx + a(x, T n (u n, v j χ s j ( u n v j χ s j dx (414
10 A YOUSSFI EJDE-2007/54 We can argue as above in order to obtain a(x, T n (u n, u n ( v j χ s j uχ s dx = ɛ 4 (n, j, a(x, T n (u n, uχ s ( u n uχ s dx = ɛ 5 (n, j, a(x, T n (u n, v j χ s j ( u n v j χ s j dx = ɛ6 (n, j Then, by (414 we have (a(x, T n (u n, u n a(x, T n (u n, uχ s ( u n uχ s dx = ɛ 7 (n, j + l udx \ s For r s, we write 0 (a(x, T n (u n, u n a(x, T n (u n, u ( u n u dx r (a(x, T n (u n, u n a(x, T n (u n, u ( u n u dx s = (a(x, T n (u n, u n a(x, T n (u n, uχ s ( u n uχ s dx s (a(x, T n (u n, u n a(x, T n (u n, uχ s ( u n uχ s dx ɛ 7 (n, j + l udx \ s Which implies by passing at first to the limit superior over n and then over j, 0 lim sup (a(x, T n (u n, u n a(x, T n (u n, u ( u n u dx n r l udx \ s Letting s + in the previous inequality, we conclude that as n, r (a(x, T n (u n, u n a(x, T n (u n, u ( u n u dx 0 (415 Let B n be defined by B n = (a(x, T n (u n, u n a(x, T n (u n, u ( u n u As a consequence of (415, one has B n 0 strongly in L 1 ( r, extracting a subsequence, still denoted by {u n }, we get B n 0 ae in r Then, there exists a subset Z of r, of zero measure, such that: B n (x 0 for all x r \ Z Using (32, we obtain for all x r \ Z, ( B n (x M 1 M(h(cM( u n (x c 1 (x 1 + M 1 M(k 4 u n (x + u n (x, where c is the constant appearing in (45 and c 1 (x is a constant which does not depend on n Thus, the sequence { u n (x} is bounded in R, then for a
EJDE-2007/54 EXISTECE OF BOUDED SOLUTIOS 11 subsequence {u n (x}, we have u n (x ξ in R, (a(x, u(x, ξ a(x, u(x, u(x (ξ u(x = 0 Since a(x, s, ξ is strictly monotone, we have ξ = u(x, and so u n (x u(x for the whole sequence It follows that u n u ae in r Consequently, as r is arbitrary, one can deduce that u n u ae in (416 Step 4: Passage to the limit Let v be a function in D( Taking v as test function in (41, one has a(x, T n (u n, u n vdx = f n v dx Lemma 42, (48 and (416 imply that a(x, T n (u n, u n a(x, u, u weakly in (L M ( for σ(πl M, ΠE M, so that one can pass to the limit in the previous equality to obtain a(x, u, u vdx = fv dx Moreover, from (45 and (48 we have u W 1 0 L M ( L ( This completes the proof of theorem 32 Remark 43 ote that the L -bound in step 1 can be proven under the weaker assumption s f m, = sup s 1 m 1 f (tdt <, s>0 0 which is equivalent to say that f belongs to the Lorentz space L(m, Indeed, one can use the inequality µn(t f n dx f (tdt (see [13, 14] in (41 to obtain: If f belongs to L(,, then h(t ( µ n(t µ n(t 1 1 0 f, and if we assume that f belongs to L(m, with m < and we obtain h(t + ( µ n(t µ n(t 1 1 ( t m m dt < +, M(t f m,, µ n(t 1 m 1 As above, starting with those inequalities we obtain the desired result Observe that when h is a constant function, this L -bound was proved in [13]
12 A YOUSSFI EJDE-2007/54 5 Appendix In this section, we prove lemma 41 based on techniques inspired from those in [13] Proof of Lemma 41 Testing by v = T k (G t (u n, which lies in W0 1 L M ( thanks to [7, Lemma 2], in (41 one has a(x, T n (u n, u n u n dx k f n dx Then (31 yields 1 k {t< u n t+k} {t< u n t+k} M 1 M(h( u n M( u n dx Letting k 0 + we obtain d M 1 M(h( u n M( u n dx dt f n dx f n dx, (51 for almost every t > 0 The hypotheses made on the -function M, which are not a restriction, allow to affirm that the function C(t = 1 B 1 (t is decreasing and convex (see [13] Hence, Jensen s inequality yields C ( {t< u M 1 n t+k} (M(h( u n M( u n dx {t< u M 1 n t+k} (M(h( u n u n dx ( {t< u = C B( u n t+k} n M 1 (M(h( u n u n dx {t< u M 1 n t+k} (M(h( u n u n dx {t< u M 1 n t+k} (M(h( u n dx {t< u M 1 n t+k} (M(h( u n u n dx M 1 (M(h(t( µ n (t + k + µ n (t M 1 (M(h(t + k {t< u n t+k} u n dx Taking into account that M 1 (M(h(t M 1 (M(1, using the convexity of C and then letting k 0 +, we obtain for almost every t > 0, M 1 (M(1 ( d M 1 (M(h(t C dt { u M 1 (M(h( u n >t} n M( u n dx M 1 (M(1( d dt { u u n >t} n dx µ n(t { u u n >t} n dx d dt ow we recall the following inequality from [13]: d u n dx C 1/ dt n(t 1 1 for almost every t > 0 (52
EJDE-2007/54 EXISTECE OF BOUDED SOLUTIOS 13 The monotonicity of the function C, (51 and (52 yield 1 M 1 (M(h(t µ n(t µ n(t 1 1 Using (21 and the fact that 0 < h(t 1, we obtain (42 f n dx µ n(t 1 1 References [1] R Adams; Sobolev spaces, Academic Press Inc, ew York, (1975 [2] C Bennett, R Sharpley; Interpolation of operators, Academic press, Boston, (1988 [3] A Alvino, L Boccardo, V Ferone, L Orsina, G Trombetti; Existence results for nonlinear elliptic equations with degenerate coercivity, Ann Mat Pura Appl, IV Ser 182, o1, (2003, 53-79 [4] A Alvino, V Ferone, G Trombetti; A priori estimates for a class of non uniformly elliptic equations, Atti Semin Mat Fis Univ Modena 46-suppl, (1998, 381-391 [5] L Boccardo, A Dall Aglio, L Orsina; Existence and regularity results for some elliptic equations with degenerate coercivity Atti Semin Mat Fis Univ Modena 46-suppl, (1998, 51-81 [6] A Elmahi, D Meskine; onlinear elliptic problems having natural growth and L 1 data in Orlicz spaces, Ann Mat Pura Appl 184, o 2, (2005, 161-184 [7] J-P Gossez; A strongly nonlinear elliptic problem in Orlicz-Sobolev spaces, Proc Sympos Pure Math 45, Amer Math Soc, (1986, 455-462 [8] J-P Gossez; Some approximation properties in Orlicz-Sobolev spaces, Stud Math 74, (1982, 17-24 [9] J-P Gossez; Surjectivity results for pseudo-monotone mappings in complementary systems, J Math Anal Appl 53, (1976, 484-494 [10] J-P Gossez, onlinear elliptic boundary value problems for equations with rapidly (or slowly increasing coefficients, Trans Amer Math soc 190, (1974, 163-205 [11] J-P Gossez, V Mustonen; Variational inequalities in Orlicz-Sobolev spaces, onlinear Anal, Theory Methods Appl 11, (1987, 379-392 [12] M Krasnosel skii, Ya Rutikii; Convex functions and Orlicz spaces, Groningen, ordhooff (1969 [13] G Talenti; onlinear elliptic equations, Rearrangements of functions and Orlicz spaces, Ann Mat Pura Appl, IV Ser 120, (1979, 159-184 [14] G Talenti; Elliptic equations and rearrangements, Ann Scuola orm Sup Pisa (4 3, (1976, 697-718 [15] Talenti, G: Linear elliptic PDE s: Level sets, rearrangements and a priori estimates of solutions, Boll Un Mat Ital 4-B(6, (1985, 917-949 Ahmed Youssfi Department of Mathematics and Informatics, Faculty of Sciences Dhar El Mahraz, University Sidi Mohammed Ben Abdallah, PB 1796 Fez-Atlas, Fez, Morocco E-mail address: Ahmedyoussfi@caramailcom