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1、?47?1K?Vol.47, No.1 2004H1?ACTA MATHEMATICA SINICAJan., 2004 ?0583-1431(2004)01-0189-08?A ? ?W? (?Ra?U?e?ey?U430071) (Fax: (027)87199291;E-mail: ) ?iu?zHDirichlet? ?u(x) = f(x,u), x , H1 0(), (P) n?f(x,t) C( R), f(x,t) t Q?tFI?Y?pGt RxQ?x ?s ?mOLT?q(x) (Dx?Af(x,t)Q?t?r?C?H).?t? M?Z?HAmbrosetti-Rabin
2、owitz?Z?XQ?H |s| MVx , 0 2, M 0?F(x,s) = ?s 0 f(x,t)dt. ?Z(AR)?vh?dH?o?L?H?S? ?N?gHvh?d?j?Z(AR)Hqb?c?lwkDirichlet?(P)? ?E?DKJ?f(x,t)Q?t?r?C?Xq(x) +Hq?. ?Dirichlet?vh?d?P? MR(2000)?35J05, 35J65 ?O175.25, O175.9 An Application of a Mountain Pass Theorem Huan Song ZHOU (Laboratory of Mathematical Physi
3、cs, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, P. R. China) (Fax: (027)87199291;E-mail: ) Abstract We are concerned with the following Dirichlet problem ?u(x) = f(x,u), x , u H1 0(), (P) where f(x,t) C( R), f(x,t)/t is nondecreasing in t R and tends to an
4、L- function q(x) uniformly in x as t + (i.e., f(x,t) is asymptotically linear in t at infi nity). In this case, Ambrosetti-Rabinowitz-type condition, that is, for some 2, M 0, 0 F(x,s) f(x,s)s, for all |s| M and x ,(AR) *?CH?U?L?(2002)18?1L2736? yQbL?1998-06-24;?zbL?2000-01-14 t?C?t?TIt?x?C 190?47?
5、is no longer true, where F(x,s) = ?s 0 f(x,t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass theorem. In this paper, without assum- ing (AR) we prove, by using a variant version of Mountain Pass theorem, that problem (P) has a positive solution under suitable
6、 conditions on f(x,t) and q(x). Our methods also work for the case where f(x,t) is superlinear in t at infi nity, i.e., q(x) +. Keywords Dirichlet problem; Mountain Pass theorem; Asymptotically linear; Resonant problem MR(2000) Subject Classifi cation 35J05, 35J65 Chinese Library Classifi cation O17
7、5.25, O175.9 1? ?D?Dirichlet? ?u(x) = f(x,u), x , u H1 0(), (1.1) Mq?Rn(N 1)q?mR?YW?f(x,t)? (H1) f(x,t) C( R)S?R?x ,?f(x,0) 0 xt 0p?f(x,t) (?)0; t 0p?f(x,t) 0; (H2)?Tzk?x , f(x,t)/t?Tt 0? (H3)?Tx Ap?limt0 f(x,t) t = p(x), limt f(x,t) t = q(x) ? 0,Mq 0 p(x), q(x) L()S|p(x)| 0?(?,H1 0() ?A?gm?p? ?f(x,
8、t)?VY?Ttv? t?uj?b?(1.1)?g?D?W?G?H1 0() j?Gd J(u) = 1 2 ? |u|2dx ? F(x,u)dx,MqF(x,s) = ? s 0 f(x,t)dt(1.2) ?Q(H1)fRv?mX?j?GdJ?J?v?(1.1)?h? ?bGd(1.2)?O?HB?vMOQAmbrosettifRabinowitz?1 (? ?2,3)q?h?J?v?MONJ?Gd(1.2)? ?lQ?1T?AR?y?AR? 2fM 0,r?t ? 0 0,?S?QAR H1 0() ?qzk?(x) 0,? ?q?J?2.1. ?Z?1(i, i = 2,3,.) 0
9、?(?,H1 0) ?A(iR)?gm?S?Lp()?F ?|p,?H1 0() QFu?u,v? = ? uvdx (u,v H 1 0() ?F? ? ?J?R?L?OC?A? ?w?ad? ?1.1 da?(H1)(H3)?R (i)? 1p?(1.1)?h? (ii)? 0,r? u = c(x)S?qzk?Rf(x,u) = q(x)u?e?(x) 0vv?n?(1.3)q ?d? ?1 fq(x) ? 0,? = 1/?,Sj? ?1.1? da?(H1), (H2)x(H3)?Sq(x) ? 0,?R (i)?(1.1)? 1p?h? (ii)?(1.1)?1 ? 0,r?qzk
10、?u(x) = c1(x)Sf(x,u) = 1u, Mq1 0?MT1?gd? ?1.2 ?1.1?l?daq(x) +Sf(x,u)?a?y? AR?r:?N 2p?r (2, 2N N2); ?N = 1,2p?r (2,+),r?v? lim u+ f(x,u) ur1 = 0 ?Tx Ap?(1.1)okRARh? ?2?1.1?(ii)?(1.1)?q(x) i, i 2p?Rh?daq(x) L() S? +,?T?1.1f?1.1?q?(ii)?(H2)?v?(?1.1? i?),?vdaq(x) +,?(H2)?v?(?1.3?i?).?(H2)? T?1.1f?1.1?q?
11、(i)f(iii)?srv?L?(1.1)? 0? liminf t f(x,t)t 2F(x,t) t k 0.(CM) ?(CM)?c?(fF). ?j?T?Td?f(x,t)?R?AR?(H1)(H3),?(CM); ?AR?(H1)(H3)Dx(fF). ?A?g(t) C1(R), g(0) = 0S?t 0p?g(t) 0, g?(t) 0;?t 0p?g(t) 0. ?T? 0,?t0 0,r?t t0p?iRg(t) ?.?F?(x,t) R,? Gd?f(x,t) g(t)t.cE?if(x,t)?(H1)(H3) (epp(x) 0, q(x) ?),?v?t 0 ?K?p
12、?Rf(x,t)t 2F(x,t) ?t2 0, e?F?(fF)?(CM)? ?B?q(x) C() L()S? qq(x) 0.l 0,?G f(x,t) = q(x)t+1 1 + t ,t 0, 0,t 0. ?E?if(x,t)?(H1)(H3),epp(x) 0.? (0,1)p?f(x,t)? ?(fF),?7. 2? ?T(1.3)q?G?,?R?J? ?2.1 fq(x) L()Sq(x) (?)0,?(1.3)q?G? 0,?S? H1 0(), r? ? | 2dx = , S ? q(x)(x) 2dx = 1. ?oR(x) 0?qz k? ?OSobolev?Qe?
13、FatouJ?xRv?mX?y?i?N? ?2.2 ?l(H1), (H3)?fq(x) +p(H2)?R (a)?, 0,r?R?u H1 0() S?u? = ,?J(u) . (b)f 0QJ?2.1T? fq(x) +,?J(t1) t+ ,e?1 0?1?M?gd? ?(a)?i?T?4qJ?3?i?e?n?(H1), (H3). (b)f 1, ?J?2.1q?d?QFatouJ?f(H3),? lim t+ J(t(x) t2 1 2 ? |(x)|2dx ? lim t+ F(x,t(x) t2(x)2 (x)2dx = 1 2 ? |(x)|2dx 1 2 ? q(x)|(x
14、)|2dx = 1 2( 1) ? |(x)|2dx 0,da 1. 1Kvq?BSi?K?NP193 I?J(t) t+ ,?(b)? 0,?AR? 0,r?min 01(x) 0 ?I?t1(x) t+ + ? 0jzk?SQ?(H2)?j?R?x ft 0,?0 2F(x,t) f(x,t)t,?F(x,t)/t2?Tt 0?yF?(H3)q?q(x) +,?R? x 0ft 0,?Tx 0,Ap? F(x,t1) t22 1 F(x,t) t22 t+ +.I?FT? K 0,?T = T(,K) 0,r?t? F(x,t1) t22 1 K 0, t T, x 0. e?n?K 0?
15、K?t Tp?R J(t1) t2 1 2 ? |1|2dx ? 0 F(x,t1) t22 1 2 1dx 1 2 ? |1|2dx K ? 0 2 1dx 1 2 ? |1|2dx 2K m0 0,?J(tun) 1+t2 2n + J(un),?R?n 1. ?8. ?i?1.1,?O?9q?rO?h?J?s?d?Mi? 11f12qb? ?2.4 (?9?I)lE?q?Banach?GdI C1(E,R)SmaxI(0), I(u1) inf?u?=I(u),Mq 0?u1 ES?u1? .?c( ) ?Gd?c = infmax01I().e? = C(0,1,E) : (0) =
16、0, (1) = u1 ?R?0fu1?wh?A?AR?un E,r?I(un) n c f (1 + ?un?)?I?(un)?E n 0. 3? ?1.1?(i)?lu H1 0() ?(1.1)?ARh?Q?(H1)(H3)? ? |u|2dx = ? f(x,u)udx ? q(x)u2dx, I? 1,?1.1 (i)?i? (ii)I? 0,r?J(t0) 0QJ?2.1T?G = C(0,1,H1 0() : (0) = 0,(1) = t0, c = inf max 0,1 J(),(3.1) ?c 0,?Q?2.4?j?un H1 0(), r? J(un) = 1 2?un
17、? 2 ? F(x,un(x)dx = c + o(1),(3.2) (1 + ?un?)?J?(un)?H1 0 () n 0.(3.3) 194?47? ?(3.3)t?c? ?J?(un),un? = ?un?2 ? f(x,un(x)undx = o(1).(3.4) e?Dx?q?O?eo(1)?n +pX?T? I?R?YW?Sf(x,t)v?a?OSobolev?Qe?x?z?HB? j?daun?H1 0() qR?Run?AR?Rx?GdJ?AR? ?i?I?i?1.1 (ii),?n?i?un?H1 0() qR? ?OEiB?f?un?H1 0() q?y?un? n +
18、,S? tn= 2c ?un?, wn = tnun= 2cun ?un? .(3.5) ?wn?H1 0() qR?b?Z?D?l?w H1 0(), r? wn n ? w?H1 0() qgx?, wn n wzk?Tq, wn n w?L2()qRx?. ? w ? 0.(3.6) uqj?Q?(H1), (H3)?j?M 0,r?R?x , t 0?|f(x,t) t | M(da?(H2)?M?O|q(x)|?).fw 0,?L2()q?wn n 0v Rx?Q(3.4)f(3.5)t? 4c = ? f(x,u+ n) u+ n w2 ndx + o(1) M ? w2 ndx
19、+ o(1) n 0, jtVc 0?Z(3.6)t?i? ?i?w(? 0)?t ? w q(x)wdx = 0,?R? H1 0(). (3.7) ? pn(x) = ? f(x,un(x)/un(x),?x Sun(x) 0; 0,?x Sun(x) 0. (3.8) Q?(H1)f(H3),d?j?j?M 0,r?R?x ,R0 pn(x) M,?bw?Z?D?l?ARd?h(x) L2(),r?pn n ? h?L2()qgx?S?zk?R0 h(x) M.e?T?R? L2(),U? wn n w?L2()qRx? ? pn(x)wn(x)(x)dx = ? pn(x)w+ n(x
20、)(x)dx n ? h(x)w+(x)(x)dx. Spnwn?L2()qR?R pnwn n ? hw+gx?TL2().(3.9) 1Kvq?BSi?K?NP195 Q(3.3)t?j?J?(un)?H1 n 0,S?un? n +,?F? H1 0(), R ? ? ? ? ? wn pn(x)wndx ? ? ? ? = tn|?J?(un),?| 2c ?un?2 ?J?(un)?H1? n 0. ?D?Q(3.9)tfwn n ? w?H1 0() qgx? ? w h(x)w+dx = 0,?R? H1 0(). (3.10) ?jtq? = w,Ej?w?2= 0,?qw w
21、+ 0,QRv?mX?y? w(x) 0?qzk?QT?un? n +S?qzk?Rwn(x) n w(x),? ?Q(3.5)t?j?da?qzk?Rw(x) 0,?un n +?qzk? ?Q(H3)?w(x) 0p?Rh(x) q(x),Z(3.10)t? ? w q(x)wdx = 0,?R? H1 0(), (3.11) eV 0,r? ? (x) v(x)dx = ? q(x)(x)v(x)dx,?R?v H1 0(). (3.12) I?dau?(1.1)?ARh?j?,?R ? u(x) (x)dx = ? f(x,u(x)(x)dx.(3.13) ?(3.12)tq?v =
22、u,?O(H2)f(H3),?(3.13)t?D? ? q(x)udx = ? u(x) (x)dx = ? f(x,u(x)(x)dx ? q(x)udx, Q? ? f(x,u(x) q(x)udx = 0. ?v?qzk?R(x) 0,?SQ? (H2), (H3)jf(x,u) q(x)u,I?q?Rf(x,u(x) = q(x)u(x)zk?y u 0?(= 1)?d?Ic?13?2?i?jA?c 0,r? u = c. Ek?fR?c 0,r?u(x) = c(x)S?qzk?Rf(x,c(x) = cq(x)(x), ?c(x)?d?Z? = 1p?u = c?(1.1)?AR?
23、i? ?1.1? N?1.1?l?i? ?1.3? QTq(x) +,?(3.1)tq?p?1,?G(4.4)q ?c,?OJ?2.2(b),?T?1.1 (ii)?i?Db?un H1 0(), r? (3.2)(3.4)t?i?1.3,n?i?un?H1 0() qR?Gwnd ?(3.5)t?wn?H1 0() qR?TARw H1 0(), R w+ n n ? w+gx?TH1 0(), w + n n w+zk?T, w+ n n w+Rx?TL2(). 196?47? f?un? n +,?Rw+(x) 0.uqj?T?5,?K? 1= x : w+(x) = 0, 2= x :
24、 w+(x) 0. ?Q(3.5)tju+ n(x) n +?2qzk?S?(H2)qq(x) +,? f(x,u+ n) u+ n n +?Tx 2zk?,ZQ(3.4)f(3.5)tDxFatouJ?D? 4c =lim n+ ?wn?2=lim n+ ? f(x,u+ n) u+ n (w+ n) 2dx lim n+ ? 2 f(x,u+ n) u+ n (w+ n) 2dx n +,dam2 0, I?m2= 0,?qiRw+ 0.?v?daw+ 0,?Rlimn+?F(x,w+ n(x)dx=0, ? J(wn) = 1 2?wn? 2 + o(1) = 2c + o(1).(3.
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