Fučík, Svatopluk

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Commentationes Mathematicae Universitatis CarolinaeSvatopluk FučíkFurther remark on a theorem by E. M. Landesman and A. C. LazerCommentationes Mathematicae Universitatis Carolinae, Vol. 15 (1974), No. 2, 259--271Persistent URL: http://dml.cz/dmlcz/105550Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1974Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCOMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE 15,2 (1974) .FURTHER REMARK ON A THEOREM BY E.M.LA.NEESMAN AND A.C LAZER Svatopluk FUČÍK, Praha Abstract: It is considered the existence of the solu­tion of the boundary value problem A-a + hu, +• c^(w) = h on SL . ̂ U A « 0 in the case Xism, a*(*) = JIMm> o-Oto) « 0 and under some other additional conditions. Key words; Equations involving nonlinear operators, boundary value problems for ordinary and partial differen­tial equations, weak and classical solutions, positive ei~ genfunctions. AMS, Primary: 47H15 Ref. 2. 7.798 Secondary 34B15, 35J60 1. Introduction. Let SL be a bounded domain in R^ (Jl 2^1) and suppose loci ш * £ (-Á)"'Ь*(UJ л (x)J>ßњ) is a uniformly e l l ipt ic , formally selfadjoint linear diffe­rential expression defined on SL , with real-valaed coeffi­cients o, m cu^ C L^CJl) , Let further <%>i JL̂ —>&̂  be a bounded continuous function. We are concerned with the solvability of the Dirichlet problem 4 1 ) ÍLÁJU) (X) s= fy(u*(x)) - Jh*(x) , X £ SL AJU(X) m 0 , x e dSL - 259 -for a given real-valued function H • In the series of papers (see e.g. [33,C63,ClOl,...) they are given sufficient conditions for the function 9* to be the boundary value problem (1) weakly solvable for each "right hand side" K e L^CJX) provided the linear boun-dary value problem f (L/xMx) m 0 , x e SI (1) 4 I AA,(x) « 0 , X C 3ii has only trivial weak solution. The question about the weak solvability of (1) in the case of the existence of nontrivial solution of the homoge-neous linearized problem (2) is more complicated. In f93 it is proved that if 0 is a single eigenvalue of (2) and the null-space of the operator L is spanned by the vector w ( 4. 0 ) then under the assumptions ( 3 ) *t£iu <%>(*>) « O'C-*- 00 ) j S-&. + OD ( 4 ) Mm, O'Cvfc) ss <^C-oo) , S-U--09 (5) o^C- co ) <: <j,C/*)<: <%>(+ 00) for each /* e Z^ the necessary and suff ic ient condition for the existence of a weak solution of the problem (1) with Jfa & L« C SL) i s the va l id i ty of the inequa l i t i es (6) o^C-oo)J \mr(x)ldx - ô C-f 00) J \w(x)\dx <f<fo(x)<w(x)dx< <o^C+430)J IwGOldx - 9,C-co)f intrMldx, - 260 -where i l + « i x e SI ; /nr Cx> ^ 0 } , Short and elementary proof i s given in [73 . This r e -su l t was generalized in [15 3 to the case of multiple eigen-value of the operator L . The assumption (5) i s replaced by (7) 9, (-co) .6 fyU)6 fy(+oo)f 9,CO.)* &(-.'.*>); <j>(0)& q,(+oo) in the paper [53 • The abstract setting of the method from the papers [93,[153 is given in [13, U3, [53 ,[113 - 1143. Moreover, in these papers the higher order equations and also the more general nonlinear perturbations are conside-red. In all previously referred papers about the solvabili-ty of (l) it is substantial that the limits $,(+&), fy(-oo) are different from zero. The purpose of this note is to pro-ve that under some assumptions in the case ^(+»)s^(-a)sO there exists a solution of (l) for each it, which is ortho-gonal in La(-1) to the eigenfunction w (see Sections 3 and 4). The proof is based on the abstract theorem which is mentioned in Section 2 (this section serves also as a pre-liminary communication for a paper C43). 2. Abstract result The proof of the result from this section immediately follows from [4, Theorem 2.3.10]. Assumptions; I. Let H be a real Hilbert space with the inner product (• , ••) and the norm II « )L , Let - 261 -(X, II • 11̂  ) , (Z } II • 11.2 ) be Banach spaces. Suppose X c c Z c H . XI, Let L : D<mv CL3 c X — * Z be a closed linear operator with the domain Jhcym, E L ] dense in the space X and closed image Iw, CL3 in the space Z and with a finitedimensional null-space Xe*/ C L J . Moreover, sup-pose Z » 3/m, CL] .9 Xe>t- I L J » (Particularly, it means that L is a Fredholm operator the index of which is equal to zero.) Ill* Let Ui X — > Z , $<m< CN3 =r X , be a completely continuous nonlinear mapping such that /wup II H CM ) IL < + co . xiX z IV, Suppose that for arbitrary /c > 0 there exists an interval < e. , fcg,), 0 <z e^ < £ 2 1 such that -{/n/f •{ C N C tttr + W) , 4AT ) • /ir £ X , II tr II ̂  £ A , , *r e Xe* CL3 , BmrB « 4 , t « < ^ , e a > J > 0 . Theorem 1* Let *H € Z . Then the e q u a t i o n L(*u) =- KC-a) - Jh, i s so lvab le i n Iterm, t L 3 provided (,to,,'tt>0*BO f o r each w c X£«v C L 3 . Remark 1* Under the same assumpt ions (and with the same proof) we can s t r e n g t h e n the a s s e r t i o n of Theorem 1 i n the fo l lowing way: For each J b 0 o r thogona l t o Xex,CL3 i n H t h e r e e x i s t s an open neighborhood lLCJfc0) c Z of t he p o i n t M,0 such t h a t the equa t ion L (xt) = N(.u.) - Jk i s s o l v a b l e i n - 262 -Ъ<ytiъ í L 1 f or any Ль c U C KQ ) . 3» Main r e s u l t Let Jl be a bounded domain in .KN « Let H .=? L^CJl) with the usual inner product. Suppose that X and Z are the Banach spaces satisfying the condition I. Moreover, let X be continuously imbedded into L^ CJ1) . (A) Let L be the linear operator satisfying II. Assume that the dimension of Xet EL 3 is one and let Xot CL3 be spanned by w0 e Xe*/ CLU , H/ur I =r 4 and the .Lebes-gue measure of the set < # e SL j ̂ ( x) -6 0 } is zero. (B) Let H be a nonlinear mapping satisfying III. Suppose that there exists a bounded continuous function <$,* "R^—>&^ such that for any *uf /v « X it is CtfC-u,), ir) as f fy(u,(x)) w(x) dx 0 Let M m K*jj*v l^(f >) > 0 . For € > 0 denote rCf)=-4vX€JX$ <wj (*) < f J . Suppose that /m-e<â» rC*) > 0 if * > 0 . (O Let the function ^ introduced in (B) satisfy the fol-lowing condition: Tћere ex i s t s *Ą > ö sucћ that M/m, mгeoъ Г (*) M "" mллA, Л x-*0+ /maafc ГCx) M *£ ,, , --лneo^ Л - 263 -where HC?) % e<£V*C t > ' $*"* ' Theorem 2» Under the assumptions (A) -'(C) the equa-tion L(u>) m H(w) -Jfa/ is solvable in Dow-CL] provided Jh, € Z and f -to* C*) ̂  Cx) dx = 0 • Proof, Prom the assumption (O it follows that tj.(f>>0 for f > 72, and $.(f) -< 0 for ...J < - i£ • According to the result from Section 2 it is sufficient to prove the following assertion (# ); Let K > 0 • Then there exist a> > 0 and an inter-val < % , €, > , 0 < ̂ 1 < ̂ 2 , such that $ fy(t<ur0(x) + <ir(x)) urQ(x)dx > o> , J %'(-ti4r0(x) + nr Cx)) tfljCx) d»x < - o> for each t e < £ 1 , e a > and v e X , II tr HL ^ jt . Denote a » II4^!)^ CJQ. . Since Jtivn, mAaA> T(x) ~ 0 x-+ 0+ there exists cT> 0 and («A> > 0 such that 4 and muafc Г( à) <= — mteл* Л 2 G(i) M /mлaъ Г(Л) /mAcuь Л - mглaò Г(Л) - 264 -H C - - І ) JИ ã + — — «б - ŕ t /wu*t* T ( A) mta* Si - /mao* TCA) where д « 2C^+^c) o/ +.^cT Set g, s — - — . £, =• 2 — T B — , C J S - T - AmuA6 J l mJUMsrCA) . Thus we have for t e < i „ S„ > , tr e JC with fl/irll. . . . ^ the following re la t ions f %>Ct<w(x) + cvCx))4^C^)ciy - / X + f A /. fe /\ > G («£-) A (mwa*> J l - mt&a* T C A )) - M A mj&** V ( A) •» fGc|) M 1 • • M « u « * J L - m i i 4 » r C A » ^ >. — hmaxv* SL mjuih PC A) » co 2 and f o-C-t'w; (*) + wCx)) /w: C#)dx a? f X + f */ £ *SL ° ° »WCA> Jrc*> ^ HC ) ^ C/meo6 i l - muto^TCA)) + M. Ammj* T( A) £ - s> . A Remark 2. If muoub .T (# ) = 0 for * € C0, o£ ) then the asser t ion of Theorem 2 i s valid i f the assumption (C) i s replaced by (C#): There ex i s t s <r̂  > 0 such that 9 * ( ? ) > 0 ; 9^C-§><0 for f > ^ • (The proof i s the same as the previous one*) - 265 -2 4. Applications Firstly we note that the investigation of the problem about the existence of simple eigenvalue of the linear map-ping such that the corresponding eigenvalue is nonnegative is included e.g. in the book tsl, where also the special cases of linear integral and differential operators are stu-died. In this section we give the applications of Theorem 2 to /|he existence of a weak solution of boundary value prob-lems for ordinary differential equations and to the existen-ce of a classical solution of boundary value problems for partial differential equations. a) Weak solutions of boundary value problems for ordi-nary differential equations. Set JC-aW^C0,3r)*Z. the Sobolev space of all absolutely continuous functions JUU on the interval ( 0 ? 3t ) with AL C0)«44,(zf) at 0 andwhich derivative u,' is square integrable on CO, *r) . We shall consider on o A ' ¥ ̂  C 0, 3t) the inner product C-u, ,^) . . = ) JUL'(*) ir'(x) dx • ' 1,2. \)Q 0 A ° A, For AJL c Yfl C 0,ar) define L*w c W^ C 0, tr ) such t h a t AJS(X) 4r'Cx)dx + J AJL(X) ^(x)dx for each i r e W^CO.ar) - ^nus we have defined a l i n e a r continuous mapping L s X — * Z such that JCe*, LL3 » * L-Uieout, <W£ <*</>I> x I . - 266 -Let fytX^-"*-* R/| be a continuous funct ion, MV$, I <%> ( § ) I s Ji . Obviously the mapping H t X—«*• 2 defined by the r e l a t i o n rvr (Hu % v y „ • o-(Au(,x>)/ir6<>d,# > 1 , 1 J 0 ^ •3 for each n/cW^CO^sr) i s completely continuous and i t sa­t i s f i e s the condition (B) from Section 3. Thus we have Theorem 3* Suppose t h a t there ex i s t s ^ > 0 such t h a t mi/n* cv^t) 2M * - * 0 + :* $Г ш f e <^Ct) 2..M ЛcW * « < - 1 ' « . - n > < _ . Then f o r e a c h Av € L ^ ( 0 , * r ) , Jo * o * t he re ex i s t s AM € W2(Q>ijr) such that the integral iden-t i t y ~ J* 4*'f*)ir'(,x)el* + f At,(x)w(x)d#-* J'qr(4*<x))*r(x)dx e s f J&vCx)nrCx>dx ô holds for each /v € W^f 0, *r) , i . e . , the boundary value problem ~AA/*+.JU, + %>(**) m Jfo AM(0) m JM(ST) « 0 is weakly solvable. Remark 3. The same result is possible to give for ge-neral Sturm-Liouville operator of the second order and for - 267 -the first eigenvalue of this operator. b) Boundary value problems for partial differential equations* The result from Section 3 it is not possible direct-ly to apply to the weak solvability of boundary value pro-blems for partial differential equations since we suppose that X c L^fJl) and JC usually means the Sobolev spa-ce n\ (SL) . Thus we give the application to the classi-cal solvability. Let Sk be an open connected bounded auhset of 3tfl with the boundary dSL , C**(SL) will denote the space of the functions which are Jfc, -times continuously differentia-ble on SL and such that the derivatives can be extended continuously onto -S . With the usual norm: ll̂ ulL • jMufi Avufv ID JU,(X)\ C (SI) is a Banach spacet C0 (SL) will denote the sub-space of C (SL) of the• functions which are zero on dJl . Finally we recall that the classical problem "Axt. + XAJU & 0 on. SL •Ian - ° has countably infinite many eigenvalues MK^\ , arranged according to increasing magnitude and considering their respective multiplicity. The last eigenvalue is simple: thus we have 0 < X^ < A^^... . Moreover, the eigenfunc-tion w0 corresponding to X^ does not vanish on SL and its values are of the same sign (this result is con-- 268 -tained in a general theorem concerning the nodes of an ei-genfunction - see e.g. £2, p.4523). Set X - C o C i i ) , Z » Z0(H) > L : w i—* Ax* + A^M, CL: C 0 Cil)—*> C (Jl)) . Since the condition (A) from Sec-tion 3 is fulfilled we have Theorem 4. Let 9.: X^—>K^ be a continuous and boun-ded function satisfying the condition CO from Section 3. Then for each JJv e C°(TL), J M,(x)<urc(x)dx *m 0 there exists a solution AP m C (Si) of the problem &M, + AA At, + 0/ CAU) * 4v on Jl *ld.a - 0 . Remark 4 . The same r e su l t as in Theorem 4 i t is pos-s i b l e to give for 1 aV * 9M,-' L : JUL, t~» £ etc • 0 0 - — s — • . s . o^fx) -=— + a,(x) u, « -*ii»4 ** a**5** *m4 *** where the coeff icients a ^ j , O^ , 0/ are sufficiently smooth and L i s uniformly e l l i p t i c operator. R e f e r e n c e s [ 1 ] M.S. BERGER and M. SCHECHTIR: On the solvabi l i ty of se-milinear operator equations and e l l i p t i c bounda-ry value problems, Bull.Amer.lath.Soc.78(1972), 741-745. [ 2 ] R. COTJRAOT and D. HJ1BERT: Hethods of Mathematical Physics, Vol .1 , lew Torlc,1953» t 3] S. FaCfe: Fredholm a l te rna t ive for nonlinear operators in Banach spaces and i t s applicat ions to diffe-* - 269 -rential and integral equations, Cas.pSst.mat. 96(1971),371-390. [4J s. FtTfifK: Nonlinear equations with noninvertible li-near part, Czechoslovak Math.Journal (to appear) [5] s. FUCxK, M. KUCERA and J. NECAS: Ranges of nonlinear asymptotically linear operators, J.Diff.Equa-tions (to appear). [6] s. FutfK, J. NEfiAS, J. SOUCEK and V. SOUCEK: Spectral Analysis of Nonlinear Operators,. Lecture Notes in Mathematics No 346,Springer Verlag 1973. [7] P# HESS: On a theorem by Landesman and Lazer, Indiana Univ.lfeth*Journal(to appear). [8] M J U KRASNOSELSKIJ: Positive Solutions of Operator's Equations (Russian), Moscow 1962. [9] E.M. LANDESMAN and A.C. LAZER: Nonlinear perturbations of linear boundary value problems at resonance, Joum.Math.Mech. 19(1970) ,609-623. [10] Jf NEfiAS: Fredholm alternative for nonlinear operators with application to partial differential equa-tions and integral equations, 5as.p§st.mat.97 (1972),65-71. [11] J. NEfiAS: On the range of nonlinear operators with li-near asymptotes which are not invertible, Comment. Math.Univ.Carolinae 14(1973),63-72. [12] L. NIRENBERG: An application of generalized degree to a class of nonlinear problems, Troisieme Colloq. Anal.Fonct. Liege Centre Beige de Recherches Mathematiques, 1971, pp.57-73. [13] L. NIRENBERG: Generalized degree and nonlinear prob-lems, "Contributions to Nonlinear Analysis" edi-ted by E. Zarantonello• Academic Press 1971, pp.1-9. [14] M. SCHECOTER: A nonlinear elliptic boundary value problem (to appear). - 270 -[153 S.A. WILLIAMS: A sharp sufficient condition for solu­tion of a nonlinear elliptic boundary value problem, J.Diff.Equations 8(1970),580-586. Matematicko-fyzikální fakulta Karlova universita Sokolovská 83, 18600 Praha 8 Československo (Oblátům 31.1.1974) - 271 -

Further remark on a theorem by E. M. Landesman and A. C. Lazer

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Further remark on a theorem by E. M. Landesman and A. C. Lazer

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Institute of Mathematics AS CR, v. v. i.