HIGHER ORDER NONLINEAR DEGENERATE ELLIPTIC PROBLEMS WITH WEAK MONOTONICITY

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1 2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 4, 2005, ISSN: URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu ft ejde.math.txstate.edu (login: ft) HIGHER ORDER NONLINEAR DEGENERATE ELLIPTIC PROBLEMS WITH WEAK MONOTONICITY YOUSSEF AKDIM, ELHOUSSINE AZROUL, MOHAMED RHOUDAF Abstract. We rove the existence of solutions for nonlinear degenerate ellitic boundary-value roblems of higher order. Solutions are obtained using seudo-monotonicity theory in a suitable weighted Sobolev sace.. Introduction and statement of results Let be an oen subset of R N with finite measure and let m be an integer and > be a real number. We will consider the degenerated artial differential oerators on where Au(x) = A m u(x) + A m u(x), (.) A m u(x) = ( ) D A (x, u,..., m u) (.2) =m is the to order art of the degenerated quasilinear oerator A. and where A m u(x) = ( ) D A (x, u,..., m u) (.3) m is the lower order art of A. The coefficients {A (x, η, ζ), m} are real valued functions defined on R Nm R Nm (with N m = card{ N N, m } and N m = card{ N N, = m}) which satisfy suitable regularity and growth assumtions (see section 2). Let V be a subsace such that W m, 0 (, w) V W m, (, w), (.4) where W m, (, w) and W m, 0 (, w) are weighted Sobolev saces associated to a vector of weights w = {w w (x), m} on satisfying some integrability conditions (see sections 2). We deal with the case where A m is affine with resect 2000 Mathematics Subject Classification. 35J40, 35J70. Key words and hrases. Weighted Sobolev saces; seudo-monotonicity; nonlinear degenerate ellitic oerators; boundary value roblems. c 2006 Texas State University - San Marcos. Published Setember 20,

2 54 Y. AKDIM, E. AZROUL, M. RHOUDAF EJDE/CONF/4 to the to order derivatives of u, i.e, A m u is of the form, A m u(x) = ( ) D L (x, u,..., m u) m + m =m ( ) D C (x, u,..., m u)d u (.5) where L (x, η) and C (x, η) are some real valued functions defined on R Nm. We will assume the following hyotheses: (H) For every u V and any multi-index m, there exists a arameter q() and a weight function σ = σ (x) such that, D u L q() (, σ ), D u(x) q(),σ c u m,,w with some constant c > 0 indeendent of u and moreover, the comact imbedding, V H m,q (, σ) (.6) holds, where H m,q (, σ) = {u, D u L q() (, σ ) for all m }. (H2) The functions {A, = m}, {L, m } and {C, m and = m} are Carathéodory functions and there exists functions g L () for all = m, g L q () () for all m, and γ L r () for all m and all = m such that (i) for all = m, A (x, η, ζ) c w / (x) (g (x) + c (ii) for all m, L (x, η) c σ =m w ζ + c ( g (x) + c (iii) for all m and all = m, C (x, η) c σ (x)w / (x) ( γ (x) + c m σ m q () σ r λ λ m σ η q() ) η q() q () ) (x) η λ q(λ) r ) for a.e. x, some ositive constants c, c and c, every (η, ζ) R Nm R Nm = R d and some exonent r such that + r + < for all m. (.7) For the existence of r see Remark 2. below. Let us consider the degenerated boundary value roblem (DBVP) associated to the equation, Au = f V, (.8)

3 EJDE/CONF/4 HIGHER ORDER NONLINEAR DEGENERATE PROBLEMS 55 where V is the dual sace of V from (.4). Recently, Drabeck, Kufner and Mustonen roved in [4] the existence result for Dirichlet degenerated roblem of second order associated to the oerator A of the form, Au(x) = N i= x i a i (x, u, u) (.9) where the Carathéodory functions a i (x, η, ζ) satisfy some simle growth conditions, that is, ( a i (x, η, ζ) c w / i (x) g(x) + w (x) η q + N w j= j ζ ) (.0) where the exonent q and the weight function w(x) verify the so called Hardy-tye inequality; i.e, N u(x) q w(x) dx c D i u w i (x) dx (.) and the comact imbedding i= W, 0 (, w) L q (, w). (.2) The authors have roved that the maing T associated to A from (.9) is seudomonotone in W, 0 (, w), by assuming only the so-called weak Leray-Lions condition N (a i (x, η, ζ) a i (x, η, ζ))(ζ i ζ i ) 0. (.3) i= Our first objective of this aer is to extend the revious result of [4] in the general class of oerators A from (.), where the lower order art A m is of the form (.5) and where the growth conditions are of the most general form (H2). More recisely, we rove the following result. Theorem.. Assume that (H), (H2) and that (A (x, η, ζ) A (x, η, ζ))(ζ ζ ) 0 (.4) =m for a.e. x, all η R Nm and all (ζ, ζ) R Nm R Nm hold. Then the maing T associated to the oerator A from (.) and (.5) is seudo-monotone in V. If in addition the degeneracy satisfies A (x, ξ)ξ c w (x) ξ, (.5) m m for a.e. x, some c > 0 and all ξ R Nm R Nm, then the DBVP associated to the equation (.8) has at least one solution u V. Remark.2. The statement of Theorem., is obviously contained in Theorem 3. below (it suffices to take J = ) where some general situation is considered. On the other hand, Drabeck, Kufner and Nikolosi in [6] have studied the existence result for the DBVP from the equation (.8) with A of the form (.) and with more

4 56 Y. AKDIM, E. AZROUL, M. RHOUDAF EJDE/CONF/4 general hyotheses (H ), (H2 ), (H3) (in section 2) and with the so-called Leray- Lions condition (A (x, η, ζ) A (x, η, ζ))(ζ ζ ) > 0. (.6) =m The authors have assumed in addition to the revious hyotheses the comact imbedding, V W m, (, w) (.7) and then, have roved that the maing T satisfies the condition (V ) (see definition 2.5) and hence used the degree theory of general maings of monotone tye. The hyotheses (.7) lay an imortant role in the work [6], because it is related to some strong converges aearing in the (V ) condition. Our second objective of this aer, is to rove the same result as in [6] without assuming the comact imbedding (.7). This is ossible by roving the seudomonotonicity of the maing T induced by the oerator A from (.). More recisely, we have the following result. Theorem.3. Assume that (H ), (H2 ), (H3) and (.6). Then the maing T associated to oerator A from (.) is seudo-monotone in V. If in addition the degeneracy (.5) is satisfied, then, the DBVP from the equation (.8) has at least one solution u V. Remark.4. Theorem.3 is obviously a consequence of the more general Theorem 3. it suffices to take J c = ). Hence, this aer can be seen as an extension of the receding aers [4, 5, 6] (where the second order case without lower order art is considered in the first aer. The degree theory is used in the two last aers) and as a continuation of the aers [2] and [3] (where the second order case with lower order art not equal to zero, is studied in the first aer and where the higher order case with A m 0 or with A m 0 but under restrictions w for all m, is considered in the last aer). Finally, note that our aroach (based on the theory of seudo-monotone maings) can be alied in the case of non reflexive Banach saces. For examle in the general settings of weighted Orlicz-Sobolev saces (see [] for this direction). This work is divided into five sections. We start with the introduction of a basic assumtions in section 2. Next, we give our main general result in section 3, which is roved in section 4. Finally, we study in section 5, some articular case (where our basic assumtion are satisfied). In our work, we shall adot many ideas introduced in [7], but the results are generalized and imroved. 2. Preinaries and basic assumtions 2.. Weighted Sobolev saces. Let be an oen subset of R N with finite measure. In the sequel we suose that the vector of weights, on, w = {w (x) : m} satisfies the integrability conditions: w L loc(), w L loc() for any m. We denote by W m, (, w) ( < < ) the sace of all real-valued functions u such that the derivatives in the sense of distributions fulfil D u L (, w ) for all m.

5 EJDE/CONF/4 HIGHER ORDER NONLINEAR DEGENERATE PROBLEMS 57 The weighted Sobolev sace W m, (, w) is normed when equied by the norm ( /. u m,,w = D u w dx) (2.) m The sace W m, 0 (, w) is defined as the closure of the set C0 () with resect to the norm (2.). Note that the conditions (2.) and (2.) imly that the saces W m, (, w) and W m, 0 (, w) are reasonably defined and are reflexive Banach saces (for more details see [6]). We recall that the dual sace of W m, 0 (, w) is equivalent to W m, (, w ) where w = {w = w : m}, with = is the Hölder s conjugate of Basic assumtions. Let J be a subset of { N N, = m} and J c its comlement. We will suose that the coefficients A of the oerator A from (.) are such that A (x, η, ζ) = B (x, η, ζ J ) J, A (x, η, ζ) = B (x, η, ζ J c) J c, A (x, η, ζ) = L (x, η, ζ J ) + J c C (x, η, ζ J )ζ m, (2.2) for a.e. x and where {B, = m}, {L, m } and {C, m and J c } are some Carathéodory functions and where ζ I denoted ζ I = {ζ, I}. We denote by N I = card{ N N, I}. Let us introduce the following modified versions of (.6) and (.4), (B (x, η, ζ J ) B (x, η, ζ J ))(ζ ζ ) > 0, (2.3) J for a.e x, all η R Nm and all ζ J ζ J R N J and J c (B (x, η, ζ J c) B (x, η, ζ J c))(ζ ζ ) 0, (2.4) for a.e x and all (η, ζ J c, ζ J c) R Nm R N J c R N J c. Let us denote by m = m N and suose that m > 0 i.e, m > N. We denote by C(, w ) the weighted saces of continuous functions, more recisely C(, w ) = {u = u(x) continuous on, u C(,w) = su x u(x)w (x) < }. (H ) Let u V. (i) For < m, there is a weight function σ = σ (x) such that, D u C(, σ ) and moreover, su D u(x)σ (x) c u m,,w (2.5) x with some constant c > 0 indeendent of u. When we denote by k(x, u(x)) the exression <m σ (x)d u(x), then, in view of (2.5), k(x, u(x)) c u m,,w for all u V. (2.6) (ii) For m m, there is a arameter q() and a weight function σ = σ (x) such that D u L q() (, σ ) and moreover, D u(x) q(),σ c u m,,w (2.7) for some constant c > 0 indeendent of u.

6 58 Y. AKDIM, E. AZROUL, M. RHOUDAF EJDE/CONF/4 (iii) The imbedding V H m,q (, σ) comact, where H m,q (, σ) = {u, D u X, for all m } with X = L q() (, σ ) for m m and X = C(, σ ) for < m. (H2 ) There exists functions g L () for = m, g L q () () for m m, ĝ L () for < m, γ L r () for all m and J c and some ositive constants c and c, moreover there exists a ositive continuous, non decreasing function G(t), t 0, such that the following estimates hold: (i) For J, B (x, η, ζ J ) G(k(x, κ))w / (ii) for J c, B (x, η, ζ J c) G(k(x, κ))w / L (x, η, ζ J ) (g (x) + c w ζ + c J (g (x) + c (iii) for m m, G(k(x, κ))σ (iv) for < m, w J c ζ + c m m m m q ( g (x) + c w () ζ q () + c J L (x, η, ζ J ) ( G(k(x, κ))σ ĝ (x) + c J w ζ + c (v) for m m and J c, C (x, η, ζ J ) G(k(x, κ))σ w / (γ (x) + c C (x, η, ζ J ) (vi) for < m and J c, G(k(x, κ))σ w / (γ (x) + c w r λ λ J w r λ λ J m m m m ζ λ r + c ζ λ r + c σ η q() ) σ η q() σ q () σ η q()) σ r λ m λ m σ r λ m λ m ) ) η q() q () ) η λ q(λ) r ) η λ q(λ) r for a.e. x, every η R Nm and every ζ I R N I where κ = {η, < m } and + r + < for any m m and any J c and with r + <

7 EJDE/CONF/4 HIGHER ORDER NONLINEAR DEGENERATE PROBLEMS 59 for any < m and any J c. Note that the exonent q () denotes the Hölder s conjugate of. Remark 2.. For all m m, the such r satisfying r + + < exists when >. And we can choose r > when < m. Remark 2.2. If m 0, then the set of multi-indices ξ with < m is emty. Then we set G(t) and since the cases iv)and vi) in (H 2) are irrelevant, we obtain the growth condition of tye C [6]. Further if we do not differ between = m and m i.e, if we take g = g L () we immediately obtain the growth conditions of tye (B) [6]. Finally if we choose q() = and σ = w, we obtain the growth condition of tye A [6]. (H3) Let G be a continuous ositive, nonincreasing function on [0, ), and let G 2 be a continuous ositive, nondecreasing function on [0, ), we will suose that for every ξ = (κ, η, ζ) R d and for a.e. x the elliticity condition holds A (x, κ, η, ζ)ζ =m G (h(x, κ)) =m w ζ G 2 (h(x, κ)) m m σ η q(), where κ = {ξ, < m } R d, η = {ξ, m m } R d2, ζ = {ξ, = m} R Nm and d + d 2 = N m. Under these assumtions, the differential oerator (.) generates a maing T from V to its dual V through the formula T u, v = B (x, η(u), ζ J ( m u))d v dx J + B (x, η(u), ζ J c( m u))d v dx J c + m + m J c L (x, η(u), ζ J ( m u))d v dx C (x, η(u), ζ J ( m u))d ud v dx, (2.8) for all u, v V and where.,. denotes the duality airing between V and V. The maing T is well defined and bounded, this can be easily seen by Hölder s inequality and the following lemma. Lemma 2.3. Let be a subset of R N with finite measure and let f L (, σ ), g L q (, σ 2 ) where σ and σ 2 are weight functions in and let h L r (, σ r σ r q 2 ) with + q + r. Then fgh L ().

8 60 Y. AKDIM, E. AZROUL, M. RHOUDAF EJDE/CONF/4 Indeed. Let s = + q + r. By Hölder s inequality we have, ( ) s ( ) s ( fgh s dx f σ dx g q q σ 2 dx h r σ r σ r q 2 dx Then fgh L s (), which imlies that, fgh L (). Let us recall the following definitions. ) s r <. Definition 2.4. A maing T from X to its dual X, is called seudo-monotone, if for every sequence {u n } X with u n u in X and su T u n, u n u 0, one has inf T u n, u n v T u, u v for all v X. Definition 2.5. Let X be a reflexive Banach sace. The maing T from X to X is said to satisfy condition (X) if the assumtions u n u in X and su T u n, u n u 0, imly u n u in X. Obviously, the class (X) of oerators is contained in the class of seudomonotone oerators. 3. Main general result The aim of this section, is to rove the following result. Theorem 3.. Assume that (H ), (H2 ), (H3), (2.3) and (2.4) hold. Then, the maing T defined by (2.8) is seudo-monotone in V. Remark 3.2. () When J =, the revious theorem alies in articular to oerators like (.) with A, m affine with resect to m u. This gives from (.4) a sufficient condition (see Theorem.). (2) When J =, m = and A 0 0, we immediately obtain [4, roosition ]. (3) When A 0 for all m and J = (res. J c = ) we obtain Theorem 8. (res. Theorem 8.3) of [] with some simle the growth conditions. Remark 3.3. Since the hyothesis (H3) concerns only the terms L with < m (see Remark 4.5 below), then the statement of Theorem 3. remains true without assuming (H3), when m 0. Remark 3.4. If we take m 0, q() = and σ = w, then X = L (, w ) for all m, hence the growth condition (H2 ) is of the tye A (see [6]) and the statement of Theorem 3. remains true without assuming (H3). Alying the revious theorem, we obtain the following existence results, which generalize the corresonding (cf. [, 4]) and extend the corresonding in [5, 6]. corollary 3.5. Assume the hytheses in Theorem 3. and the condtion on the degeneracy (.5). Then the DBVP from the equation (.8) has at least one solution u V. Remark 3.6. If the exression, ( u V = =m ) / w (x) D u dx

9 EJDE/CONF/4 HIGHER ORDER NONLINEAR DEGENERATE PROBLEMS 6 is a norm in V equivalent to the usual norm (2.) (see section 5 where this fact is verified for V = W m, 0 (, w)), then we can relace in Corollary 3.5, the degeneracy (.5) by the weaker condition A (x, ξ)ξ c w ξ. (3.) m =m 4. Proof of Theorem 3. For this goal, we need the following lemmas. Lemma 4.. Let (g n ) n be a sequence of L (, σ) and let g L (, σ) ( < < ), where σ is a weight function in. If g n g in measure (in articular a.e in ) and it is bounded in L (, σ), then g n g in L q (, σ q ) for all q <. Proof. Let ε > 0 and set A n = {x / g n (x) g(x) σ / (x) ( 2 meas() )/q }. We have g n g q σ q dx = g n g q σ q dx + g n g q σ q dx A n A c n ε 2 + g n g q σ q dx. By Hölder inequality, one can see that A c n A c n ( g n g q σ q dx g n g σ dx ( M meas(a c n) ) q ) q ( meas(a c n), ε ) q where M is a constant does not deend on n. On the other hand, since g n g in measure, meas(a c n) 0 as n. Then there exists some n 0 N such that for all n n 0, g n g q σ q ε dx A 2. c n The following lemma is a generalization of [9, Lemma 3.2] in weighted saces. Lemma 4.2. Let g L q (, σ) and let g n L q (, σ), with g n q, σ c ( < q < ). If g n (x) g(x) a.e. in, then g n g in L q (, σ), where denotes weak convergence. Proof. Since g n σ q is bounded in L q () and g n (x) σ q (x) g(x) σ q (x), a.e. in, then by [9, lemma 3.2], g n σ q g σ q in L q (). Moreover, for all ϕ L q (, σ q ), we have ϕ σ q L q (). Then g n ϕ dx gϕ dx; i.e. g n g in L q (, σ).

10 62 Y. AKDIM, E. AZROUL, M. RHOUDAF EJDE/CONF/4 Proof of Theorem 3.. Let (u n ) n be a sequence in V such that: u n u in V and i.e., su T u n, u n u 0, (4.) { su B (x, η(u n ), ζ J ( m u n ))(D u n D u) dx J + B (x, η(u n ), ζ J c( m u n ))(D u n D u) dx J c + L (x, η(u n ), ζ J ( m u n ))(D u n D u) dx + m m (a) We shall rove that } C (x, η(u n ), ζ J ( m u n ))D u n (D u n D u) dx 0. J c T u n, v T u, v asn v V. (4.2) By (H )(iii), the comact imbedding imlies that for a subsequence D u n D u in X D u n D u a.e. in m. (4.3) Ste () We shall rove that L (x, η(u n ), ζ J ( m u n ))(D u n D u) = 0. (4.4) (i) We show that m m m L (x, η(u n ), ζ J ( m u n ))(D u n D u) dx = 0. (4.5) Let m m be fixed. Thanks to (H2 ), we have L (x, η(u n ), ζ J ( m u n ))(D u n D u) dx G(k(x, u n (x)))σ (D u n D u) g dx + c G(k(x, u n (x)))σ (D u n D q u) w () D u n (x) J + c m m G(k(x, u n (x)))σ (D u n D u) σ q () q () dx D u n (x) q() q () dx. By (2.6) we have, G(k(x, u n (x)) G(c u n m,,w ).

11 EJDE/CONF/4 HIGHER ORDER NONLINEAR DEGENERATE PROBLEMS 63 Alying the Hölder s inequality with exonents and q () we obtain L (x, η(u n ), ζ J ( m u n ))(D u n D u) dx G(c u n m,,w ) D u n D u,σ ( g q () + c D q u n (),w + c J m m D u n q() q () q(),σ ). Thanks to (2.7), we have D u n q(),σ c u n m,,w for all m m. Since D u n,w u n m,,w for all J, we conclude that L (x, η(u n ), ζ J ( m u n ))(D u n D u) dx with, D u n D u,σ R ( u n m,,w ) ( R (t) = G(c t) g q () + c 2 t q () + c3 m m ) t q() q () which is a ositive continuous function, hence R ( u n m,,w ) is bounded. Moreover, by (4.3) we have, then which yields (4.5). (ii) We show that <m D u n D u,σ 0 as n. L (x, η(u n ), ζ J ( m u n ))(D u n D u) dx 0, L (x, η(u n ), ζ J ( m u n ))(D u n D u) dx = 0. (4.6) Let < m be fixed. Similarly by virtue of (H2 ), L (x, η(u n ), ζ J ( m u n ))(D u n D u) dx G(c u n m,,w ) su( (D u n D u)σ ) ( ĝ + c D u n,w x J + c D u n q() q(),σ ). m m It follows from (2.7) and D u n,w u n m,,w for all J that L (x, η(u n ), ζ J ( m u n ))(D u n D u) dx where D u n D u C(,σ) R ( u n m,,w ), R (t) = G(c t) ( ĝ + c 2 t + c 3 m m t q()).

12 64 Y. AKDIM, E. AZROUL, M. RHOUDAF EJDE/CONF/4 This function is also ositive and continuous, hence R ( u n m,,w ) is bounded. Since, by (4.3) D u n D u C(,σ) 0 as n, it follows that L (x, η(u n ), ζ J ( m u n ))(D u n D u) dx 0, which yields (4.6). Thus, due to (4.5) and (4.6) we conclude (4.4). Ste (2) We shall rove that C (x, η(u n ), ζ J ( m u n ))D u n (D u n D u) dx = 0. (4.7) m J c (i) Let m m and J c be fixed. And let s such that, s = + + r <. By Hölder s inequality, we have C (x, η(u n ), ζ J ( m u n ))D u n (D u n D u) s dx ( C (x, η(u n ), ζ J ( m u n )) r σ ) r w r s r dx ( ) s D u n ( ) s w dx D u n D u σ dx. By (H2 ) the sequences {C (x, η(u n ), ζ J ( m u n )), m m, J c } (res {D u n, J c }) remain bounded in L r (, σ r w r ) (res L (, w )). Moreover, D u n D u s,σ 0 as n. Then C (x, η(u n ), ζ J ( m u n ))D u n (D u n D u) s dx = 0. Consequently, i.e, m m J c C (x, η(u n ), ζ J ( m u n ))D u n (D u n D u) dx = 0, C (x, η(u n ), ζ J ( m u n ))D u n (D u n D u) = 0. (4.8) (ii) Let < m and J c be fixed. By (H2 ) the sequences {σ C (x, η(u n ), ζ J ( m u n ))D u n, < m, J c } remain bounded in L s () with s = + r <. Indeed, σ C (x, η(u n ), ζ J ( m u n ))D u n s dx ( ) C (x, η(u n ), ζ J ( m u n )) r σ r w r s r ( s dx D u n w dx). The right hand side is bounded because {C (x, η(u n ), ζ J ( m u n ))} is bounded in L r (, σ r w r ) and {D u n } is bounded in L (, w ).

13 EJDE/CONF/4 HIGHER ORDER NONLINEAR DEGENERATE PROBLEMS 65 Thanks to s, the sequences {σ C (x, η(u n ), ζ J ( m u n ))D u n, < m, J c } remain bounded in L (). Since C (x, η(u n ), ζ J ( m u n ))D u n (D u n D u) dx it follows that su( (D u n D u)σ ) x C (x, η(u n ), ζ J ( m u n ))D u n σ dx C (x, η(u n ), ζ J ( m u n ))D u n (D u n D u) dx = 0 (because su x ( (D u n D u)σ ) 0). Which gives C (x, η(u n ), ζ J ( m u n ))D u n (D u n D u) dx = 0. (4.9) <m J c Combining (4.8) and (4.9) we obtain (4.7). Ste (3) We shall rove that B (x, η(u n ), ζ J ( m u n )) and that J B (x, η(u n ), ζ J ( m u)))(d u n D u) dx = 0 (B (x, η(u n ), ζ J c( m u n )) J c B (x, η(u n ), ζ J c( m u)))(d u n D u) dx = 0. Combining (4.), (4.4) and (4.7) one obtain su B (x, η(u n ), ζ J ( m u n ))(D u n D u) dx J + B (x, η(u n ), ζ J c( m u n ))(D u n D u) dx 0. J c (4.0) (4.) (4.2) Thanks to (4.3) and (H2 ) one deduce that B (x, η(u n ), ζ J ( m u)) B (x, η(u), ζ J ( m u)) in L (, w ), J B (x, η(u n ), ζ J c( m u)) B (x, η(u), ζ J c( m u)) in L (, w ), J c. Since D u n D u in L (, w ) for all = m, one can write B (x, η(u n ), ζ J ( m u))(d u n D u) dx = 0, J J c B (x, η(u n ), ζ J c( m u))(d u n D u) dx = 0. (4.3) Combining (4.2), (4.3), (2.3) and (2.4) we conclude the assertions (4.0) and (4.).

14 66 Y. AKDIM, E. AZROUL, M. RHOUDAF EJDE/CONF/4 Ste (4) To rove the relation (4.2), it suffices to show the following assertions: (i) For every v V, B (x, η(u n ), ζ J ( m u n ))D v dx = (ii) For every v V, = (iii) For every v V, J B (x, η(u), ζ J ( m u))d v dx. J m m m m = (iv) For every v V, = L (x, η(u n ), ζ J ( m u n ))D v dx L (x, η(u), ζ J ( m u))d v dx. L (x, η(u n ), ζ J ( m u n ))D v dx <m L (x, η(u), ζ J ( m u))d v dx. <m m m (v) For every v V, = C (x, η(u n ), ζ J ( m u n ))D u n D v J c C (x, η(u), ζ J ( m u))d ud v. J c (B (x, η(u n ), ζ J c( m u n ))D v dx J c (B (x, η(u), ζ J c( m u))d v dx. J c (4.4) (4.5) (4.6) (4.7) (4.8) Proof of assertions (i)and (ii). Invoking Landes [8, lemma 6], we obtain from (4.0) and the strict monotonicity (2.3) that which gives D u n D u a.e in for each J, (4.9) B (x, η(u n ), ζ J ( m u n )) B (x, η(u), ζ J ( m u)) a.e. in J, L (x, η(u n ), ζ J ( m u n )) L (x, η(u), ζ J ( m u)) a.e in m m. From the growth condition (H2 ), the sequence {B (x, η(u n ), ζ J ( m u n )), J} (res {L (x, η(u n ), ζ J ( m u n )) m m }) are bounded in L (, w ) (res. L q () (, σ )), hence by Lemma 4.2 we have B (x, η(u n ), ζ J ( m u n )) B (x, η(u), ζ J ( m u))

15 EJDE/CONF/4 HIGHER ORDER NONLINEAR DEGENERATE PROBLEMS 67 in L (, w ) for all J and L (x, η(u n ), ζ J ( m u n )) L (x, η(u), ζ J ( m u)) in L q () (, σ ) for all m m, which imlies (i) and (ii). Proof of assertion (iii). In virtue of the growth condition (H2 ) we have for all v V and all < m ( L (x, η(u n ), ζ J ( m u n ))D v D v G(k(x, η(u n )))σ ĝ (x) + c w D u n + c m m σ D u n q()). Since G(k(x, η(u n ))) c and su x ( D vσ ) c 2 for all < m, where c i (i =, 2) are some ositive constants, it follows that L (x, η(u n ), ζ J ( m u n ))D v c (ĝ (x) + c w D u n + c J It follows from (4.3) and (4.9) that m m J σ D u n q()) = g n. L (x, η(u n ), ζ J ( m u n )) L (x, η(u), ζ J ( m u)) a.e. in < m and g n g = c (ĝ (x) + c w D u + c J m m Lemma 4.3. D u n D u as n in L (, w ) for all J. By (4.3) and Lemma 4.3 we obtain g n dx g dx. σ D u q()) a.ea.e. in. By the generalized Lebesgue theorem we have, L (x, η(u n ), ζ J ( m u n ))D v dx L (x, η(u), ζ J ( m u))d v dx for all < m which imlies (4.6). Proof of assertion (iv). By (4.3) and (4.9) we have for each m and each J c, C (x, η(u n ), ζ J ( m u n )) C (x, η(u), ζ J ( m u)) a.e. in. So, from (H2 ) the sequences {C (x, η(u n ), ζ J ( m u n )), m m and J c } (res. {C (x, η(u n ), ζ J ( m u n )), < m and J c }) remain bounded in L r (, σ r q in L q (, σ 4. also yields w r ) (res. L r (, σ r w r )). Then Lemma 4. yields C (x, η(u n ), ζ J ( m u n )) C (x, η(u), ζ J ( m u)) w q ) for all q < r, all m m and all J c. Lemma C (x, η(u n ), ζ J ( m u n )) C (x, η(u), ζ J ( m u))

16 68 Y. AKDIM, E. AZROUL, M. RHOUDAF EJDE/CONF/4 in L q (, σ q w q ) for all q < r, all < m and all J c.. Remark that r > s = Let s such that s = + m and since < r for < m one has s in L s (, σ C (x, η(u n ), ζ J ( m u n )) C (x, η(u), ζ J ( m u)) w s ) for all m m. Also one has C (x, η(u n ), ζ J ( m u n ))σ in L (, w ) for all < m. Lemma 4.4. For all v V, one has s s for m C (x, η(u), ζ J ( m u))σ (4.20) s () D u n D v D ud s v in L s (, σ w ) for each m m and each J c. (2) D u n D vσ D ud vσ in L (, w ) for each < m and each J c. In view of (4.20) and Lemma 4.4 we conclude (4.7). Proof of assertion (v). First we show that J c (B (x, η(u), ζ J c(v)) h )(v D u) dx 0 (4.2) for all v = (v ) =m L (, w ), where h stands for the weak it of {B (x, η(u n ), ζ J c( m u n )), J c } in L (, w). Indeed by (4.) we have, su B (x, η(u n ), ζ J c( m u n ))(D u n D u) dx 0, J c imlies su B (x, η(u n ), ζ J c( m u n ))D u n dx J c J c h D u dx (4.22) and from weak Leray-Lions condition (2.4), for any v = (v ) =m L (, w ), we obtain B (x, η(u n ), ζ J c( m u n ))D u n dx J c B (x, η(u n ), ζ J c( m u n ))v dx J c + B (x, η(u n ), ζ J c(v))(d u n v ) dx. J c Letting n we conclude by (4.22) that h D u dx h v dx + B (x, η(u), ζ J c(v))(d u v ) dx J c J c J c and hence (4.2) follows. Choosing v = D u + tŵ with t > 0, ŵ = (ŵ ) =m L (, w ) and letting t 0 we obtain h = B (x, η(u), ζ J c( m u)) a.e. in which imlies (4.8).

17 EJDE/CONF/4 HIGHER ORDER NONLINEAR DEGENERATE PROBLEMS 69 (b) We shall rove that inf T u n, u n T u, u. (4.23) In view of monotonicity condition (2.3) and (2.4) we have B (x, η(u n ), ζ J ( m u n ))D u n + B (x, η(u n ), ζ J c( m u n ))D u n J J c B (x, η(u n ), ζ J ( m u n ))D u J B (x, η(u n ), ζ J ( m u))(d u n D u) J B (x, η(u n ), ζ J c( m u n ))D u J c B (x, η(u n ), ζ J c( m u))(d u n D u). J c Letting n and using (4.4) and (4.7)we obtain (4.23). Proof of Lemma 4.3. Let E be a measurable subset of, in view of stes, 2, 3 and 4 in [6, Lemma 2.7], we obtain w D u n (x) dx = 0 (4.24) meas E 0 E J uniformly with resect to n N, i.e the sequence (w D u n ) is equi-integrable. And due to (4.9) we have w D u n w D u a.e. J. Since meas() <, by Vitali s theorem we obtain D u n D u in L (, w ) for all J. Proof of Lemma 4.4. () Let m m and J c fixed. Let ϕ s L s (, σ w s ). Since, = s +, by Hölder s inequality we obtain ( ) s D vϕ w D v ( ) s σ ϕ s w ) s σ < (because s ( ) L (, w ), we have i.e., = s D u n D vϕ ). Then D vϕ L (, w ) and since D u n D u in D u n D vϕ D u n D v D ud v L s (, σ for all m m and all J c. s for all ϕ L s (, σ s s w ) w s );

18 70 Y. AKDIM, E. AZROUL, M. RHOUDAF EJDE/CONF/4 (2) Let < m and J c and let ϕ L (, w ). Thanks to D v C(, σ ) v V, we have D vσ ϕ L (, w ). Since D u n D u in L (, w ), we have D u n D vσ ϕ dx D ud vσ ϕ dx for all ϕ L (, w). Remark 4.5. Note that, the elleticity condition (H3) is only used to rove (4.24) (see ste 3 in [6, lemma 2.7], which concerns only the equality (4.6) corresonding to a terms L with < m ). 5. Secific case Let be a bounded oen subset of R N satisfying the cone condition. In the sequel we assume in addition that the collection of weight functions w = {w (x), m} satisfies w (x) = for all m, and the integrability condition: There exists ν ] N P, [ [ P, [ such that w ν L () = m. (5.) Note that (5.) is stronger than (2.). Assumtions (2.) and (5.) imly ( ) / u V = D u w (x) dx =m is a norm defined on V = W m, 0 (, w) and it s equivalent to (2.). Let m = mν N(ν + ) ν = m N with = ν ν +. (5.2) Remark 5. ([5]). Under the above assumtion the following continuous imbeddings hold: (i) For k < m, W m, (, w) C k ( ). (ii) For k = m, with arbitrary r, < r <, (iii) For k > m, W m, (, w) W k,r (). W m, (, w) W k,r k () νn N(ν+) ν(m k). where r k satisfies < r k q k = Moreover the imbedding (i) and (ii) are comact and (iii) is comact if r k < q k. Now, we define H m,q (, σ) = m where X = L q() (, σ ), q() > for m m and X = C (, σ ) for < m. Also we define the assumtion X,

19 EJDE/CONF/4 HIGHER ORDER NONLINEAR DEGENERATE PROBLEMS 7 (H4) Let νn < q() < q = N(ν + ) ν(m ) for m < m and q() arbitrary if = m and σ for all m. Remark 5.2. If (H4) is satisfied, then by Remark 5., W m, (, w) H m,q () which imlies immediately that (H2 )(iii) with σ. Theorem 5.3. Let be a bounded oen subset of R d. And assume that (2.), (5.), (H ), (H2 )(,i,ii), (H3), (H4), (2.3) and (2.4) are satisfied. Then the oerator T defined in (2.8) is seudo-monotone in V = W m, 0 (, w). If in addition the degeneracy (3.) is satisfied, then the degenerate boundaryvalue roblem from (.8) has at least one solution u V. References [] E. Azroul, Sur certains roblèmes ellitiques (non linéaires dégénérés ou singuliers) et calcul des variations, Thèse de Doctorat d Etat, Faculté des Sciences Dhar-Mahraz, Fès, Maroc, Janvier [2] Y. Akdim, E. Azroul, A. Benkirane; Pseudo-monotonicity and degenerated ellitic oerators of second order, (submitted). [3] Y. Akdim, E. Azroul, A. Benkirane; Pseudo-Monotone and Nonlinear Degenerated Ellitic Boundary-value roblems of higher order I, (submitted). [4] P. Drabek, A. Kufner, V. Mustonen; Pseudo-monotonicity and degenerated or singular ellitic oerators, Bull. Austral. Math. Soc. Vol. 58 (998), [5] P. Drabek, A. Kufner, F. Nicolosi; On the solvability of degenerated quasilinear ellitic equations of higher order, Journal of differential equations, 09 (994), [6] P. Drabek, A. Kufner, F. Nicolosi; Non linear ellitic equations, singular and degenerate cases, University of West Bohemia, (996). [7] J. P. Gossez, V. Mustonen; Pseudo-monotoncity and the Leray-Lions condition, Differential Integral Equations 6 (993), [8] R. Landes, On Galerkin s method in the existence theory of quasilinear ellitic equations, J. Functional Analysis 39 (980), [9] J. Leray, J. L. Lions; Quelques résultats de Vi sik sur des roblèmes ellitiques non linéaires ar les méthodes de Minty-Browder, Bul. Soc. Math. France 93 (965), Youssef Akdim Déartement de Mathématiques et Informatique, Faculté des Sciences Dhar-Mahraz, B. P. 796 Atlas Fès, Maroc address: akdimyoussef@yahoo.fr E. Azroul Déartement de Mathématiques et Informatique, Faculté des Sciences Dhar-Mahraz, B. P. 796 Atlas Fès, Maroc address: azroul elhoussine@yahoo.fr Mohamed Rhoudaf Déartement de Mathématiques et Informatique, Faculté des Sciences Dhar-Mahraz, B. P. 796 Atlas Fès, Maroc address: rhoudaf mohamed@yahoo.fr