RADIAL POSITIVE SOLUTIONS FOR A NONPOSITONE PROBLEM IN AN ANNULUS

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1 Electonic Jounal of Diffeential Equations, Vol. 04 (04), o. 9, pp. 0. ISS: UL: o ftp ejde.math.txstate.edu ADIAL POSITIVE SOLUTIOS FO A OPOSITOE POBLEM I A AULUS SAID HAKIMI, ABDEAHIM ZETITI Abstact. The main pupose of this aticle is to pove the existence of adial positive solutions fo a nonpositone poblem in an annulus when the nonlineaity is supelinea and has moe than one zeo.. Intoduction In this aticle we study the existence of adial positive solutions fo the boundayvalue poblem u(x) = λf(u(x)) x Ω, (.) u(x) = 0 x Ω, whee λ > 0, f : [0, + ) is a continuous nonlinea function that has moe than one zeo, and Ω is the annulus: Ω = C(0,, ) = {x : < x < } ( >, 0 < < ). When f is a nondeceasing nonlineaity satisfying f(0) < 0 (the nonpositone case) and has only one zeo, poblem (.) has been studied by Acoya and Zetiti [ and by Hakimi and Zetiti in a ball when f has moe than one zeo [5. We obseve that the existence of adial positive solutions of (.) is equivalent to the existence of positive solutions of the poblem u () u () = λf(u()) < < (.) u() = u( ) = 0. Ou main objective in this aticle is to pove that the esult of existence of adial positive solutions of the poblem (.) emains valid when f has moe than one zeo and is not inceasing entiely on [0, + ); see [, Theoem.4. emak.. In this aticle, we assume (without loss of geneality) that f has exactly thee zeos. We assume that the map f : [0, + ) satisfies the following hypotheses: (F) f C ([0, + ), ) such that f has thee zeos β < β < β 3, with f (β i ) 0 fo all i {,, 3}. Moeove, f 0 on [β 3, + ). (F) f(0) < Mathematics Subject Classification. 35J5, 34B8. Key wods and phases. onpositone poblem; adial positive solutions. c 04 Texas State Univesity - San Macos. Submitted Apil, 03. Published Apil 5, 04.

2 S. HAKIMI, A. ZETITI EJDE-04/9 f(u) (F3) lim u + u = +. (F4) The function h(u) = F (u) f(u)u is bounded fom below in [0, + ), whee F (x) = x 0 f()d. emak.. We obseve that ou aguments also wok in the case Ω = B(O, ), impoving slightly the esults in [5. In fact in [5, besides imposing that f is inceasing, we need (F), (F), (F3) and that Fo some k (0, ), lim d + ( d f(d) ) / ( F (kd) df(d)) = +. On the othe hand, it is clea that ou hypothesis (F4) is moe geneal than this assumption. Fo a nonexistence esult of positive solutions fo supelineaities satisfying (F), (F) and (F3) see [6. Also see [3 fo existence and nonexistence of positive solutions fo a class of supelinea semipositone systems, and [4 fo existence and multiplicity esults fo semipositone poblems.. Main esult In this section, we give the main esult in this wok. Moe pecisely we shall pove the following theoem. Theoem.. Assume that the hypotheses (F) (F4) ae satisfied. Then thee exists a positive eal numbe λ such that if λ < λ, poblem (.) has at least one adial positive solution. To pove Theoem., we need the next fou technical lemmas. The fist lemma assues the existence of a unique solution u(., d, λ) of (.) in [, + ) fo all λ, d > 0. The thee last lemmas concen the behaviou of the solution of (.). emak.. In this aticle we follow the wok of Acoya and Zetiti [, and we note that the poofs of Lemmas.4 and.7 ae analogous with those of [, Lemmas. and.3. On the othe hand, the poofs of the second and thid lemmas ae diffeent fom that of [, Lemma. and.. This is so because ou f has moe than one zeo. So we apply the Shooting method. Fo this we conside the auxiliay bounday-value poblem u () u () = λf(u()), > (.) u() = 0, u () = d, whee d is the paamete of Shooting method. emak.3. Fo suitable d, poblem (.) has a solution u := u(., d, λ) such that u > 0 on (, ) and u( ) = 0. So, such solution u of (.) is also a positive solution of (.). In this sequel, we suppose that the nonlineaity f C ([0, + )) is always extended to by f (,0) f(0). Lemma.4. Let λ, d > 0 and f C ([0, + )) a function which is bounded fom below. Then poblem (.) has a unique solution u(., d, λ) defined in [, + ), In addition, fo evey d > 0 thee exist M = M(d) > 0 and λ = λ(d) > 0 such that max u(, d, λ) M, [, b λ (0, λ(d)).

3 EJDE-04/9 ADIAL POSITIVE SOLUTIOS 3 Poof. The poof of the existence is given in two steps. In fist, we show the existence and uniqueness of a local solution of (.); i.e, the existence a ε = ε(d, λ) > 0 such that (.) has a unique solution on [, + ε. In the second step we pove that this unique solution can be extended to [, + ). Step : (Local solution). Conside the poblem u () u () = λf(u()), > u( ) = a, u ( ) = b, (.) whee. Let u be a solution of (.). Multiplying the equation by and using the initial conditions, we obtain fom which u satisfies u() = a + b u () = { b λ ( ) λ s f(u(s))ds }. (.3) t s f(u(s))ds dt. (.4) Convesely, if u is a continuous function satisfying (.4), then u is a solution of (.). Hence, to pove the existence and uniqueness of a solution u of (.) defined in some inteval [, + ε, it is sufficient to show the existence of a unique fixed point of the opeato T defined on X (the Banach space of the eal continuous functions on [, + ε with the unifom nom), whee (T v)() = a+ b T : X = C([, + ε, ) X ( v T v, ) λ t s f(v(s))ds dt, (.5) fo all [, + ε and v X. To check this, Let δ > 0 such that δ > a and B(0, δ) = {u X : u δ}. Fo all u, v B(0, δ), we have (T u T v)() = λ t s {f(v(s)) f(u(s))}ds dt, then t (T u T v)() λ t [ λ Howeve, t t [ s dsdt = t s sup ζ (0,δ s ds t [t dt t dt f (ζ) v(s) u(s) dsdt dt sup ζ (0,δ dt t f (ζ) u v.

4 4 S. HAKIMI, A. ZETITI EJDE-04/9 theefoe, Hence Similaly, = ( ) ( ( ) ) ( ) = + ( ). ( ) ( + ε), because [, + ε = ε( + ε) ; T u T v ε( + ε) λ sup f (ζ) u v ζ [0,δ T u T v λ ε( + ε) λ sup f (ζ) u v. ζ [0,δ sup f (ζ) ε( + ε) u v. (.6) ζ (0,δ ( T u a + b ) ( + ε) + λ sup f(ζ) ε( + ε). (.7) ζ [0,δ ow, by (.6) and (.7), we can choose ε = ε(δ) > 0 (depending on δ) sufficiently small such that T is a contaction fom B(0, δ) to B(0, δ). Consequently, T has a fixed point u in B(0, δ). The fixed point u is unique in X fo a δ as lage as we wanted. Step : Let u(.) = u(., d, λ) be the unique solution of (.) (we take a = 0, b = d and = in (.)), and denote by [, (d, λ)) its maximal domain. We shall pove by contadiction that (d, λ) = +. Fo it, assume := (d, λ) < +. u is bounded on [, ). In fact, using (.4) and that f is bounded fom below, we have d d = u() + λ u() + λ ( ) t inf f(ξ) ξ [0,+ ) s f(u(s))ds dt t s ds dt, [, ), then, thee exists K > 0 such that u() K fo all [, ). On the othe hand, using again (.4), we obtain ( u() d ) λ max f(ξ) ξ [0,K K, [, ), fo convenient K > 0. Hence u is bounded. t s ds dt

5 EJDE-04/9 ADIAL POSITIVE SOLUTIOS 5 By using this and (.3) and (.4), we deduce that {u( n )} and {u ( n )} ae the Cauchy sequence fo all sequence ( n ) [, ) conveging to. This is equivalent to the existence of the finite limits ow, conside the poblem lim u() = a and lim u () = b. v () v () = λf(v()), v( ) = a, v ( ) = b < (.8) and by step, we deduce the existence of a positive numbe ε > 0 and a solution v of this poblem in [, + ε. It is easy to see that { u(), if < w() = v(), if + ε, is a solution of (.) in [, + ε which is a contadiction, so = +. To pove the second pat of the lemma, we conside the opeato T defined by (.5) on X 0 = C([,, ) with =, a = 0 and b = d. Taking M = δ > d and λ(d) = min { M M max ξ [0,M f(ξ), M max ξ [0,M f (ξ) } with M = e t s ds dt. By (.6) and (.7), we deduce that T is a contaction fom B(0, M, X 0 ) into B(0, M, X 0 ), whee B(0, M, X 0 ) = {u X 0 : max u() M}. [, b So, the unique fixed point of T belongs to B(0, M, X 0 ). The lemma is poved. Lemma.5. Assume (F), (F) and let d 0 > 0. Then thee exists λ = λ (d 0 ) > 0 such that the unique solution u(, d 0, λ) of (.) satisfies Poof. Fo λ > 0, we conside the set u(, d 0, λ) > 0, (,, λ (0, λ ). Ψ = { (, ) : u(.) = u(., d 0, λ)is nondeceasing in (, )}. Since u () = d 0 > 0, Ψ is nonempty, and clealy bounded fom above. Let = sup Ψ (which depends on λ). We have two cases: Case. If =, the poof is complete. Case. If <, we shall pove u(.) = u(., d 0, λ) > 0, fo all (, fo all λ sufficiently small. In ode to show it, assume that <. Then u ( ) = 0, and since u () = [ d 0 λ s f(u(s))ds, then u( ) > β. Hence the set Γ = { [, : u(t) β and u (t) 0, t [, } is nonempty and bounded fom above. Let = sup Γ >. We shall pove that fo λ sufficiently small =. We obseve that u () 0 fo all Γ, then

6 6 S. HAKIMI, A. ZETITI EJDE-04/9 u() u( ), fo all [,. Theefoe, by the mean value theoem, thee exists c (, ) such that u( ) = u( ) + u (c)( ), but then u (c) = λ c u( ) > u( ) λ c t f(u(t))dt, sup f(ζ) ( ). [β,u( ) If M = M(d 0 ) > 0 and λ(d 0 ) > 0 (defined in Lemma.4), then β < u( ) M, λ (0, λ(d 0 )). Let K = K(d 0 ) > 0 such that f(ζ) < K(ζ β ) fo all ζ (β, M. We deduce that λk u( ) > u( ) ( )(u( ) β ), λ (0, λ(d 0 )), Thus, if λ (0, λ ) with λ = min{λ(d 0 ), K( b ) b } we have u( ) > β, which implies that =. Lemma.6. Assume (F) (F3). Let λ > 0. Then (i) lim d + (d, λ) = (ii) lim d + u(, d, λ) = + Poof. If (i) is not tue, then thee exists ε > 0 so that fo all n thee exists d n such that (d n, λ) ε, fom which (d n, λ) + ε (because (d n, λ) ), then thee exists 0 (, ) and a sequence (d n ) (0, + ) conveging to such that u n := u(., d n, λ) satisfies Let = +0. By the equality u n () = d n u n () > 0, u n() 0, (, 0, n. ( ) λ t s f(u n (s))ds dt, we obseve that (u n ()) is unbounded. Passing to a subsequence of (d n ), if it is necessay, we can suppose lim n + u n () = +. ow, conside M n = inf { f(u n ()) u n () : (, 0 ) }. By (F3), lim n + M n = +. Let n 0 such that λm n0 > µ 3 whee µ 3 is the thid eigenvalue of [ d d + d d in (, 0) with Diichlet bounday conditions. We take a nonzeo eigenfunction φ 3 associated to µ 3 ; i.e., φ 3() + φ 3() + µ 3 φ 3 () = 0, < < 0 φ 3 () = 0 = φ 3 ( 0 ).

7 EJDE-04/9 ADIAL POSITIVE SOLUTIOS 7 Since φ 3 has two zeos in (, 0 ), we deduce fom the Stum compaison Theoem [7 that u n0 has at least one zeo in (, 0 ). Which is a contadiction (because u n () > 0 fo all (, 0 and all n ). (ii) Let be the same numbe as in the poof of lemma.5. we have u ( ) = 0. Howeve, u ( ) = [ d λ t f(u(t))dt, then Hence d = λ t f(u(t))dt. lim u(, d, λ) = +. d + Lemma.7. Assume (F) (F4) and let γ be a positive numbe. Then thee exists a λ > 0 such that: (a) Fo all λ (0, λ ) the unique solution u(, d, λ) of (.) satisfies u (, d, λ) + u (, d, λ) > 0, [,, d γ. (b) Fo all λ (0, λ ), thee exists d > γ such that u(, d, λ) < 0 fo some (,. Poof. (a) Let λ, d > 0 and u(.) = u(., d, λ) the unique solution of (.). We define the auxiliay function H on [, + ) by setting H() = u () + λf (u()) + u()u (), [, + ). We can pove, as in [, 5 the next identity of Pohozaev-type: H() = t H(t)+λ t s [F (u(s)) f(u(s))u(s)ds, Taking t =, in this identity we obtain H() = d + λ s [ F (u(s)) f(u(s))u(s) ds, t [,. hence H() d ( ) + λm, (.9) whee m is a stictly negative eal such that F (u) f(u)u m fo all u, so H() γ ( ) + λm, [,, d γ. We note that m exists by (f 4 ). Hence thee exists λ > 0 such that Theefoe, H() > 0, [,, d γ, λ (0, λ ). (.0) u (, d, λ) + u (, d, λ) > 0, [,, d γ, λ (0, λ ). (b) We ague by contadiction: fix λ (0, λ ) and suppose that u(, d, λ) 0, [,, d γ.

8 8 S. HAKIMI, A. ZETITI EJDE-04/9 Choose ϱ > 0 such that thee exists a solution of ω + ω + ϱω = 0, whee ω(0) =, ω (0) = 0, is the fist zeo of ω. 4 We note (see [8) that ω() 0 and ω () < 0, fo all (0, b 4. By (F3), thee exists d 0 = d 0 (λ) > γ such that f(u) u ϱ λ, u d 0. (.) On the othe hand, let = (d, λ) and = (d, λ) be the same numbes as in the poof of Lemma.5. By Lemma.6, we can assume that = (d, λ) < + < 4 and u(, d, λ) > d 0, d d 0, the definitions of and imply u (, d, λ) 0, [,, d d 0. (.) Define v() = u( )ω( ), hence v () + v () + ϱv() = 0, fo all (, + b 4 ) with u( ) = v( ), v ( ) = 0, v( + b 4 ) = 0, v() > 0 and v () 0, fo all (, + b 4 ), thus v () + v () + ϱv() 0, (, + ), 4 if u() d 0, fo all (, + b 4 ), hence by (.) and the Stum compaison theoem (see [7), u have a zeo in (, + b 4 ). Which is a contadiction. Hence, thee exists (, + b 4 ) such that u(, d, λ) = d 0. ow, conside the enegy function E(, d, λ) = u (, d, λ) + λf (u(, d, λ)),. By (.9), (.) and the equality H() = E() + u()u (), we obtain E(, d, λ) H(, d, λ) d ( + λm hence, thee exists d = d (λ) d 0 such that Howeve, hence thus E(, d, λ) λf (d 0 ) + ), [,, ( ) d 0, [,, d d. E () = u () 0, E( ) E(), [,, [,, u () d 0 ( ), [,, d d,

9 EJDE-04/9 ADIAL POSITIVE SOLUTIOS 9 and by (.), we deduce u () d 0, [,, d d. The mean value theoem implies that thee exists a c (, + b ) such that ( u + ) u( ) = u (c). Hence ( u + ) 0. Which is a contadiction (because u ( + b ) < 0). Poof of theoem.. Let d 0 > 0. By Lemmas.5 and.7, thee exists λ > 0 such that, if λ (0, λ ) then (i) u(, d 0, λ) > 0 fo all (, (ii) u (, d, λ) + u(, d, λ) > 0 fo all [, and all d d 0, (iii) thee exist d > d 0 and (, such that u(, d, λ) < 0. Define Γ = {d d 0 u(, d, λ) > 0, (, ), d [d 0, d}. By (i), d 0 Γ then Γ is nonempty. In addition, by (iii) Γ is bounded fom above by d. Take d = sup Γ. it is clea that u(, d, λ) 0, Since d < d, we deduce (using (ii)) that [,. u(, d, λ) > 0, (, ). (.3) u(., d, λ) will be a solution seaching, if we pove u(, d, λ) = 0. Assume that u(, d, λ) > 0. Then by (.3) and the fact that u (, d, λ) = d > 0, we have that u(, d, λ) > 0, (,, d [d, d + δ, whee δ is sufficiently small. Hence d + δ Γ, which is a contadiction. Theefoe, u(, d, λ) = 0. efeences [ D. Acoya, A. Zetiti; Existence and non-existence of adially symmetic non-negative solutions fo a class of semi-positone poblems in annulus, endiconti di Mathematica, seie VII, Volume 4, oma (994), [ A. Casto,. Shivaji; onnegative solutions fo a class of nonpositone poblems, Poc. oy. Soc. Edin., 08(A)(988), pp [3 M. Chheti, P. Gig; Existence and and nonexistence of positive solutions fo a class of supelinea semipositone systems, onlinea Analysis, 7 (009), [4 D. G. Costa, H. Tehani, J. Yang; On a vaiational appoach to existence and multiplicity esults fo semi positone poblems, Electon. J. Diff. Equ., Vol. (006), o., -0. [5 Said Hakimi, Abdeahim Zetiti; adial positive solutions fo a nonpositone poblem in a ball, Eletonic Jounal of Diffeential Equations, Vol. 009(009), o. 44, pp. -6. [6 Said Hakimi, Abdeahim Zetiti; onexistence of adial positive solutions fo a nonpositone poblem, Eletonic Jounal of Diffeential Equations, Vol. 0(0), o. 6, pp. -7. [7 Hatman; Odinay Diffeential equations, Baltimoe, 973. [8 B. Gidas, W. M. i, L. ienbeg; Symmety and elated popeties via the maximum pinciple, Commun. Maths Phys., 68 (979),

10 0 S. HAKIMI, A. ZETITI EJDE-04/9 Said Hakimi Univesité Sultan Moulay Slimane, Faculté polydisciplinaie, Dépatement de Mathématiques, Béni Mellal, Moocco addess: h saidhakimi@yahoo.f Abdeahim Zetiti Univesité Abdelmalek Essaadi, Faculté des sciences, Dépatement de Mathématiques, BP, Tétouan, Moocco addess: zetitia@hotmail.com

Existence and Uniqueness of Positive Radial Solutions for a Class of Quasilinear Elliptic Systems

JOURAL OF PARTIAL DIFFERETIAL EQUATIOS J Pat Diff Eq, Vol 8, o 4, pp 37-38 doi: 48/jpdev8n46 Decembe 5 Existence and Uniqueness of Positive Radial Solutions fo a Class of Quasilinea Elliptic Systems LI