Anisotropic Elliptic Equations in L m

Transcription

1 Journal of Convex Analysis Volume 8 (2001), No. 2, Anisotroic Ellitic Equations in L m Li Feng-Quan Deartment of Mathematics, Qufu Normal University, Qufu , Shandong, China lifq079@ji-ublic.sd.cninfo.net Received June 20, 2000 In this aer, we rove the existence of solutions to anisotroic nonlinear ellitic equations with right hand side term in L m () and obtain the aroriate function sace for the weak solutions. This aer gives a generalization of some results given in [1] and [3]. Keywords: Anisotroic ellitic equations, L m data 1991 Mathematics Subject Classification: 35D05, 35D10, 35J65 1. Introduction Let be an oen bounded set of R N (N 2), i > 1, (i = 1, 2,, N) and a : R R N R N be a Carathéodory function. We assume that there exist two real ositive constants α, β and a nonnegative function h L 1 () such that for any s R, ξ R N, η R N and for almost every x, every comonent a j (x, s, ξ) of a, a(x, s, ξ)ξ α N ξ i i, (1) a j (x, s, ξ) β(h(x) + s + N ξ i i ) 1 1 j, (2) where satisfies 1 = 1 N N 1 i. [a(x, s, ξ) a(x, s, η)][ξ η] > 0, ξ η. (3) The aim of this aer is to obtain a solution of the anisotroic ellitic equation (P ) { div(a(x, u, Du)) = f in, u = 0 on, in the sense of the distributions. When f L m () with m satisfies 1 < m < m = N N N +. (4) This work is suorted by NSF of Shandong rovince(noy98a09012, NoQ99A05). ISSN / $ 2.50 c Heldermann Verlag

2 418 Li Feng-Quan / Anisotroic ellitic equations in L m We also assume 2 1 N < i < (N 1), and < N, i = 1, 2,, N. (5) N Set and m = Nm N m W 1,(r i) 0 () = {u W 1,1 0 () D i u L r i ()}, (r i 1, i = 1, 2,, N). (7) If a does not deend on x and s, namely a(x, s, ξ) a(ξ), a(ξ) is the vector field whose comonents are ξ i i 2 ξ i (i = 1, 2,, N; i > 1). In [1], it has been roved that there exists a weak solution u N W 1,(r i) 0 () with 1 r i < i( 1)N when f M (N 1) b (), and there exists a weak solution u N W 1,() 0 () with = i( 1)N when f L 1 log L 1 () (N 1) too. If 1 = 2 = = N =, the existence results have been roved in [3] when f M b (),f L 1 log L 1 () and f L m () with 1 < m < N N N+. We consider the existence of weak solutions to roblem (P ) when f L m () (m > 1) here. If = N, then m = 1, and if f is in L m (), then m > m = 1, and roblem (P ) is known to have a weak solution in N 0 () by [4] (since f ( N 0 ()) ). Let us now assume that < N. Then m > 1 and if f is in L m (), m m, Problem (P ) is known to have a weak solution in N 0 () by [4] (since f ( N 0 ()) ). The only case of interest is when f is in L m () with 1 < m < m, and we rove the following theorem. Theorem 1.1. Assume that (1) (3) and (5). Let 1 < m < m = N N N+ (6) and f be in L m (). Then roblem (P ) exists a weak solution u N W 1,() 0 (), with = i( 1)m. Remark 1.2. The Theorem extends the results of Proosition 1 in [2] and Theorem 3 in [3]. Furthermore it can be even as a regularity theorem regarding the solution u obtained in Theorem 1 in [1]. 2. Proof of Theorem 1.1 In order to rove the Theorem 1.1, we need the following nonisotroic Sobolev inequality (cf. [1, 5]). Lemma 2.1. If u N W 1,(r i) 0 (), r i 1(i = 1, 2,, N), then u L s () C 1 ( D i u L r i() ) 1 N, (8) where s = r = Nr if r < N, r satisfies 1 = 1 N 1 N r r N r i, C 1 is a ositive contant deending only on N and r i, (i = 1, 2,, N); if r N, then (8) is satisfied for every s [1, + ) and C 1 deends also on s and meas.

3 Li Feng-Quan / Anisotroic ellitic equations in L m 419 By the density roerty, we may choose a sequence {f k } C 0 (), f k f strongly in L m (), as k, (9) such that f k L m () f L m (), k = 1, 2,. (10) We consider the following aroximation roblem: (P k ) { div(a(x, u k, Du k )) = f k in, u k = 0 on. In the following, we will give a generalization of Estimate 3 in [3]. Lemma 2.2. Assume (1) (3), (9) (10) and (5). Let 1 < m < m, then for any given k 1, there exists a weak solution u k N 0 () to roblem (P k ), moreover, we have D i u k L () C 2, = i( 1)m, i = 1, 2,, N (11) and where q = Nq N q, q = u k L q () C 2, (12) N N, C 1 2 is a ositive constant indeendent of k. Proof. For any given k 1, by [4], it is easy to rove that roblem (P k ) admits a weak solution u k N 0 () such that a(x, u k, Du k )Dvdx = f k vdx, v N W 1,( i) 0 (). (13) To rove Lemma 2.2, we use a choice of a test functions as in [6]. For 0 < s < 1, define φ as φ(y) = y 0 (1 + t ) s dt, y R. (14) It is easy to see that φ(u k ) N 0 (), taking v = φ(u k ) in (13), we obtain Noting (1) and (14), (15) yields N a(x, u k, Du k )φ Du k dx = f k φ(u k )dx. (15) D i u k i (1 + u k ) dx 1 f s k (1 + u k ) 1 s dx. (16) α(1 s)

4 420 Li Feng-Quan / Anisotroic ellitic equations in L m For any < i and 1 i N, Hölder s inequality and (16) imly that If D i u k dx ( D i u k i qi dx) (1 + u k ) s [α(1 s)] i ( (10), Hölder s inequality and (17) yield i ( (1 + u k ) s i dx) 1 i f k (1 + u k ) 1 s dx) i ( q = D i u k dx [α(1 s)] q q i i i f k i L m () ( (1 + u k ) (1 s)m dx) m i ( [α(1 s)] q q i i i f i L m () ( (1 + u k ) (1 s)m dx) m i ( = C 3 ( (1 + u k ) (1 s)m dx) m i ( (1 + u k ) q dx) 1 where C 3 = [α(1 s)] If we get i f i L m (), m = (1 + u k ) s i dx) 1 i. (17) s i, (18) m. m 1 i, (1 + u k ) q dx) 1 i (1 + u k ) q dx) 1 i (19) m (1 s) = q, (20) D i u k dx C 4 + C 5 ( u k q dx) 1 where C 4 and C 5 are two ositive constant indeendent of k. By (18) and (20), we obtain + i m i (21) q = ( 1)m, = i ( 1)m. i = 1, 2,, N. (22) Taking r i =, s = q in Lemma 2.1, we have ( u k q dx) C q 1 ( D j u k L q j ()) q N (23) where C 1 is a ositive constant deending only on N and (i = 1, 2,, N), but indeendent of k. Putting (23) into (21), we get for any i, with 1 i N D i u k dx C 4 + C 5 C q (1 qi + i m ) i 1 ( D j u k L q j ()) q N (1 m i ). (24)

5 Li Feng-Quan / Anisotroic ellitic equations in L m 421 Therefore, there exist two ositive constants C 6 and C 7 indeendent of k, such that Let By (25), we get D i u k L () C 6 + C 7 ( D j u k L q j ()) q N ( 1 1 m ) i, i = 1, 2,, N. (25) d = D j u k L q j (). (26) d C 8 + C 9 d q N N ( 1 1 m ) i = C 8 + C 9 d q ( 1 q 1 m ) (27) where C 8 and C 9 are two ositive constants indeendent of k. By (22) and the conditions satisfied by m and, we have q ( 1 q 1 ) < 1. (28) m By (28) and (27), there exists a ositive constant C 10 indeendent of k, such that d C 10. (29) Thus (11) follows from (29) and (25). Lemma 2.1 (taking r i = ) and (11) yield (12), and by (5), we have > 1 and > 1. This finishes the roof of Lemma 2.2. i 1 Proof of Theorem 1.1. Using Lemma 2.1 and Lemma 2.2, Theorem 1.1 can follow as in [3]. In fact, by (11) and (12), there exists a subsequence of {u k }(still denoted by {u k }) such that D i u k D i u weakly in L (), i = 1, 2,, N, (30) Using the same method as [3], we can rove u k u strongly in L q (), (31) u k u a. e. in. (32) D i u k D i u a. e. in, i = 1, 2,, N. (33) Since a is a Carathéodory function in R R N, by (32) and (33), we get a i (x, u k (x), Du k (x)) a i (x, u(x), Du(x)), a. e. in. (34) By (2), (11) and (12), there exists a ositive constant C 11 indeendent of k, such that By (34) and (35), we obtain a i (, u k, Du k ) i ( 1)m C 11. (35) L ( i 1) () a i (, u k, Du k ) a i (, u, Du) weakly in L i ( 1)m ( i 1) (). (36) By (36) and (9), let k in (13), we get a(x, u, Du)Dvdx = fvdx, v C 0 (). (37) Therefore u is a weak solution to roblem (P ) and u N W 1,() 0 () with = i( 1) m. Thus Theorem 1.1 is roved.

6 422 Li Feng-Quan / Anisotroic ellitic equations in L m The author would like to thank the referee for his comments and sug- Acknowledgements. gestions. References [1] L. Boccardo, T. Gallouët, P. Marcellini: Anisotroic equations in L 1, Differential and Integral Equations 9(1) (1996) [2] L. Boccardo, T. Gallouët: Nonlinear ellitic and arabolic equations involving measure data, J. Funct. Anal. 87 (1989) [3] L. Boccardo, T. Gallouët: Nonlinear ellitic equations with right hand side measures, Comm. Partial Differential Equations 17(3-4) (1992) [4] J. L. Lions: Quelques Méthodes de Résolution des Problémes aux Limites Nonlinéaires, Dunod, Paris, [5] M. Troisi: Theoremi di inclusione er sazi di Sobolev nonisotroi, Ricerche Mat. 18 (1969) [6] L. Boccardo, T. Gallouét, J. L. Vazquez: Nonlinear ellitic equations in R N without growth restrictions on the data, J. Differential Equations 105 (1993)