TRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL

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1 GLASNIK MATEMATIČKI Vol , TRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL p-laplacian Haihen Lü, Donal O Regan and Ravi P. Agarwal Academy of Mathematic and Sytem Science, Beijing, China, National Univerity of Ireland, Ireland and Florida Intitute of Technology, USA Abtract. We give condition on f involving pair of lower and upper olution which lead to the exitence of at leat three olution to the two point boundary value problem u p u = q t f t, u, u on, 1, u = u 1 =. 1. Introduction In thi paper we conider a two point boundary value problem for the one-dimenional p Laplace equation of the form ϕ p u = q t f t, u, u, < t < 1, u = u 1 = ; here ϕ p = p, p > 1, and we aume the following two condition hold: H1 q C, 1 with q > on, 1 and q d <, and H f : [, 1] R R i continuou. By a olution of we mean a function u C 1 [, 1], with ϕ p u C 1, 1, atifying 1.1 on, 1 and u = u1 =. In thi paper we aume there exit two lower olution α 1, α and two upper olution β 1, β for problem 1.1 and 1. atifying α 1 α, β 1 β and we how that there are three olution. For the pecial cae f t, u, u = f u we give growth condition on f which lead to the exitence of three poitive Mathematic Subject Claification. 34B15. Key word and phrae. Lower and upper olution, three olution, p Laplace equation, Nagumo condition. 73

2 74 H. LÜ, D. O REGAN AND R. P. AGARWAL olution. In [1], J. Henderon and H. B. Thompon conidered with p =. In thi paper C k J will denote the pace of function f : J R which are k time continuouly differentiable. For u C [, 1], u = max t [,1] u t, while for u C 1 [, 1], u = max { u, u }. Definition 1.1. A function α C 1 [, 1], ϕ p α C 1, 1 will be called a lower olution of if ϕ p α q t f t, α t, α t for t, 1, with α, α 1. A function β C 1 [, 1], ϕ p β C 1, 1 i a upper olution of if the revere inequalitie hold. Definition 1.. We ay that f atifie a Nagumo condition relative to the pair α and β, with α, β C [, 1], α β in [, 1], if there exit a function Ψ : [,, continuou, uch that 1.3 f t, y, z Ψ z for all t, y, z E, where E = { t, y, z [, 1] R : α t y β t }, and alo that 1.4 here ϕ pv du Ψ p u > q t dt; v = max { β α 1, β 1 α }.. General Reult Theorem.1. Suppoe H1 and H are atified. Aume that there exit two lower olution α 1 and α and two upper olution β 1 and β for problem atifying i α 1 α β, ii α 1 β 1 β, iii α β 1, iv if u i a olution of with u α, then u > α on, 1, and v if u i a olution of with u β 1, then u < β 1 on, 1. If f atifie the Berntern-Nagumo condition with repect to α 1, β, then problem ha at leat three olution u 1, u and u 3 atifying α 1 u 1 β 1, α u β, and u 3 β 1 and u 3 α. Suppoe that hypothee H1, H and the Nagumo condition relative to a lower olution α 1 and upper olution β are atified. We tart with the contruction of the modified problem. Define P αβ t, x = max {α t, min {x, β t}} for all x R. One can find the next reult, with it proof, in [5].

3 TRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL p-laplacian 75 and Lemma.. For each u C 1 [, 1] the next two propertie hold: a dp αβ t, u t exit for a.e. t [, 1], and dt b if u, u m C 1 [, 1] and u m u in C 1 [, 1] then d dt P αβ t, u m t d dt P αβ t, u t, for a.e. t [, 1]. From Definition 1., we can find a real number, L >, uch that v < L, L < α 1 t, β t < L for all t [, 1] ϕpl ϕ pv du Ψ p u > q t dt. We conider the following modified problem, ϕ p u = q t k t, u, d.1 dt P α 1β t, u t, < t < 1,. with u = u 1 =, k t, x, y = f t, P α1β t, x, h y + tanh x P α1β t, x, where h i defined by h y = max { L, min {y, L}} for all y R. Thu k i a continuou function on [, 1] R and atifie.3 k t, x, y Ψ y + π, for y L, and.4 k t, x, y M, for t, x, y [, 1] R, for ome contant M. Moreover, we may chooe M o that α 1, β < M. Firt, we how that every olution of.1. i a olution Lemma.3. If u i a olution of.1., then u [α 1, β ]. Proof. We prove α 1 t u t for t [, 1]. Similar reaoning how u t β t for t [, 1]. By definition of α 1 and β we have that α 1 u β and α 1 1 u 1 β 1. If there exit t, 1 uch that u t α 1 t = min t [,1] {u α 1 t} <,

4 76 H. LÜ, D. O REGAN AND R. P. AGARWAL then, ince u α 1 C 1 [, 1], we have u α 1 t =. Furthermore, there exit t 1 < t < t 1 uch that u < α 1 in t 1, t and u α 1 t 1 = u α 1 t =. Thu, ϕ p u t ϕ p α 1 t q t f t, α 1 t, α 1 t +q t tanh [u t α 1 t] q t f t, α 1 t, α 1 t = q t tanh [u t α 1 t] <, for all t t 1, t. A a reult ϕ p u t ϕ p α 1 t < ϕ p u t ϕ p α 1 t = for all t t 1, t, o u t < α 1 t for all t t, t. Thu u α 1 t < u α 1 t <, which i a contradiction. Lemma.4. If u i a olution of.1. then L < u t < L for every t [, 1]. Proof. Let u C 1 [, 1] be a olution of.1.. From Lemma.3 we have u [α 1, β ], and o ϕ p u t = q t f t, u t, h u t for t, 1. By the mean-value theorem, there exit t, 1 with and a a reult u t = u 1 u L < v α 1 1 β u t β 1 α 1 v < L. Let v = u t. Suppoe that there exit a point in the interval [, 1] for which u > L or u < L. From the continuity of u we can chooe t 1 [, 1] uch that one of the following ituation hold: i u t = v, u t 1 = L and v u t L for all t t, t 1, ii u t 1 = L, u t = v and v u t L for all t t 1, t, iii u t = v, u t 1 = L and L u t v for all t t, t 1, and iv u t 1 = L, u t = v and L u t v for all t t 1, t. Without lo of generality, uppoe L v u t L for all t t, t 1. Then ϕ p u t = q t f t, u t, h u t = q t f t, u t, u t for t t, t 1,

5 TRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL p-laplacian 77 and o ϕ p u t = q t f t, u t, u t q t Ψ u t for t t, t 1. A a reult ϕpl ϕ pv du Ψ p u = t1 t t1 t ϕp u t Ψ u dt t q t dt. Note alo that p for [ϕ p v, ϕ p L], o we have v v and thu ϕ p v ϕ p v, which lead a contradiction. ϕpl ϕ pv du Ψ p u > ϕpl ϕ pv q t dt, p u Ψ p u du Proof of the Theorem.1. From Lemma.3-.4 it i enough to how.1. ha three olution a decribed in the tatement of Theorem.1. Solving.1. i equivalent to finding a u C 1 [, 1] which atifie.5 u t = t p p A u q τ k u τ dτ d, where k u τ k τ, u, d dτ P α 1β τ, u τ for a.e. τ [, 1], and A u atifie.6 A u q τ k u τ dτ d =. The argument in [] guarantee that A u exit and i unique for u C 1 [, 1]. Now define the following operator T : C 1 [, 1] C 1 [, 1] here u C 1 [, 1] and t [, 1] by.7 T u t = t p A u q τ k u τ dτ d, where A u atifie.6. We claim that T : C 1 [, 1] C 1 [, 1] i continuou. Suppoe u n u in C 1 [, 1]. Let A un correpond to u n and A u correpond to u, and we will now

6 78 H. LÜ, D. O REGAN AND R. P. AGARWAL how that lim n A un = A u. We know A un.8 p p A u q τ k un τ dτ d q τ k u τ dτ d =. The mean value theorem implie that there exit η n, 1 uch that.9 p A un q τ k un τ dτ p A u q τ k u τ dτ =, η n η n and o.1 A un A u = η n q τ k un τ k u τ dτ. On the other hand, ince u n u in C 1 [, 1] and k i a continuou function we have from Lemma. that k un t k u t for a.e. t [, 1], o.4 and the dominated convergence theorem yield Moreover, and o.11 lim n qk un qk u in L 1, 1. q τ k un τ k u τ dτ η n q k un k u L 1 for all n N, Thi together with.1 yield Furthermore, A un Alo ince A un t η n q τ k un τ k u τ dτ =. q τ k un τ dτ A u t lim A u n = A u. n q τ k un τ dτ A u t q τ k u τ dτ for all t [, 1]. t q τ k u τ dτ A un A u + q k un k u L 1 for all t [, 1],

7 TRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL p-laplacian 79 the convergence i uniform in [, 1]. In addition the uniform continuity of p on compact interval yield and a a reult T u n T u uniformly on [, 1] T u n T u uniformly on [, 1]. We next claim that T C 1 [, 1] i a relatively compact et in C 1 [, 1]. We firt how that there exit a contant N with Since A u N for all u C 1 [, 1]. p A u q τ k u τ dτ d =, the Mean Value theorem for integral implie that there exit ξ [, 1] with A u q τ k u τ dτ =. Conequently, which implie p A u = ξ ξ A u M q τ k u τ dτ, q τ τ N where M i defined in.4. Next we how that T C 1 [, 1] i bounded. Thi follow from the following inequalitie: or where T u t J = max and J d and T u t J for t [, 1] p Q T u t p Q for t [, 1], { p N M Q = N + M q u du, ϕ 1 p q u du. N + M } q u du We next how the equicontinuity of T C 1 [, 1] on [, 1]. For u C 1 [, 1] and t, [, 1] we have t T u t T u J v dv.

8 8 H. LÜ, D. O REGAN AND R. P. AGARWAL Finally, to ee that T C 1 [, 1] { = y : y T C 1 [, 1] } i equicontinuou on [, 1], we ue the fact that p i uniformly continuou on [ Q, Q] and.4. By Arzela-Acoli Theorem, T : C 1 [, 1] C 1 [, 1] i compact. Let { } Ω = u C 1 [, 1] : u < M + L + J + J d. It i immediate from the argument above that T Ω Ω. Thu Let d I T, Ω, = 1. Ω α = {u Ω : u > α on, 1} and Ω β1 = {u Ω : u < β 1 on, 1}. Since α β 1, α > M, and β 1 < M i.e. we chooe M with α, β 1 < M it follow that Ω β1 Ω α, Ω β1 Ω α =, and Ω\{Ω β1 Ω α }. By aumption iv and v, there are no olution in Ω β1 Ω α. Thu d I T, Ω, = d I T, Ω\{Ω β1 Ω α }, +d I T, Ω α, + d I T, Ω β1,. We how that d I T, Ω α, = d I T, Ω β1, = 1. Then d I T, Ω\{Ω β1 Ω α }, = 1, and there are olution in Ω\{Ω β1 Ω α }, Ω α and Ω β1, a required. We how d I T, Ω α, = 1. The proof that d I T, Ω β1, = 1 i imilar and hence omitted. We define I W, the extenion to Ω of the retriction of I T to Ω α a follow. Let w t, x, y = f t, P αβ t, x, h y + tanh x P αβ t, x, where P αβ replace α 1 by α and h are defined previouly. Thu w i a continuou function on [, 1] R and atifie w t, x, y Ψ y + π, for y L, and w t, x, y M 1, for t, x, y [, 1] R, for ome contant M 1. Moreover, we may chooe M 1 o that α, β < M 1. Conider the problem: ϕ p u = q t w t, u, d.1 dt P α β t, u t, < t < 1,.13 u = u 1 =.

9 TRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL p-laplacian 81 Solving.1.13 i equivalent to finding a u C 1 [, 1] which atifie t u t = B u q τ w u τ dτ d, p p where w u τ w τ, u, d dt P α β t, u t for a.e. τ [, 1], and B u atifie.14 B u q τ w u τ dτ d =. A before B u exit and i unique for u C 1 [, 1]. Now define the following operator W : C 1 [, 1] C 1 [, 1] here u C 1 [, 1] and t [, 1] by W u t = t p B u q τ w u τ dτ d, where B u atifie.14. Again it i eay to check from a previou argument and v that u i a olution of.1.13 if u Ω α and W u = u note W : C 1 [, 1] C 1 [, 1] i compact. Thu d I W, Ω\Ω α, =. Moreover it i eay to ee that W Ω Ω. By aumption iv and v, there are no olution in Ω α Ω β1. Thu d I T, Ω α, = d I W, Ω α, = d I W, Ω\Ω α, + d I W, Ω α, = d I W, Ω, = 1. Thu there are three olution, a required. A light modification of the argument in Theorem.1 yield the next reult. Theorem.5. Suppoe H1 and H atified. Aume that there exit two lower olution α 1 and α and two upper olution β 1 and β for problem atifying i α 1 < α β, ii α 1 β 1 < β, iii there exit < ε < min t [,1] {α t α 1 t, β t β 1 t} uch that all ε, ε], the function α t ε and β 1 + ε are, repectively, lower and upper olution of , and iv α ε β 1 + ε. If f atifie the Berntern-Nagumo condition with repect to α 1, β, then problem ha at leat three olution u 1, u and u 3 atifying α 1 u 1 β 1, α u β, and u 3 β 1 and u 3 α.

10 8 H. LÜ, D. O REGAN AND R. P. AGARWAL Proof. In the proof of Theorem.1, define Ω α = {u Ω : u > α ε on, 1} and Ω β1 = {u Ω : u < β 1 + ε on, 1}, where Ω i defined in Theorem Conider the problem ϕ p u + f u =, for all t [, 1], u = u 1 =. Theorem.6. Aume there exit real number a, b, c with < a < b, < a < c and { c > max b + p 1 b ρ 1 p } 1 p M e Mb, h e and uppoe there i a continuou nonnegative function f uch that i f y < a M, y [, a], ii f y ρ b M, y [b, b + p ρ 1 1 e p ], and iii f y c M, y [, c]. Then problem ha at leat three olution u 1, u and u 3 atifying u 1 < a, α u, and u 3 > a and u 3 α, where α i given by 1 1 b M t, b 1 b ρ p M α t = b 1 b ρ p M 1 1 [ 1 [ t 1 for all t [, e], t p 1 e p ], for all t [ e, ] 1, p 1 e p ], for all t [ 1, 1 e], b M 1 t, for all t [1 e, 1]; here e = p, ρ = b 1 e and M = max t [,1] h t where [ 1 p p 1 h t = t p ], for t [, ] 1, [ 1 p p t 1 p ], for t [ 1, 1]. Proof. It i eaily proved that h atifie ϕ p u + 1 =, t [, 1], u = u 1 =. Let α 1 t, β 1 t = a M h t, and β t = c M h t for t 1, and let α be a above. It i eay to check that β t c and ϕ p β = c M for t 1. It follow that β 1 i a trict upper olution and β i an upper olution for problem with β 1 < β on, 1.

11 TRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL p-laplacian 83 Since α i ymmetric in t = 1, α e = α 1 e, and α e = α 1 e, o it follow that α i in C 1 [, 1]. Moreover α atifie ϕ p α = f α on, e 1 e, 1, and ϕ p α = ρ b M f α on e, 1 e, o α i a lower olution for problem = b + b ρ 1 1 p M e p > b > a = β 1 1. Moreover α 1 Alo ince α = 1 1 b M 1 1 c M = β, ch e α e = b M = β e, 1 α = b + p 1 b ρ 1 p 1 p M e 1 < c = β, it follow that α < β on, 1. We how that there i no olution u of problem with u α on [, 1] and u t = α t for ome t, 1. Aume thi i fale and that there i uch a olution. Conider the cae t, e. Since u t = α t and u α and ϕ p u ϕ p α on [, e], it follow that u = α for all t [, e]. Thu = ϕ p u e = f u e = f α e = f b, a contradiction, o t /, e. Similarly t [1 e, 1 lead to the contradiction that ϕ p u 1 e =, o t / [1 e, 1. Aume that t [e, 1 e. Again u t = α t and y α and ϕ p u ρ b M = ϕp α on [e, 1 e]. Thu u = α on [e, 1 e] o that ϕ p u 1 e ρ b M < and u 1 e = 1 1 b M. It follow that u x < α x for any x 1 e, 1 e + δ for ome δ >, a contradiction. Thu u t α t for any t, 1, a required. Thu the condition of Theorem.1 are atified and there are three olution of problem.15.16, a required. Reference [1] J. Henderon and H. B. Thompon, Exitence of multiple olution for econd order boundary value problem, J. Diff. Equ. 166, [] D. O Regan, Some general exitence principle and reult for φ y = qf t, y, y, < t < 1, SIAM J. Math. Anal , [3] D. O Regan, Exitence theory for φ y = qf t, y, y < t < 1, Commun. Appl. Anal , [4] C. De Coter, Pair of poitive olution for the one-dimenional p Laplacian, Nonl. Anal , [5] M. X. Wang, A. Cabada and J. Nieto, Monotone method for nonlinear econd order periodic boundary value problem with Carathéodory function, Ann. Polon. Math , [6] Q. Yao and H. Lü, Poitive olution of one-dimenional ingular p Laplace equation, Acta Mathematica Sinica ,

12 84 H. LÜ, D. O REGAN AND R. P. AGARWAL [7] H. Lü and D. O Regan, A general exitence theorem for ingular equation ϕ p y + g t, y =, Math. Ineq. Appl. 5, H. Lü Intitute of Applied Mathematic Academy of Mathematic and Sytem Science Chinee Academy of Science Beijing 18 China D. O Regan Department of Mathematic National Univerity of Ireland Galway Ireland R. P. Agarwal Department of Mathematical Science Florida Intitute of Technology Melbourne, FL USA Received: