The Existence Of A Class Of Semi-linear Solution With The Circle Equation With Multiple,

The existence and mlltiplicity are st11died fOr solutions of The Dirichlet bo11ndaryvaIue problem fOr semhalear elliptic equations-- fi u = l(x.u) w h(x) for a.c. x 6 fl, u = 0 t-IIL 0ft. (11b} tllr variaLioIlal 1nethods. All existence tlleorem is obLained fOr problem (l) li1 LllecriTical growth case. Some m11ltiplicity resl1lts are obtained tbr problem (1 ) in thesuLlcritical growth case a11d the subAnear Srowth case at resollaIlce. Tlle mali1 resultsare the fOllowing theorems.Theorem 2.1 Suppose that l 5atidies the critical growth e(Indition, that is, thereexist a constani C1;t 0 and a real function ry e L'(O) soch thatl(x. tj 1 s C,1t"' -- 1 - t,(x)fOr all f F R and a.e. x G n, tvhere 2. = 2N/(N -- 2) if N 2 3 and 2' may be replacedhJ an} number in (2, +co) if Y = l or 2, and that,,(z. t) --;.\,t2 -- --'ro (2)tas tE - co uniforiIily fOr a.e. z E fl. Assunle Lhat h E L'lfl) batisfles that[ h(x)v(x)dx = 0.j Cttt![ere }-' iR tI1e tlle nr'rinal eigenf1nletiou cl-lrresptflnl1ing t":\ l w1t11 \-(x ) w.. f-,r RlI:i' t \7. Tli':ll plLll,l*:1I1 l 1 ) l1as;It l;.ast (,llt! s',luti4,l1s iI1 H{ \7 ).Theorem 3.1 S11ppose That h = Ii, f satisfies (2 ) 3nrl The s1lbcrlticai growTh,:')l,=liIi.,ll. tllat is. tllerc cxist G;: 0 al1d 2 <(p <(2' suclt tl1utjI(x. t)l 5 C:(it1;--' + 1)fOr all t E R and a.e. z E n. If there exists an integer m 2 l.6:> 0 and 8 f> 0 suchthatx. 5 Uyt 5 A.+1 -- b (3)1for a1l 0 < it1 < b and a.e. z 6 O, problem (1) has at least two nonzero sol11tions inHl(fl)., -Theorem 4.1 Let f(x, t) = A1t + g(x, t) and h = D. Suppose that there existf) 5 a c 1 and C1:x 0 5ueh that19(z,t)I 5 f:'1(it1" + 1 ) (4)l7Jr all t 6 R and a.e- x 5 \?. andi]vif--'" /' G(x, v)dx ~ --(x)j,,a5 JJrJ! -:x. in E(.\l). ir (3) holds, prl>blen1 (1 ) has at least two nonzero soltlti<'ns inHl.(fl).Theorem 4.2 Let j(x, t) = .\1 t + g(x, t) and h = 0. SuPPose that there ealstsa E L'(O) with a(x) 5 0 fOr a. e. x E fl and Ai a(x)dx < 0, such thatlimsup ee i,'(z)ltl -,co ludrirmly fOr a+ e. x e fl If (3) and (4) holds, problem (1 ) has at least two nolltrivaialsolution in Hj(fl).