Existence Results for Quasilinear Degenerated Equations Via Strong Convergence of Truncations

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1 Existence Results for Quasilinear Degenerated Equations Via Strong Convergence of Truncations Youssef AKDIM, Elhoussine AZROUL, and Abdelmoujib BENKIRANE Déartement de Mathématiques et Informatique, Faculté des Sciences Dhar-Mahraz, B.P Atlas, Fès, Maroc. Recibido: 29 de Noviembre de 2001 Acetado: 5 de Febrero de 2004 ABSTRACT In this aer we study the existence of solutions for quasilinear degenerated ellitic oerators A(u) + g(x, u, u) = f, where A is a Leray-Lions oerator from W 1, 0 (, w) into its dual, while g(x, s, ξ) is a nonlinear term which has a growth condition with resect to ξ and no growth with resect to s, but it satisfies a sign condition on s. The right hand side f is assumed to belong either to W 1, (, w ) or to L 1 (). Key words: Weighted Sobolev saces, Hardy inequality, Quasilinear degenerated ellitic oerators, Truncations 2000 Mathematics Subject Classification: 35J15, 35J20, 35J Introduction In this aer, we shall be concerned with the existence of solutions for quasilinear degenerated ellitic equations of the tye (P) { Au + g(x, u, u) = f in D (), u W 1, 0 (, w), g(x, u, u) L 1 (), g(x, u, u)u L 1 (), Rev. Mat. Comlut. 359 ISSN:

2 where Au = div(a(x, u, u)) is a weighted Leray-Lions oerator from the weighted Sobolev sace X = W 1, 0 (, w) into X = W 1, (, w ) and where g is a nonlinear lower order term having natural growth ( g(x, s, ξ) b( s )(c(x) + N w i ξ i )) which satisfies the sign condition g(x, s, ξ)s 0. The right hand side f is assumed to belong to X or L 1 (). In the last case, we also assume that g(x, s, ξ) has an exact natural growth i.e., g(x, s, ξ) ρ N w i ξ i. It will turn out that for any solution u, g(x, u, u) L 1 (), but for a general v X, g(x, v, v) can be more singular. Drabek and Nicolosi in [10] roved the existence of bounded solution for the degenerated roblem (P) where g(x, u, u) = c 0 u 2 u, more recisely for the roblem, Au c 0 u 2 u = f(x, u, u) with some more general degeneracy, but under some other assumtions on f and a(x, s, ξ). The existence result for the roblem (P) (resectively, unilateral roblem) where f lies in the dual sace W 1, (, w ) is also studied in [1] (resectively, [2]), namely, the authors obtain the existence results by roving that the ositive art u + ε (res. u ε ) of u ε strongly converges to u + (res. u ), where u ε is a solution of the aroximate roblem. Our first aim of this aer is to rove (in Theorem 3.7) the same existence result as in [1] by using another aroach based on the strong convergence of the truncations T k (u ε ) in W 1, 0 (, w). Moreover, we assume only the weak integrability condition σ 1 q L 1 loc () (see assumtion (H 1) below) instead of the stronger one which is σ 1 q L 1 () as in [1]. For that, we aroximate the term g(x, s, ξ) by some functions involving χ ε where ε is a sequence of comacts covering the bounded oen set and χ ε is a characteristic function, i.e., g ε (x, s, ξ) = g(x,s,ξ) 1+ε g(x,s,ξ) χ ε (x). The second aim of this aer is to rove (in Theorem 3.12) the existence result for the following roblem { ( P) Au + g(x, u, u) = f in D (), u W 1, 0 (, w), g(x, u, u) L 1 (), where f L 1 (), under some added hyothesis (see (35) below). Note that in the non weighted case Boccardo, Gallouët and Murat in [6] have roved the existence of at least one solution for the roblem (P) and ( P). Let us oint out that another work in this direction can be found in [4] where the right hand side f is assumed to belong to W 1, () and in [5] with f L 1 (). Our results (Theorem 3.7 and Theorem 3.12) generalize those obtained in [4], [5] and [6], in the weighted case. The resent aer is organized as follows: In Section 2, we give some reliminaries. In the first art of Section 3, we rove some technical lemmas concerning some Revista Matemática Comlutense 360

3 convergences in weighted Sobolev saces and determinate basic assumtions. And in the second art we study the first main result (where f X ). In the third art, we study the case where f L 1 (). The fifth art is devoted to an examle which illustrates our abstract hyotheses. Note that, in the roof of our main general results, many ideas have been adoted from the work of [6]. 2. Preliminaries Let be a bounded oen subset of R N (N 1), let 1 < <, and let w = { w i (x), 0 i N } be a vector of weight functions; i.e. every comonent w i (x) is a measurable function which is strictly ositive a.e. in. Furthermore, we suose in all our considerations that w i L 1 loc() (1) and w 1 1 i L 1 loc() (2) for any 0 i N. We define the weighted sace L (, γ), where γ is a weight function on, by L (, γ) = {u = u(x), uγ 1 L ()} with the norm ( u,γ = u(x) γ(x) dx. We denote by W 1, (, w) the sace of all real-valued functions u L (, w 0 ) such that the derivatives in the sense of distributions satisfies u x i L (, w i ) for all i = 1,..., N, which is a Banach sace under the norm ( u 1,,w = u(x) w 0 (x) dx + u(x) x i w i (x) dx Since we shall deal with the Dirichlet roblem, we shall use the sace X = W 1, 0 (, w). (3) defined as the closure of C0 () with resect to the norm (3). Note that, C0 () is dense in W 1, 0 (, w) and (X, 1,,w ) is a reflexive Banach sace. We recall that the dual sace of the weighted Sobolev saces W 1, 0 (, w) is equivalent to W 1, (, w ), where w = { wi = w 1 i, i = 0,..., N }, and is the conjugate of, i.e. = 1. For more details, we refer the reader to [9]. Now we state the following assumtion: 361 Revista Matemática Comlutense

4 Assumtion (H 1 ). The exression ( N u X = u(x) x i w i (x) dx (4) is a norm defined on X and it is equivalent to the norm (3). There exist a weight function σ on and a arameter q, 1 < q <, such that with q = q q 1. The Hardy inequality, ( u(x) q σ dx q σ 1 q L 1 loc(), (5) ( N c u(x) x i w i (x) dx, (6) holds for every u X with a constant c > 0 indeendent of u. Moreover, the imbedding exressed by the inequality (6) is comact. X L q (, σ), (7) Note that (X, X ) is a uniformly convex (and thus reflexive) Banach sace. Remark 2.1. Assume that w 0 (x) 1 and in addition the integrability condition: There exists ν ] N, [ [ 1 1, [ such that w ν i L 1 () for all i = 1,..., N (which is stronger than (2)). Then ( N u X = u(x) x i w i (x) dx is a norm defined on W 1, 0 (, w) and it is equivalent to (3). Moreover W 1, 0 (, w) L q () for all 1 q < 1 if ν < N(ν + 1) and for all q 1 if ν N(ν + 1), where 1 = ν ν+1 and 1 = N1 Nν N 1 = N(ν+1) ν is the Sobolev conjugate of 1 (see [9]). Thus the hyotheses (H 1 ) is satisfied for σ 1. Definition. Let X be a reflexive Banach sace. An oerator B from X to its dual X satisfies roerty (M) if for any sequence (u n ) X satisfying u n u in X weakly, B(u n ) χ in X weakly and lim su Bu n, u n χ, u then one has χ = B(u). n Revista Matemática Comlutense 362

5 3. Main results Let A be the nonlinear oerator from W 1, 0 (, w) into its dual W 1, (, w ) defined as Au = div(a(x, u, u)) where a : R R N R N is a Carathéodory vector-function satisfying the following assumtions: Assumtion (H 2 ). a i (x, s, ξ) βw 1 i (x)[k(x) + σ 1 s q + 1 w j (x) ξ j 1 ] for i = 1,..., N, (8) j=1 [a(x, s, ξ) a(x, s, η)](ξ η) > 0, for all ξ η R N, (9) a(x, s, ξ) ξ α w i ξ i, (10) where k(x) is a ositive function in L () and α, β are ositive constants. Let g(x, s, ξ) be a Carathéodory function satisfying the following assumtions: Assumtion (H 3 ). g(x, s, ξ)s 0 (11) ( N ) g(x, s, ξ) b( s ) w i ξ i + c(x), (12) where b : R + R + is a continuous increasing function and c(x) a ositive function which is in L 1 () Some technical lemmas Let us give and rove the following lemmas which are needed below. Note that lemmas 3.1, 3.2, 3.4 and 3.6 are roved in [1]. Nevertheless, for the sake of comleteness, we rovide their roofs. Lemma 3.1. Let g L r (, γ) and let g n L r (, γ), with g n r,γ c, 1 < r <. If g n (x) g(x) a.e. in, then g n g in L r (, γ), where denotes weak convergence and γ is a weight function on. Proof. Since g n γ 1 r is bounded in L r () and g n (x)γ 1 r (x) g(x)γ 1 r (x), a.e. in, then by the Lemma 3.2 [13], we have g n γ 1 r gγ 1 r in L r (). 363 Revista Matemática Comlutense

6 Moreover, for all ϕ L r (, γ 1 r ), we have ϕγ 1 r L r (). Then g n ϕ dx gϕ dx, i.e. g n g in L r (, γ). Lemma 3.2. Assume that (H 1 ) holds. Let F : R R be uniformly Lischitz, with F (0) = 0. Let u W 1, 0 (, w). Then F (u) W 1, 0 (, w). Moreover, if the set D of discontinuity oints of F is finite, then { (F u) F (u) u = x i a.e. in { x : u(x) D }, x i 0 a.e. in { x : u(x) D }. Remark 3.3. The revious lemma is a generalization of the corresonding in [11] ( ) (where w 1 and F C 1 (R) and F L (R)) and of the corresonding in [3] (where w 0 w 1 w N is some weight function and F C 1 (R) and F L (R)). Also note that the revious lemma imlies that functions in W 1, 0 (, w) can be truncated. Proof of Lemma 3.2. First, note that the roof of the second art of Lemma 3.2 is identical to the corresonding in the non-weighted case (see [11]). Consider firstly the case F C 1 (R) and F L (R). Let u W 1, 0 (, w). Since C0 () is dense in W 1, 0 (, w), there exists a sequence u n C0 () such that u n u in W 1, 0 (, w). Passing to a subsequence, we can assume that Then, u n u a.e. in and u n u a.e. in. F (u n ) F (u) a.e. in. (13) On the other hand, from the relation F (u n ) w 0 F u n w 0 and F (u n ) x i w i = F (u n ) u n x i w i M u n x i w i, we deduce that the function F (u n ) remains bounded in W 1, 0 (, w). Thus, going to a further subsequence, we obtain Thanks to (13), (14) and (7) we conclude that F (u n ) v in W 1, 0 (, w). (14) v = F (u) W 1, 0 (, w). Revista Matemática Comlutense 364

7 We now turn our attention to the general case. Taking convolutions with mollifiers ρ n in R, we have F n = F ρ n, F n C 1 (R) and F n L (R). Then, by the first case we have F n (u) W 1, 0 (, w). Since F n F uniformly in every comact, we have F n (u) F (u) a.e. in. On the other hand, (F n (u)) is bounded in W 1, 0 (, w), then for a subsequence F n (u) v in W 1, 0 (, w) and a.e. in (due to (7)). Hence, v = F (u) W 1, 0 (, w). The following lemmas follow from the revious lemma. Lemma 3.4. Assume that (H 1 ) holds. Let u W 1, 0 (, w) and let T k (u), k R +, the usual truncation. Then T k (u) W 1, 0 (, w). Moreover, we have T k (u) u strongly in W 1, 0 (, w). Lemma 3.5. Assume that (H 1 ) holds. Let (u n ) be a sequence of W 1, 0 (, w) such that u n u weakly in W 1, 0 (, w). Then, T k (u n ) T k (u) weakly in W 1, 0 (, w). Proof. Since u n u in W 1, 0 (, w) and by (7) we have for a subsequence u n u strongly in L q (, σ) and a.e. in. On the other hand, N T k (u n ) X = T k (u n ) x i w i = T k(u n ) u n x i u n x i w i = u n X. Then (T k (u n )) is bounded in W 1, 0 (, w). Hence, using (7), we have T k (u n ) T k (u) weakly in W 1, 0 (, w). The following lemma generalizes to the weighted case the analogous Lemma 5 in [7]. For that, we use the method of [7] and [13] which gives the strong convergence of u n. Lemma 3.6. Assume that (H 1 ) and (H 2 ) are satisfied, and let (u n ) be a sequence in W 1, 0 (, w) such that u n u weakly in W 1, 0 (, w) and [a(x, u n, u n ) a(x, u n, u)] (u n u) dx 0. (15) Then, u n u in W 1, 0 (, w). Proof. Let D n = [a(x, u n, u n ) a(x, u n, u)] (u n u). Then, by (9), D n is a ositive function and by (15), D n 0 in L 1 (). Extracting a subsequence, still denoted by u n, and using (7) we can write u n u a.e. in, D n 0 a.e. in. w i 365 Revista Matemática Comlutense

8 Then, there exists a subset B of, of zero measure, such that for x \B, u(x) <, u(x) <, k(x) <, w i (x) > 0 and u n (x) u(x), D n (x) 0. We set ξ n = u n (x) and ξ = u(x). Then i.e., D n (x) = [a(x, u n, ξ n ) a(x, u n, ξ)](ξ n ξ) α w i ξn i + α w i ξ i βw 1 i βw 1 i [ k(x) + σ 1 u n q + [ k(x) + σ 1 u n q + j=1 1 w j ξj n ] ξ 1 i j=1 1 w j ξj ] ξ 1 n i j=1 D n (x) α w i ξn i 1 ] c x [1 + w j ξj n 1 + w 1 i ξi n where c x is a constant which deends on x, but does not deend on n. Since u n (x) u(x), we have u n (x) M x, where M x is some ositive constant. Then, by a standard argument ξ n is bounded uniformly with resect to n. Indeed, (16) becomes D n (x) ( ξn i αw i c x N ξn i c xw ξn i 1 i c xw 1 ) i ξn i 1. If ξ n (for a subsequence), there exists at least one i 0 such that ξ i0 n, which imlies that D n (x) which gives a contradiction. Let now ξ be a cluster oint of ξ n. We have ξ < and by the continuity of a with resect to the two last variables we obtain (a(x, u(x), ξ ) a(x, u(x), ξ))(ξ ξ) = 0. In view of (9) we have ξ = ξ. The uniqueness of the cluster oint imlies u n (x) u(x) a.e. in. Since the sequence a(x, u n, u n ) is bounded in N L (, w i ) and a(x, u n, u n ) a(x, u, u) a.e. in, Lemma 3.1 imlies (16) a(x, u n, u n ) a(x, u, u) in N L (, wi ) and a.e. in. Revista Matemática Comlutense 366

9 We set ȳ n = a(x, u n, u n ) u n and ȳ = a(x, u, u) u. As in the roof of Lemma 5 in [7] we can write ȳ n ȳ in L 1 (). By (10), we have α w i u n x i a(x, u n, u n ) u n. Let z n = N w i un x i, z = N w i u x i, y n = ȳn α lemma we obtain i.e. hence, This imlies 2y dx lim inf y + y n z n z dx, n 0 lim su z n z dx, n 0 lim inf z n z dx lim su z n z dx 0. n n u n u in which with (4) comletes the resent roof Case where f W 1, (, w ) In this subsection we assume that N L (, w i ), and y = ȳ α. Then, by Fatou s f W 1, (, w ). (17) Consider the nonlinear roblem with Dirichlet boundary conditions, { Au + g(x, u, u) = f in D () (P) u W 1, 0 (, w), g(x, u, u) L 1 (), ug(x, u, u) L 1 (). Our main result is then the following: Theorem 3.7. Under the assumtions (H 1 ) (H 3 ) and (17), there exists a solution of (P). Remarks 3.8. (i) The revious result is also roved in [1] by using another aroach based on the strong convergence both of ositive and negative arts of the solution u ε of the aroximate roblem (see also [4] in non weighted case). (ii) Theorem 3.7, generalizes to weighted case the analogous statement in [6]. 367 Revista Matemática Comlutense

10 (iii) Note that in [1] the authors have assumed that σ 1 q L 1 () which is stronger than (5). Proof of Theorem 3.7. Ste (1) (The aroximate roblem and riori estimates) Let ε be a sequence of comact subsets of such that ε is increasing to as ε 0. We consider the sequence of aroximate equations, { A(u ε ) + g ε (x, u ε, u ε ) = f (P ε ) u ε W 1, 0 (, w) where g ε (x, s, ξ) = g(x, s, ξ) 1 + ε g(x, s, ξ) χ ε (x). and where χ ε is the characteristic function of ε. Note that g ε (x, s, ξ) satisfies the following condition g ε (x, s, ξ)s 0, g ε (x, s, ξ) g(x, s, ξ) and g ε (x, s, ξ) 1 ε. We define the oerator G ε : X X by G ε u, v = g ε (x, u, u)v dx. Thanks to Hölder s inequality, we have for all u X and v X ( g ε (x, u, u)v dx 1 ε ( ( ) g ε (x, u, u) q σ q q 1 q dx v q q σ dx σ 1 q ε c ε v X. q dx v q,σ (18) For the above inequality we have used (5) and (7). Lemma 3.9. The oerator A + G ε : X X is bounded, coercive, hemicontinuous and satisfies roerty (M). This lemma will be roved below. In view of Lemma 3.9, Problem (P ε ) has a solution by a classical result (cf. Theorem 2.1 and Remark 2.1 in Chater 2 of [12]). Since g ε verifies the sign condition, using (10) we obtain u ε α w i x i f, u ε, Revista Matemática Comlutense 368

11 i.e. α u ε f X u ε, then u ε β 1, (19) where β 1 is some ositive constant. Ste (2) (Strong convergence of T k (u ε )) Note that many ideas in this ste and ste (3) have been adated from the one used in [6]. Thanks to (19) and (7), we can extract a subsequence, still denoted by u ε, such that u ε u weakly in W 1, 0 (, w) and u ε u a.e. in. (20) Let k > 0, by Lemma 3.5 we have Our objective is to rove that T k (u ε ) T k (u) weakly in W 1, 0 (, w) as ε 0. T k (u ε ) T k (u) strongly in W 1, 0 (, w) as ε 0. (21) Fix k, and make the notation z ε = T k (u ε ) T k (u). We use as a test function in (P ε ) v ε = ϕ λ (z ε ) where ϕ λ (s) = se λs2. Since v ε is bounded in X and converges to zero a.e. in and using (7), we have v ε 0 in X as ε 0, then This imlies that f, v ε 0. (22) η 1 (ε) = Au ε, v ε + G ε u ε, v ε = f, v ε 0 as ε 0. (23) Since g ε (x, u ε, u ε )v ε 0 in the subset { x, u ε (x) k }, statement (23) yields Au ε, v ε + g ε (x, u ε, u ε )v ε dx η 1 (ε). (24) { u ε k} We study each term in the left hand side of (24). We have Au ε, v ε = a(x, u ε, u ε ) (T k (u ε ) T k (u))ϕ λ(z ε ) dx = a(x, T k (u ε ), T k (u ε )) (T k (u ε ) T k (u))ϕ λ(z ε ) dx a(x, u ε, u ε ) T k (u)ϕ λ(z ε ) dx { u ε >k} = (a(x, T k (u ε ), T k (u ε )) a(x, T k (u ε ), T k (u))) (T k (u ε ) T k (u))ϕ λ(z ε ) dx + η 2 (ε), (25) 369 Revista Matemática Comlutense

12 where η 2 (ε) = a(x, T k (u ε ), T k (u)) (T k (u ε ) T k (u))ϕ λ(z ε ) dx a(x, u ε, u ε ) T k (u)ϕ λ(z ε ) dx { u ε >k} which converges to 0 as ε 0. On the other hand, g ε (x, u ε, u ε )v ε dx { u ε k} b(k) c(x) ϕ λ (z ε ) dx where { u ε k} { u ε k} [ b(k) c(x) + ] u ε x i w i v ε dx + b(k) a(x, u ε, u ε ) u ε ϕ λ (z ε ) dx α { u ε k} = η 3 (ε) + b(k) a(x, T k (u ε ), T k (u ε )) T k (u ε ) ϕ λ (z ε ) dx α = b(k) (a(x, T k (u ε ), T k (u ε )) a(x, T k (u ε ), T k (u))) (T k (u ε ) α T k (u)) ϕ λ (z ε ) dx + η 4 (ε) η 3 (ε) = b(k) c(x) ϕ λ (z ε ) dx 0 as ε 0 { u ε k} and η 4 (ε) = η 3 (ε) + b(k) a(x, T k (u ε ), T k (u)) (T k (u ε ) T k (u)) ϕ λ (z ε ) dx+ α b(k) a(x, T k (u ε ), T k (u ε )) T k (u) ϕ λ (z ε ) dx 0 as ε 0. α ) 2 Note that, when λ we have ( b(k) 2α ϕ λ(s) b(k) α ϕ(s) 1 2. Combining this with (24), (25) and (26) we obtain ( a(x, Tk (u ε ), T k (u ε )) a(x, T k (u ε ), T k (u)) ) (T k (u ε ) T k (u)) dx (26) η 5 (ε) = 2(η 1 (ε) η 2 (ε) + η 4 (ε)) 0 as ε 0. Revista Matemática Comlutense 370

13 Finally, Lemma 3.6 imlies (21). Ste (3) (Passing to the limit) In virtue of (21) we have for a subsequence u ε u a.e. in, which with (20) yields a(x, u ε, u ε ) a(x, u, u) a.e. in, g ε (x, u ε, u ε ) g(x, u, u) a.e. in g ε (x, u ε, u ε )u ε g(x, u, u)u a.e. in. (27) On the other hand, thanks to (8) and (19), we have that a(x, u ε, u ε ) is bounded in N (, w L i ). Then, by Lemma 3.1, we obtain It remains to rove that a(x, u ε, u ε ) a(x, u, u) weakly in N L (, wi ). (28) g ε (x, u ε, u ε ) g(x, u, u) strongly in L 1 (). (29) By (27), alying Vitali s theorem it suffices to rove that g ε (x, u ε, u ε ) is uniformly equi-integrable. Indeed, multilying (P ε ) by u ε and thanks to (10), (11) and (19), we obtain 0 g ε (x, u ε, u ε )u ε dx β, (30) where β is some ositive constant. For any measurable subset E of and any m > 0, we have g ε (x, u ε, u ε ) dx = g ε (x, u ε, u ε ) dx + g ε (x, u ε, u ε ) dx where E E X ε m E Y ε m X ε m = { x, u ε (x) m }, Y ε m = { x, u ε (x) > m } (31) From these exressions, (12) and (30), we have g ε (x, u ε, u ε ) dx g ε (x, u ε, T m (u ε )) dx E E X ε m + 1 m b(m) E g ε (x, u ε, u ε )u ε dx ( N w i T m (u ε ) x i ) + c(x) dx + β 1 m. (32) 371 Revista Matemática Comlutense

14 Since the sequence ( T m (u ε )) converges strongly in N L (, w i ), the above inequality imlies the equi-integrability of g ε (x, u ε, u ε ). From (28) and (29), we can ass to the limit in Au ε, v + g ε (x, u ε, u ε )v = f, v and we obtain, Au, v + g(x, u, u)v = f, v for any v W 1, 0 (, w) L (). (33) Moreover, since g ε (x, u ε, u ε )u ε 0 a.e. in, by (27), (30) and Fatou s lemma we have g(x, u, u)u L 1 (). (34) This concludes the roof of Theorem 3.7. Remark Note that the statement of (33) holds true for v = u, i.e., Au, u + g(x, u, u)u = f, u. Indeed, utting v = T k (u) in (33) and using Lemma 3.4, we have Au f, T k (u) Au f, u. On the other hand, using Lebesgue s dominated convergence theorem, since g(x, u, u)t k (u) g(x, u, u) u L 1 () (due to (34)) and we conclude that g(x, u, u)t k (u) g(x, u, u)u a.e. in. g(x, u, u)t k (u) g(x, u, u)u in L 1 (). Proof of Lemma 3.9. We set B ε = A + G ε. Using (8) and Hölder s inequality we can show that A is bounded [8]. Thanks to (18) we have B ε bounded. The coercivity follows from (10) and (11). To show that B ε is hemicontinuous, let t t 0, and rove that B ε (u + tv), w B ε (u + t 0 v), w as t t 0 for all u, v, w X. Since for a.e. x, a i (x, u + tv, (u + tv)) a i (x, u + t 0 v, (u + t 0 v)) as t t 0, thanks to the growth condition (8), Lemma 3.1 imlies a i (x, u + tv, (u + tv)) a i (x, u + t 0 v, (u + t 0 v)) in L (, w 1 i ) as t t 0. Revista Matemática Comlutense 372

15 Finally for all w X, A(u + tv), w A(u + t 0 v), w as t t 0. On the other hand, g ε (x, u + tv, (u + tv)) g ε (x, u + t 0 v, (u + t 0 v)) as t t 0 for a.e. in. Also (g ε (x, u + tv + (u + tv))) t is bounded in L q (, σ 1 q ) because ( 1 ) q g ε (x, u + tv, (u + tv)) q σ 1 q c ε. ε Then, Lemma 3.1 gives ε σ 1 q g ε (x, u + tv, (u + tv)) g ε (x, u + t 0 v, (u + t 0 v)) in L q (, σ 1 q ) as t t 0. Since w L q (, σ) for all w X, G ε (u + tv), w G ε (u + t 0 v), w as t t 0. Next we show that B ε satisfies roerty (M); i.e. for a sequence u j in X satisfying (i) u i u in X, (ii) B ε u j χ in X and (iii) lim su j B ε u j, u j u 0, we have χ = B ε u. Indeed, by Hölder s inequality and (7), ( g ε (x, u j, u j )(u j u) 1 ε ( q q dx u j u q,σ 0 as j, g ε (x, u j, u j ) q σ q q dx ( σ q q ε u j u q q σ dx i.e., G ε u j, u j u 0 as j. Combining the last convergence with (iii), we obtain lim su Au j, u j u 0. j And by the seudo-monotonicity of A (see Proosition 1 [8]), we have Au j Au in X and lim j Au j, u j u = 0. On the other hand, 0 = lim a(x, u j, u j ) (u j u) dx j = lim (a(x, u j, u j ) a(x, u j, u)) (u j u) dx j + a(x, u j, u) (u j u) dx. 373 Revista Matemática Comlutense

16 The last integral in the right hand tends to zero since a(x, u j, u) a(x, u, u) in N L (, w 1 i ) as j. Hence, by Lemma 3.6 we have u j u a.e. in. Then g ε (x, u j, u j ) g ε (x, u, u) a.e. in as j. And since gε (x, u j, u j )σ 1 q q 1 ε σ 1 q q χ ε L q () (due to (5)), by Lebesgue s dominated convergence theorem, we obtain g ε (x, u j, u j ) g ε (x, u, u) in L q (, σ 1 q ) as j, which with (7) imlies g ε (x, u j, u j )v dx i.e., G ε u j G ε u in X. Finally, g ε (x, u, u)v dx as j, for all v X, B ε u j = Au j + G ε u j Au + G ε u = B ε u = χ. Remark The assumtion (5) aears necessary in order to rove the boundedness of G ε in W 1, 0 (, w). Thus, when g 0, we don t need to assume (5) The case where f L 1 () In this subsection we assume that f L 1 (). There exists ρ 1 > 0 and ρ 2 > 0 such that N for s ρ 1, g(x, s, ξ) ρ 2 w i ξ i. (35) We relace (H 1 ) by the following assumtion Assumtion (H 1 ). (H 1 ) with σ L 1 (). Consider the nonlinear roblem with Dirichlet boundary conditions { (P Au + g(x, u, u) = f in D () ) u W 1, 0 (, w), g(x, u, u) L 1 (). In this case we have the following existence theorem: Theorem Under the assumtions (H 1), (H 2 ), (H 3 ) and (35), there exists at least one solution of ( P). Revista Matemática Comlutense 374

17 Remark Under the assumtion (17), ug(x, u, u) belongs to L 1 (), that is not the case in general when we assume the hyothesis (35) (cf. Remark 3 [5]). Remark Theorem 3.12 generalizes to the weighted case the analogous statement in [5] and [6]. Proof of Theorem Let f ε be a sequence of smooth functions which converges strongly to f in L 1 () and f ε L1 () c 1 for some constant c 1. Now, consider the following aroximate roblem { ( P A(u ε ) + g ε (x, u ε, u ε ) = f ε ε ) u ε W 1, 0 (, w). with g ε is defined as in the roblem (P ε ). The existence of the solution u ε of this roblem is verified as in the roblem (P ε ). Note that the stes of the roof of Theorem 3.12 are similar to those of Theorem 3.7, assuming that the following assertions are verified: Assertion 1 (Estimate (19)). There exist a constant c such that where u ε is a solution of ( P ε ). u ε X c, Assertion 2 (Convergence (22)). For v ε = ϕ λ (T k (u ε ) T k (u)) where ϕ λ (s) = se λs2, we have f ε v ε 0 as ε 0 Assertion 3 (Equi-integrability of g ε (x, u ε, u ε )). The sequence (g ε (x, u ε, u ε )) ε is uniformly equi-integrable in. By alying the assertions described above we deduce the result as in the case when f X. Proof of the assertion 1. Multilying ( P ε ) by T k (u ε ) X and since g ε (x, u ε, u ε )T k (u ε ) 0, we obtain In view of (10), we have a(x, u ε, T k (u ε )) T k (u ε ) T k (u ε ) w i x i dx k α c 1, f ε T k (u ε ) kc Revista Matemática Comlutense

18 i.e. { u ε >k} T k (u ε ) c 2. (36) On the other hand, we have k g ε (x, u ε, u ε ) f ε T k (u ε ) dx kc 1. (37) Then, by (35), (36), (37) and for k > ρ 1, we obtain N u ε X = 1 ρ 2 { u ε>k } { u ε >k} u ε w i x i dx + g ε (x, u ε, u ε ) + c 3 + c 2 c 4, where c i, i = 1, 2,..., are various ositive constants. Then, u ε c. T k (u ε ) w i x i dx Proof of the assertion 2. Since v ε converges to zero weakly in L (). and f ε converges strongly to f in L 1 (). Then, f ε v ε 0 as ε 0. Proof of the assertion 3. For any measurable subset E of and any m > 0, we have, as in (32), E g ε (x, u ε, u ε ) dx = b(m) E ( N E X ε m g ε (x, u ε, u ε ) dx + g ε (x, u ε, u ε ) dx E Ym ε ω i T m (u ε ) x i ) + c(x) dx + Y ε m g ε (x, u ε, u ε ) dx (38) For fixed m, the first integral of the right hand side of (38) is small uniformly in ε when the measure of E is small (due to T m (u ε ) converges strongly in Π N L (, w i )). We now discuss the behaviour of the second integral of the right hand side of (38). We use in ( P ε ) the test function ψ m (u ε ), where for m > 1 ψ m (s) = 0 if s m 1, ψ m (s) = 1 if s m, ψ m (s) = 1 if s m, ψ m(s) = 1 if m 1 s m. Revista Matemática Comlutense 376

19 This yields a(x, u ε, u ε ) u ε ψ m(u ε ) dx + and thus { u ε >m} g ε (x, u ε, u ε ) dx g ε (x, u ε, u ε )ψ m (u ε ) dx = { u ε >m 1} f ε dx. f ε ψ m (u ε ) dx, From the condition σ L 1 (), it is easy to verify that { x, u ε > m 1 } 0 uniformly in ε when m +, and since f ε f strongly in L 1 (), we have { u ε >m 1} f ε dx is small uniformly in ε when m, which imlies that the second term of the right hand side of (38) is small uniformly in ε and in E, when m is sufficiently large. This comletes the roof of the uniform equi-integrability of g ε (x, u ε, u ε ). Remark The hyothesis σ L 1 () in (H 1) aears in order to rove that {x, u ε > m 1} 0 uniformly in ε Examle Some ideas of this examle come from [8]. Let be a bounded domain of R N (N 1), satisfying the cone condition. Let us consider the Carathéodory functions: a i (x, s, ξ) = w i ξ i 1 sgn(ξ i ) for i = 1,..., N g(x, s, ξ) = ρs s r N w i ξ i, ρ > 0, r > 0 where w i (x) (i = 0, 1,..., N) are given weight functions, strictly ositive almost everywhere in. We shall assume that the weight functions satisfy, w i (x) = w(x), x, for all i = 0,..., N. Then, we can consider the Hardy inequality (6) in the form, ( u(x) q σ(x) dx q c ( u(x) w. It is easy to show that the a i (x, s, ξ) are Carathéodory functions satisfying the growth condition (8) and the coercivity (10). Also the Carathéodory function g(x, s, ξ) satisfies the conditions (11), (12) and (35) with s ρ 1 = 1 and ρ 2 = ρ > 0. On the other hand, the monotonicity condition is satisfied, in fact, ( ai (x, s, ξ) a i (x, s, ˆξ) ) (ξ i ˆξ i ) ( = w(x) ξi 1 sgn ξ i ˆξ i 1 sgn ˆξ ) i (ξi ˆξ i ) > Revista Matemática Comlutense

20 for almost all x and for all ξ, ˆξ R N with ξ ˆξ, since w > 0 a.e. in. In articular, let us use the secial weight functions w and σ exressed in terms of the distance to the boundary. Denote d(x) = dist(x, ) and set w(x) = d λ (x), σ(x) = d µ (x). In this case, the Hardy inequality reads ( u(x) q d µ (x) dx q c ( The corresonding imbedding is comact if: (i) For, 1 < q <, u(x) d λ (x) dx. λ < 1, N q N + 1 0, µ q λ + N q N + 1 > 0. (39) (ii) For, 1 q < <, λ < 1, µ q λ + 1 q > 0. (40) Remark Conditions (39) or (40) are sufficient for the comact imbedding (7) to hold (see for examle [8, Examle 1], [9, Examle 1.5,. 34], and [14, theorems and 19.22]). Finally, the hyotheses of Theorem 3.7 (res. Theorem 3.12) are satisfied, therefore the roblem (P) (res. ( P)) has at least one solution. References [1] Y. Akdim, E. Azroul, and A. Benkirane, Existence of solutions for quasilinear degenerate ellitic equations, Electron. J. Differential Equations (2001), [2], Existence of solution for quasilinear degenerated ellitic unilateral roblems, Ann. Math. Blaise Pascal 10 (2003), [3] B. O. Turesson, Nonlinear otential theory and weighted Sobolev saces, Lecture Notes in Mathematics, vol. 1736, Sringer-Verlag, Berlin, 2000, ISBN [4] A. Bensoussan, L. Boccardo, and F. Murat, On a nonlinear artial differential equation having natural growth terms and unbounded solution, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), (English, with French summary) [5] L. Boccardo and T. Gallouët, Strongly nonlinear ellitic equations having natural growth terms and L 1 data, Nonlinear Anal. 19 (1992), [6] L. Boccardo, T. Gallouët, and F. Murat, A unified resentation of two existence results for roblems with natural growth, Progress in Partial Differential Equations: the Metz Surveys, 2 (1992), Pitman Res. Notes Math. Ser., vol. 296, Longman Sci. Tech., Harlow, 1993, Revista Matemática Comlutense 378

21 [7] L. Boccardo, F. Murat, and J.-P. Puel, Existence of bounded solutions for nonlinear ellitic unilateral roblems, Ann. Mat. Pura Al. (452 (1988), (English, with French and Italian summaries) [8] P. Drábek, A. Kufner, and V. Mustonen, Pseudo-monotonicity and degenerated or singular ellitic oerators, Bull. Austral. Math. Soc. 58 (1998), [9] P. Drábek, A. Kufner, and F. Nicolosi, Non linear ellitic equations, singular and degenerate cases, University of West Bohemia, [10] P. Drábek and F. Nicolosi, Existence of bounded solutions for some degenerated quasilinear ellitic equations, Ann. Mat. Pura Al. (465 (1993), [11] D. Gilbarg and N. S. Trudinger, Ellitic artial differential equations of second order, Sringer- Verlag, Berlin, 1977, ISBN [12] J.-L. Lions, Quelques méthodes de résolution des roblèmes aux limites non linéaires, Dunod, [13] J. Leray and Jacques-Louis Lions, Quelques résulatats de Višik sur les roblèmes ellitiques nonlinéaires ar les méthodes de Minty-Browder, Bull. Soc. Math. France 93 (1965), [14] B. Oic and A. Kufner, Hardy-tye inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Longman Scientific & Technical, Harlow, 1990, ISBN Revista Matemática Comlutense