MULTIPLE SOLUTIONS FOR ASYMPTOTICALLY LINEAR RESONANT ELLIPTIC PROBLEMS. Francisco O. V. de Paiva. 1. Introduction. Let us consider the problem (1.

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1 Topological Mehods in Nonlinear Analysis Journal of he Juliusz Schauder Cener Volume 1, 003, 7 47 MULTIPLE SOLUTIONS FOR ASYMPTOTICALLY LINEAR RESONANT ELLIPTIC PROBLEMS Francisco O. V. de Paiva Absrac. In his paper we esablish he exisence of muliple soluions for he semilinear ellipic problem (1.1) u = g(x, u) in, on, where R N is a bounded domain wih smooh boundary, a funcion g: R R is of class C 1 such ha g(x, 0) = 0 and which is asympoically linear a infiniy. We considered boh cases, resonan and nonresonan. We use criical groups o disinguish he criical poins. Le us consider he problem 1. Inroducion (1.1) u = g(x, u) in, on, where R N is a open bounded domain wih smooh boundary and a funcion g: R R be of class C 1 such ha g(x, 0) = 0, which implies ha (1.1) 000 Mahemaics Subjec Classificaion. 35J65, 35B38. Key words and phrases. Cerami condiion, mulipliciy of soluions, double resonance, sign changing soluion. The auhor was suppored by CAPES, Brazil. c 003 Juliusz Schauder Cener for Nonlinear Sudies 7

2 8 F. O. V. de Paiva possesses he rivial soluion. We will be ineresed in nonrivial soluions. Assume ha g(x, ) lim sup l, l R. The classical soluions of he problem (1.1) correspond o criical poins of he funcional F defined on H = H0 1 (), by (1.) F (u) = 1 u dx G(x, u) dx, u H, where G(x, ) = 0 g(x, s) ds. Under he above assumpions F C. Denoe by 0 < λ 1 <... λ j... he eigenvalues of (, H 1 0 ). We wrie a(x) b(x) o indicae ha a(x) b(x) wih sric inequaliy holding on a se of posiive measure. We firs assume he followings hypoheses on g. (g1) g C 1 ( R) and g(x, 0) = 0. g(x, ) is sricly increasing for 0, a.e. x, and (g) g(x, ) is sricly decreasing for 0, a.e. x. (g3) λ j L(x) = lim inf in. (g4) λ j L(x) and g(x, ) lim sup g(x, ) lim [g(x, ) G(x, )] =, a.e. x. = K(x) λ j+1, uniformly Theorem 1.1. Le g: R R be a funcion which saisfies (g1) and (g). Moreover, suppose ha here exis k and m 1 such ha λ k 1 g (x, 0) < λ k λ k+m lim inf g(x, ) lim sup g (x, ) λ k+m+ for all x and R, g(x, ) λ k+m+1, where he limis are uniform for x in. If eiher (g3) or (g4) hold wih j = k+m. Then problem (1.1) has a leas wo nonrivial soluions. Theorem 1.. Le g: R R be a funcion which saisfies (g1). Suppose ha here exiss m 1 such ha g (x, 0) < λ 1 < λ m+1 lim inf g(x, ) lim sup g(x, ) λ m+, where he limis are uniform for x in. Suppose ha eiher (g3) or (g4) hold wih j = m + 1. Moreover, if (g4) hold assume ha here exiss C(x) L 1 () such ha g(x, ) G(x, ) C(x) for all R, a.e. x. Then problem (1.1) has a leas hree nonrivial soluions.

3 Muliple Soluions 9 Theorem 1.3. Le g: R R be a funcion which saisfies (g1) and (g). Suppose ha here exiss m such ha g (x, 0) < λ 1 < λ m+1 lim inf g(x, ) lim sup g(x, ) λ m+, where he limis are uniform for x in. If eiher (g3) or (g4) hold wih j = m+1. Then problem (1.1) has a leas four nonrivial soluions, one of hose changing sign, anoher one posiive and a hird one negaive. Theorem 1.4. Le g: R R be a funcion which saisfies (g1) and (g). If here exiss k such ha λ k 1 g g(x, ) (x, 0) < λ k < lim < λ k+1, ± g (x, ) λ k+1 for all x and R, where he limis are uniform for x in. If 0 is an isolaed criical poin hen problem (1.1) has exacly wo nonrivial soluions. Remark 1.5. We call (1.1) a resonan or double resonan problem when i happens, respecively, ha λ j lim inf g(x, ) lim = λ j, g(x, ) lim sup g(x, ) λ j+1, for some j 1, uniform for a.e. x (cf. [], [5]). Mulipliciy for double resonan problems were reaed by recen papers [3] [5]. In [3], he auhor considered only he auonomous case and assume srong resonan hypoheses. Theorem 1.6 below, gives a example of a funcion ha saisfies he hypoheses of Theorem 1.1 and does no saisfy he hypoheses in [4], [5]. Under he condiions of Theorem.5, bu assuming resonance only a one eigenvalue, Dancer and Zhang ([13]) proved ha problem (1.1) has a leas one sign-changing soluion, one posiive soluion, and one negaive soluion. (1.3) Now consider he auonomous problem u = g(u) in, on, where g: R R is a funcion of class C 1 such ha g(0) = 0. Casro and Lazer (see [6]), and Ambrosei and Mancini (see [1]) proved ha if g C, g () > 0 a.e. in R, and λ k 1 < g (0) < λ k < lim ± g () < λ k+1

4 30 F. O. V. de Paiva for some k 1, hen (1.3) has exacly wo nonrivial soluions. In Mizoguchi ([19]), i was shown ha if g C, g () > 0 a.e. in R, and λ k 1 g (0) < λ k λ k+1 < lim ± g () < λ k+, hen here exis a leas wo nonrivial soluions of he problem (1.3). Our nex resul exends he previous resuls in his auonomous case. Theorem 1.6. Le g: R R be a funcion of class C 1, g(0) = 0, which saisfies { g() is convex if 0 and g() is concave if 0. If here exis k and m 1 such ha λ k 1 g (0) < λ k... λ k+m < l ± = hen problem (1.3) has a leas wo nonrivial soluions. lim ± g () λ k+m+1, In fac, he above hypohesis on he convexiy of g implies ha (Proposiion 3.1) lim [g() G()] =. Hence he previous heorem is corollary of Theorem 1.1. Remark 1.7. In [3], Barsch, Chang and Wang showed ha if g () > g()/ for all 0 and g (0) < λ 1 < λ λ k < lim g () < λ k+1, (k > ), hen problem (1.3) has a leas four nonrivial soluion, wo of hese soluions change sign, one is posiive and anoher one is negaive. They observe ha he nonresonance a infiniy in he above resul can be removed using argumens like in [4] and [9]. In [9], he auhor assumes a Ladesmam Lazer condiion and ha g() λ k+1 is bounded for all R. In [4], he auhors suppose ha g() λ k+1 c( r + 1), for some r (0, 1), wih he purpose of compuing he criical groups a infiniy. The nex resul is a corollary of Theorem 1.3. Theorem 1.8. Le g: R R be a funcion of class C 1, g(0) = 0, which saisfies { g() is convex if 0, and g() is concave if 0. Suppose ha here exis k > such ha g (0) < λ 1 < λ λ k < g() lim = λ k+1. ±

5 Muliple Soluions 31 Then problem (1.3) has a leas four nonrivial soluions, one of hose change sign, one is posiive and anoher one is negaive. Remark 1.9. The funcional in he nonresonan case saisfies he Palais Smale Condiion, (PS) in shor, and he difficuly in he resonan case is he lack of a (PS) condiion. Bu if he funcion g saisfies lim [g(x, ) G(x, )] = uniformly in, hen in [11], Cosa and Magalhães showed ha his condiion is sufficien o obain a weak version of he (PS) condiion, namely he (C) condiion, which was inroduced by Cerami in [7]. The (C) condiion was used by Barolo, Benci and Forunao in [] o prove a general minimax heorem (see [0] for his resuls wih he (PS) condiion). The so called Second Deformaion Lemma, proved by Chang (see [9]), has a version wih he Cerami condiion replacing he usual (PS) condiion, as proved by Silva and Teixeira in []. In Secion, we collec some resuls on Morse Theory, wih he funcional saisfying he Cerami condiion. In Secion 3, we prove some lemmas abou he geomery of he funcional and a compacness condiion. In Secion 4 we prove he main heorems.. Remarks on Criical Poin Theory In his secion some classical definiions and resuls in Morse Theory are recalled. These resuls will be used in he proofs of main heorems. In [9] he (PS) condiion is used, whereas we use here a weaker compacness condiion on he funcionals. This resuls can be found in [18] wih ohers hypoheses. Le H be a Hilber space and f: H R be a funcional of class C 1. Denoe he se of criical poins of f by K. Given c R, we se f c = {x H : f(x) c} and K c = f 1 (c) K. Definiion.1. Given f C 1 (H, R) and c R, we say ha f saisfies he Cerami condiion a level c R, denoed by (C) c, if every sequence {x n } H saisfying f(x n ) c and (1 + x n ) f (x n ) 0, n, has a converging subsequence. If f saisfies he (C) c condiion for every c R, we say ha i saisfies he (C) condiion. I is clear ha a funcional saisfying he (PS) condiion also saisfies he (C) condiion.

6 3 F. O. V. de Paiva Definiion.. f C 1 (H, R) is said o possess he deformaion propery if i saisfies he following condiion (i) for every a < b such ha K f 1 (a, b) =, hen f a is a srong deformaion rerac of f b \ K b. The nex resul is a version of a deformaion lemma (for references see [9]) proved in [1] (see also []). Lemma.3 (Deformaion Lemma). Suppose ha f C 1 (H, R) saisfies he (C) condiion and a is he only possible criical value of f in he inerval [a, b). Assume ha he conneced componens of K a are only isolaed poins. Then, f a is a srong deformaion rerac of f b \ K b. This lemma is an imporan ool in Criical Poin Theory. Now we sae some known resuls, which are also rue under he (C) condiion insead of he usual (PS). For proofs of hese resuls assuming (PS) see [9]. Where furher references can be found. These proofs can be early adaped for he case when (C) condiion is assumed. Le Y X be opological spaces, denoe by H (X, Y ) he singular relaive homology groups wih coefficiens in Z. Definiion.4. Le x 0 be an isolaed criical poin of f, and le c = f(x 0 ). We call C p (f, x 0 ) = H p (f c U x0, (f c \ {x 0 }) U x0 ) he p-h criical group of f a x 0, p = 0, 1,..., where U x0 is a neighbourhood of x 0 such ha K (f c U x0 ) = {x 0 }. Theorem.5. Assume ha α H j (f b, f a ) is nonrivial, and c = inf sup f(x). τ α x τ Suppose ha f possesses he deformaion propery. Then here exiss x 0 K c such ha C j (f, x 0 ) 0. Definiion.6. Le D be a j-opological ball in H, and S be a subse in H. We say ha D and S homologically link, if D S = and τ S, for each singular j chain τ wih τ = D where τ is he suppor of τ. The following proposiion provides examples of ses homologically linking, heir proofs are a consequence of Examples, 3 and Theorem 1. in Chaper II of [9]. Proposiion.7. Le H 1 and H be wo closed subspaces of a Hilber space H. Suppose ha H = H 1 H and dim H 1 <. Then, if D 1 = B R H 1 and S 1 = H, D 1 and S 1 homologically link.

7 Muliple Soluions 33 Proposiion.8. Le H 1 and H be wo closed subspaces of a Hilber space H. Suppose ha H = H 1 H and dim H 1 <. Le e H, e = 1, and R, r, ρ > 0 wih ρ < R. Se D = {x + se : x H 1 B r, s [0, R]} and S = H B ρ. Then D and S homologically link. Theorem.9 ([9, Chaper II, Theorem 1.1 ]). Assume ha D and S homologically link, where D is a j-opological ball. If f C(H, R 1 ) saisfies f(x) > a for all x S and f(x) a for all x D, hen H j (f b, f a ) 0, for b > max{f(x) : x D}. We inend o compue he criical groups of an isolaed criical poin. For his purpose we presen he Shifing Theorem. Firs, consider he Spliing Theorem Theorem.10 (Spliing Theorem). Suppose ha U is a neighbourhood of x 0 in a Hilber space H and ha f C (U, R). Assume ha x 0 is he only criical poin of f and ha A = d f(x 0 ) wih kernel N. If 0 is eiher an isolaed poin of he specrum σ(a) or no in σ(a), hen here exiss a ball B δ, δ > 0, cenered a 0, an ordering-preserving local homomorphism φ defined on B δ, and a C 1 mapping h: B δ N N such ha f φ(z + y) = 1 (Az, z) + f(h(y) + y) for all x B δ, where y = P N x, z = P N x, and P N is he orhogonal projecion ono he subspace N. We call N = φ(u N). The following heorem ses up he relaionship beween he criical poins of f and hose of f := f N. I is proved in [9]. Theorem.11 (Shifing Theorem). Suppose he hypoheses of he Spliing Theorem. Assume ha he Morse index of f a x 0 is µ, hen we have C p (f, x 0 ) = C p µ ( f, x 0 ), p = 0, 1,.... In addiion, if d f(x 0 ) has finie dimensional kernel, hen we have Corollary.1. Suppose ha N is finie dimensional wih dimension ν and x 0 is (i) a local minimum of f, hen C p (f, x 0 ) = δ pµ Z, (ii) a local maximum of f, hen C p (f, x 0 ) = δ p(µ+ν) Z, (iii) neiher a local maximum nor a local minimum of f, hen C p (f, x 0 ) = 0 for p µ and p µ + ν.

8 34 F. O. V. de Paiva 3. Preliminary lemmas Le g: R R be a funcion of class C 1 such ha g(x, 0) = 0. Suppose ha here exis k and m 1 such ha (3.1) λ k+m L(x) = lim inf λ k 1 g (x, 0) < λ k, g(x, ) lim sup g(x, ) λ k+m+1, where he limis are uniform for x in. Le H = H 1 0 () and denoe he norms in H 1 0 () and L () by and, respecively. Le H 1, H and H 3 be he subspaces of H spanned by he eigenfuncions corresponding o he eigenvalues {λ 1,..., λ k 1 }, {λ k,..., λ k+m } and {λ k+m+1,... }, respecively. The nex resul is similar o Lemma 1 in [19]. The proof given here is a variaion of he one found in [19]. Le F be defined as in (1.). Lemma 3.1. Under he assumpions above and he hypohesis (g), he following saemens hold: (i) There are r > 0 and a > 0 such ha F (u) a for all u H H 3 wih u = r, (ii) F (u), as u, for u H 1 H, (iii) F (u) 0 for all u H 3, and (iv) F (u) 0 for all u H 1. Proof. By (g) and (3.1), we can ake he posiive numbers α, δ saisfying ha g(x, ) λ k 1 α < λ k for all R wih δ. Moreover, g(x, )/ < l = λ k+m+1 for all R from (g) and (3.1). Le H l = ker( li). Then H H 3 = V W, where V = H H l. For u H H 3, pu u = v + w, v V and w W. Since V is spanned by a finie number of eigenfuncions which are L -funcions. Then here exiss r > 0 such ha sup v(x) γ l x γ α δ if v r, where γ > l and w γ w for all w W. Suppose ha v r. If v(x) + w(x) δ, hen 1 λ k v γ w G(x, v + w) 1 λ k v γ w 1 α(v + w)

9 Muliple Soluions 35 = 1 λ k v γ w 1 αv 1 αw αvw = 1 4 α w (γ α)w + 1 (λ k α)v αvw 1 4 α w + 1 (λ k α)v αvw 1 4 l w + 1 (λ k α)v αvw. If v(x) + w(x) > δ, we have and hence G(x, v + w) 1 l(v + w) 1 (l α)δ 1 λ k v γ w G(x, v + w) 1 λ k v γ w 1 l(v + w) + 1 (l α)δ = 1 λ k v γ w 1 l v 1 l w lvw + 1 (l α)δ = 1 4 l w + 1 (λ k α) v αvw (γ l) w + (α l)vw + 1 (α l) v + 1 (l α)δ 1 4 l w + 1 (λ k α) v αvw, (in order o see he las inequaliy, consider 1 4 (γ l) w + (α l)vw + 1 (α l) v + 1 (l α)δ as a quadraic form in w and prove ha i is posiively defined). Therefore, we obain F (u) = 1 v + w G(x, u) dx 1 4 w 1 4 l w + 1 { ( 1 (λ k α) v min 1 l ), λ } k α u. 4 γ l This implies saemen (i). By he hypohesis λ k+m L(x), he Proposiion in [1] saes ha here exiss δ 1 > 0 such ha u L(x)u dx δ 1 u for all u H 1 H. To show (ii), le δ 1 be given above and ε > 0 be such ha ε < λ 1 δ 1. By he definiion of L(x), here exiss M = M(ε) such ha F (x, ) (L(x) ε) M for all R, a.e. x.

10 36 F. O. V. de Paiva Therefore, for u H 1 H we have F (u) u L(x)u dx + ε u + M ( ) δ 1 + ελ1 u + M, as u, since ε/λ 1 δ 1 < 0. We prove (iii) and (iv) by sraighforward calculaions. Le u 0 be a criical poin of F, defined by Definiion.1. The Morse index µ(u 0 ) of u 0 measures he dimension of he maximal subspace of H = H 1 0 () on which F (u 0 ) is negaive definie. We denoe he dimension of he kernel of F (u 0 ) by ν(u 0 ). The nex lemma evaluaes ν(u 0 ) for a nonzero criical poin of F. Similar ideas used in he proof below can be seen in [19] and [6]. Lemma 3.. Under he hypoheses of Lemma 3.1, ν(u 0 ) m + 1 provided ha u 0 is a nonzero criical poin of F defined in (1.). Proof. Le u 0 be a nonzero criical poin of F, ha is, a nonrivial weak soluion of problem (1.1). We denoe g(x, u 0 ) = g(u 0 ) and g (x, u 0 ) = g (u 0 ). Noe ha F (u 0 ) if and only if u = g (u 0 )u in, on. From g(x, 0) = 0, he problem (1.1) can be rewrien in he form u = q(x)u in, on, where q(x) = g(u 0 )/u 0 if u 0 (x) 0 and q(x) = g (0) if u 0 (x) = 0. I is a sandard resul ha u 0 is a classical soluion of (1.1). Then u 0 canno vanish idenically on every open subse of, by he unique coninuaion propery (see [15]). Le α 1 < α... α n... and β 1 < β... β n... be eigenvalues of he problems (3.) and (3.3) u = αq(x)u in, on, u = βg (u 0 )u in, on, respecively. Le {ψ n } and {φ n } denoe he corresponding eigenfuncions of problems (3.) and (3.3) saisfying, for all n, m N, qψ n ψ m dx = δ nm and g (u 0 )φ n φ m dx = δ nm.

11 Muliple Soluions 37 Claim. β n < α n for all n N In fac, by (g) we have g (u 0 (x)) g(u 0 (x))/u 0 (x) and again by he unique coninuaion propery ({ m x : g (u 0 (x)) > g(u 0(x)) u 0 (x) }) > 0. Then we use Proposiion 1.1 A in [14], and so he Claim is proved. Nex, suppose ha {ν n } and {δ n } denoe he eigenvalues of he problems and u = νλ k+m+1 u in, u = δλ k 1 u in, on, on, respecively. Immediaely, his implies ν n = λ n /λ k+m+ and δ n = λ n /λ k 1. By (g) and (3.1), we have λ k 1 < g(u 0(x)) u 0 (x) g (u 0 (x)) < λ k+m+ for all x such ha u 0 (x) 0. By a mehod similar o he proof of β n < α n, we obain α k 1 < δ k 1 = 1, 1 = ν k+m+ < α k+m+ and 1 = ν k+m+ < β k+m+. From u 0 0, 1 is an eigenvalue of (3.). Therefore, i holds ha α k = 1, or α k+1 = 1,..., or α k+m+1 = 1. If α k+m+1 = 1, he fac β k+m+1 < α k+m+1 = 1 = ν k+m+ < β k+m+ implies ha 1 is no an eigenvalue of (3.3), i.e. ν(u 0 ) = 0. If α k+m = 1, he fac β k+m < α k+m = 1 = ν k+m+ < β k+m+ implies ha ν(u 0 ) 1. Analogously, if α k+m = 1 hen ν(u 0 ),..., if α k = 1 hen ν(u 0 ) m + 1. This complees he proof of he lemma. Now we observe a compacness condiion for he funcional F defined by (1.), in he resonan case. Consider g: R R be a C 1 -funcion and G(x, ) = g(x, s) ds such ha 0 (3.4) λ j lim inf g(x, ) lim sup here exiss C(x) L 1 () such ha g(x, ) λ j+1 uniformly in, (3.5) g(x, ) G(x, ) C(x) for all R, a.e. x,

12 38 F. O. V. de Paiva and (3.6) lim [g(x, ) G(x, )] =, a.e. x. In [11] i was shown ha he assumpions (3.4) (3.6) are enough o prove ha funcional he F, defined by (1.), saisfies he Cerami condiion (see [16]). Noe ha he hypohesis (g) implies (3.5) wih C(x) = 0. In order o prove ha Theorems 1.6 and 1.8 follow from Theorems 1.1 and 1.4, respecively, we have o prove ha he funcion g saisfies (3.6). Proposiion 3.3. Le g: R R be a nonlinear funcion of class C 1, such ha g(0) = 0, and which saisfies { g() is convex if 0, and g() is concave if 0. Moreover, assume ha g()/ is bounded. Then Proof. Fix > 0, and noe ha lim [g() G()] =. 1 [g() G()] = 0 ( g() ) s g(s) ds. The convexiy of g gives ha (g()/)s > g(s) for s (0, ). Denoe by A he region of he plane beween he line s (g()/)s and s g(s) in (0, ). Le s() (0, ) defined by ( g() g() s() g(s()) = max s (0,) ) s g(s), and he riangle wih verices (0, 0), (s(), g(s())) and (, g()). A by convexiy of g, hence 1 [g() G()]. Therefore he proposiion follows of Claim., as. In fac, he heigh of, wih reference o base b = [(0, 0), (, g())], is [ ] ( ( )) g() g() h() = s() g(s()) cos arcan. We have Hence lim inf h() > 0, since g()/ is bounded; and b as. The claim is proved. The argumen wih < 0 is enirely similar and he proof of proposiion is complee.

13 Muliple Soluions 39 Lemma 3.4. Le g: R R be a coninuous funcion saisfying g(x, )/ is bounded for > 0, g(x, ) = 0 for all 0, and g(x, ) (3.7) λ j L(x) = lim λ j+1, j. Then he C 0 -funcional F + : H0 1 R defined by F + (u) = 1 u dx G(x, ) dx, saisfies he (PS) condiion. Proof. Le {u n } H0 1 be a sequence such ha {F + (u n )} is bounded, and F +(u n ) 0 as n. I follows ha for all ϕ H0 1 we have (3.8) F +(u n ), ϕ = u n ϕ g(x, u n )ϕ dx 0 as n. Se ϕ = u n. We have u n g(x, u n )u n dx + O( u n ) C u n + O( u n ). Therefore, we need o show ha { u n } is bounded, which implies ha { u n } is bounded. Since is bounded and g is subcriical, hen if { u n } is bounded, by he compacness of Sobolev embedding and by he sandard processes we know ha here exiss a subsequence of {u n } in H 1 0 which converges srongly, hence he Lemma 3.4 is proved. Assume by conradicion ha u n as n. Le v n = u n / u n. Then v n = 1 and { v n } is bounded. We can assume ha v n v weakly in H 1 0, srongly in L and a.e. in. Thus, u n (x) a.e. x. From (3.7) and (3.8) i follows ha [ v ϕ L(x)v + ϕ] dx for all ϕ H0 1, where v + (x) = max{0, v(x)}. By he regulariy heory we have v = L(x)v + in. By he maximum principle and by he unique coninuaion propery, v = v + 0 and L λ j or L λ j+1. Since, j, v 0, which conradics v = 1. The proof is compleed. 4. Proofs of main heorems I follows from [11] ha he funcional F, defined by (1.), saisfies he (C) condiion (or he (PS) condiion on he nonresonan case). Then we can use he heorems in Secion. Wihou loss of generaliy, we assume ha F has only a finie number of criical poins.

14 40 F. O. V. de Paiva Proof of Theorem 1.1. The cases (g3) and (g4) are considered simulaneously. Le H i, i = 1,, 3 be as in Lemma 3.1. Consider S 1 = B r (H H 3 ) and D 1 = {v + e : v H 1, 0 R, v + e R}, where B r denoes he closed ball wih radius r cenered of 0, and e H is chosen such ha (4.1) F (u) > 0 for all u S 1 and F (u) 0 for all u D 1, his is possible by (i) and (iii) in Lemma 3.1. Since D 1 and S 1 homologically link and D 1 is a k-opological ball, by (4.1) we have H k (F b, F 0 ) 0, where b > max{f (u) : u D} (see Theorem.9). Hence we can conclude, by Theorem.5, ha here exis u 1 criical poin of F, such ha (4.) C k (F, u 1 ) 0. Nex, se S = H 3 and D = B R (H 1 H ). By (ii) and (iv) in Lemma 3.1, we have F (u) 0 for all u S and F (u) < 0 for all u D. Again, since D and S homologically link and D is a (k +m)-opological ball, we have ha here exis u criical poin of F, such ha (4.3) C k+m (F, u ) 0. Now we have o prove ha u 1 u, and are nonrivial. Noe ha 0 is a criical poin of F and µ(0)+ν(0) k 1. By Shifing Theorem (Theorem.11), C p (F, 0) = 0 for all p k. So u 1 and u are nonrivial, by (4.) and (4.3). Again by Shifing Theorem (Corollary.1) we have, eiher (i) C p (F, u 1 ) = δ pµ(u1), or (ii) C p (F, u 1 ) = δ p(µ(u1)+ν(u 1)), or (iii) C p (F, u 1 ) = 0 if p µ(u 1 ) and p µ(u 1 ) + ν(u 1 )). If (i) or (ii) hold, hen C k+m (F, u 1 ) = 0 by (4.). If (iii) hold hen k > µ(u 1 ) by (4.) and hence k + m µ(u 1 ) + ν(u 1 ) by Lemma 3., again C k+m (F, u 1 ) = 0 by (iii). Therefore u 1 u by (4.3). The proof of Theorem 1.1 is finished. Proof of Theorem 1.. Se { g(x, ) for 0, g + (x, ) = 0 for 0, and consider he problem u = g + (x, u) in, on.

15 Muliple Soluions 41 Define F + (u) = 1 u dx G + (x, u) dx, u H0 1 (). Then F + C 0 and, by Lemma 3.4, saisfies (PS) condiion. Since g (x, 0) < λ 1, is a sricly local minimum of F +. Le ϕ 1 > 0 o be he firs eigenfuncion of (, H0 1 ), and consider γ > λ 1 such ha G + (x, ) (γ/) C for > 0. Then F + (sϕ 1 ) = s ϕ 1 dx G + (x, sϕ 1 ) dx λ 1s ϕ 1 dx γs ϕ 1 dx + C = s (λ 1 γ) ϕ 1 dx + C, as s. By he Mounain Pass Theorem, F + has a nonrivial criical poin u +. By he maximum principle, u + > 0. Therefore u + is a criical poin of he funcional F defined by (1.). Similarly, we ge a negaive criical poin u of F. Moreover, as in [10], we have rank C p (F ± C 1 0, u ± ) = δ p1. Thus, rank C p (F C 1 0, u ± ) = rank C p (F ± C 1 0, u ± ) = δ p1 for all p = 0, 1,,... By he proof of he previous heorem, here exiss a nonrivial soluion u such ha C m+1 (F, u) 0, where m 1. By Theorem 1 in [8], we have C m+1 (F C 1 0, u) = C m+1 (F, u). Therefore u is a hird nonrivial soluion. Proof of Theorem 1.3. By he proof of he previous heorem, problem (1.1) has a leas hree nonrivial soluions one is posiive, anoher is negaive and a hird soluion u is such ha C m+1 (F, u) 0, wih m. So he heorem follows of nex claim. Claim. (1.1) has a sign changing soluion w such ha C p (F, w) = δ p Z. Proof. We use he noaion as in [3]. Le P = {u X = C0() 1 : u 0}, D = P ( P ) and ϕ i he normalized eigenfuncion associaed o λ i, i = 1,. We have ϕ 1 in(p ). The main ingredien in he proof of he Claim is he negaive gradien flow ϕ of F in H, ha is, d d ϕ = F ϕ, ϕ 0 = id. We have ha ϕ (u) X for u X and ϕ induces a coninuous (local) flow on X which we coninue o denoe by ϕ. The main order relaed propery of ϕ is

16 4 F. O. V. de Paiva ha P and P are posiively invarian (by g(x, ) 0). F has he reracing propery on X (see [13]). Now he proof follows as in Theorem 3.6 in [3]. We skech i briefly for compleeness. Here we denoe by F a = {u X : F (u) a}. As k > by (ii) in Lemma 3.1 here exiss R > 0 such ha F (u) < 0 for any u span{ϕ 1, ϕ } wih u R. Now we se B = {sϕ 1 + ϕ : s R, 0 R}, B = {sϕ 1 + ϕ : s = R or {0, R}}. We have B F 0 D. Le β = max F (B) so ha (B, B) (F β D, F 0 D). Le ξ β H (F β D, F 0 D) be he image of 1 Z = H (B, B) under he homomorphism induced by he inclusion. For γ β le Z = H (B, B) H (F β D, F 0 D) j γ : H (F γ D, F 0 D) H (F β D, F 0 D) be also induced by he inclusion. Now we define Γ = {γ β : ξ β image(j γ )} and c = inf Γ. I is a criical value by he nex lemma and sandard deformaion argumens. Lemma 4.1. ξ β 0. In fac, le e 1 (P ) be he firs eigenvalue of u g (x, 0)u = λu in, on, and se X 1 = span{e 1 }, X = X1 some ρ > 0 small. This implies X. We have inf F (X B ρ ) α > 0 for (B, B) (F β D, F 0 D) (X, X \ X B ρ ). Therefore he lemma follows of ha he homeomorphism H (B, B) H (X, X \ X B ρ ) induced by inclusion is nonrivial (i is showed in [3]). As a consequence of previous lemma we have 0 / Γ because j 0 = 0. As F 0 D is a srong deformaion rerac of F γ D for γ > 0 small enough (see

17 Muliple Soluions 43 Remark 5.), we have c > 0. Clearly β Γ, hence c (0, β]. We choose ε > 0 small enough. Consider he commuaive diagram H (F c ε D, F 0 D) j H (F c+ε D, F 0 D) H (F c+ε D, F c ε D) jc ε j c+ε H (F β D, F 0 D) Since c + ε Γ here exiss ξ c+ε H (F c+ε D, F 0 D) wih j c+ε (ξ c+ε ) = ξ β. Now ξ c+ε / image(j) because c ε / Γ. Therefore he exacness of he lef column yields H (F c+ε D, F c ε D) 0. This implies ha here exiss a criical poin w such ha w / D and C (F, w) 0 (see he Appendix, below). Le w + = max{w, 0} and w = w + w. By (g) we have F (w)w +, w + = ( w + g (x, w)w+) = (w + g(x, w) g (x, w)w+) ( ) ( ) g(x, w) g(x, = w+ g (x, w) = w+ w+ ) g (x, w + ) < 0. w + w + Similarly F (w)w, w < 0. As w + and w are orhogonal, we have F (w)u, u < 0 for all u span{w +, w }, ha is, he Morse index of w is. By he Shifing Theorem we have C p (F, w) = δ p Z. Proof of Theorem 1.4. Le a < b such ha F (K) (a, b) (see [1]). Then by he hypoheses (g) and (4.4) λ k 1 g g(x, ) (x, 0) < λ k < lim < λ k+1, ± where he limis are uniform for x, I is proved in [17], ha C p (F, 0) = δ p,k 1 Z and H p (F b, F a ) = δ pk Z. Moreover, noe ha if u 0 is a nonrivial criical poin of F by (4.4), he Lemma 3. saes ha u 0 is nondegenerae and he Morse index of u 0 is k. Therefore C p (F, u 0 ) = δ pk Z. Le m he number of nonrivial criical poins of F, by he Morse ideniy, we have ( 1) k = ( 1) k 1 + m( 1) k. I follows ha m =. Then problem (1.1) has exacly wo soluions.

18 44 F. O. V. de Paiva 5. Appendix In his secion we prove ha if u 1,..., u r be all he sign changing criical poins of F a he level c, hen we can choose ε > 0 such ha H (F c+ε D, F c ε D) r C (F, u i ). Again F a = {u X : F (u) a}. Le N N be wo closed neighbourhoods of {u 1,..., u r } saisfying i=0 dis(n, N) 7 δ, δ > 0. 8 By he (C) condiion here exis consans b and ε posiive, such ha Define a smooh funcion: F (u) b for all u F c+ε \ (F c ε N ), { 1 0 < ε < min 4 δb, 1 } 8 δb. p(s) = { 0 for s / [c ε, c + ε], 1 for s [c ε, c + ε], wih 0 p(s) 1 and 0 < ε < ε/. Le A = H \ (N ) δ/8, where (N ) δ = {u H : dis(u, N ) δ}, and B = N. Le d(u) = dis(u, B) dis(u, A) + dis(u, B). We see ha 0 d(u) 1, d = 0 on N and d = 1 ouside (N ) δ/8. Define { 1 for 0 s 1, q(s) = 1/s for s 1. Denoe h(u) = d(u)p(f (u))q( F (u) ). Consider he ODE σ(τ) = h(σ(τ))f (σ(τ)), σ(0) = u 0 for all u 0 X. The global exisence and uniqueness of he flow σ() on R are known. η(u, ) = σ() wih σ(0) = u. Then η C([0, 1] X, X) saisfies Le η(1, F c+ε \ N) F c ε. This resul can be found in [9, Chaper I, Theorem 3.3]. We use i o prove he nex resul.

19 Muliple Soluions 45 Lemma 5.1. Suppose ha here are only finiely many sign changing criical poins u 1,..., u r, of F a he level c. Then we can choose ε > 0 and neighborhoods N i X \ D of u i wih he following properies: (i) N i N j = for i j, (ii) u i = N i K, (iii) F c ε N i is posiively invarian under ϕ, and (iv) here exiss T > 0 wih ϕ T (F c+ε ) F c ε N 1 N r. Proof. Le be u 0 F 1 [c ε, c+ε] X. By he (C) condiion, we have ha here is a δ > ε such ha 0 < h(u) is bounded when u F 1 [c δ, c + δ] H. Le ω(τ, u 0 ) = τ 0 h(η(ζ, u 0 )) dζ, τ [0, 1], le = ω(τ, u 0 ): [0, 1] [0, ), and le ϕ(, u 0 ) = η(τ, u 0 ). Then dϕ d = dη dτ dτ d = F (η(τ, u 0 )) = F (ϕ(, u 0 )). Now we choose he (N i ) s saisfying (i) (iii), ε as in above resul and we define T = max{ω(1, u 0 ) : u 0 F 1 [c ε, c + ε] X} <. Hence, by he previous resul, we have and using (iii) we have (iv). ϕ T (F c+ε \ N) F c ε, Seing N = N 1... N r properies (iii) and (iv), in he above lemma, imply ha F c ε N D is a srong deformaion rerac of F c+ε D, hence H (F c ε N D, F c ε D) H (F c+ε D, F c ε D). The excision propery of homology implies H (N, N F c ε ) H (F c ε N, F c ε ) H (F c ε N D, F c ε D). Now properies (i) and (ii) yield H (N, N F c ε ) How we wan o prove. r H (N i, N i F c ε ) i=0 r C (F, u i ). Remark 5.. The same idea, in he Lemma 5.1, can be used o show ha F 0 D is a srong deformaion rerac of F γ D for γ > 0 small enough. In fac, we can prove ha he flow used in [1] have he same orbis of he flow ϕ. i=0

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21 Muliple Soluions 47 [3] J. Su, Semilinear ellipic boundary value problems wih double resonance beween wo consecuive eigenvalues, Nonlinear Anal. 48 (00), [4] W. Zou, Muliple soluions for ellipic equaions wih resonance, Nonlinear Anal. 48 (00), [5] W. Zou and J. Q. Liu, Muliple soluions for resonan ellipic equaions via local linking heory and Morse heory, J. Differenial Equaions 170 (001), Manuscrip received Ocober 10, 00 Francisco O. V. de Paiva IMEEC, UNICAMP Caixa Posal Campinas-SP, BRAZIL address: TMNA : Volume N o