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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 791762, 18 pages doi:10.1155/2008/791762 Research Article Existence of Solutions for a Class of Weighted p t -Laplacian System Multipoint Boundary Value Problems Qihu Zhang,1, 2, Zheimei Qiu,2 and Xiaopin Liu2 Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China School of Mathematical Science, Xuzhou Normal University, Xuzhou, Jiangsu 221116, China College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710062, China Correspondence should be addressed to Zheimei Qiu, zhimeiqiu@yahoo.com.cn Received 12 June 2008; Accepted 22 October 2008 Recommended by Alberto Cabada This paper investigates the existence of solutions for weighted p t -Laplacian system multipoint boundary value problems When the nonlinearity term f t, ·, · satisfies sub-p− −1 growth condition or general growth condition, we give the existence of solutions via Leray-Schauder degree Copyright q 2008 Qihu Zhang et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Introduction In this paper, we consider the existence of solutions for the following weighted p t -Laplacian system: −Δp t ,w t u δf t, u, w t 1/ p t −1 u 0, t ∈ 0, , 1.1 with the following multipoint boundary value condition: m−2 βi u ηi u i m−2 e0 , u αi u ξi e1 , 1.2 i where p ∈ C 0, , R and p t > 1, −Δp t ,w t u − w t |u |p t −2 u is called the weighted p t -Laplacian; w ∈ C 0, , R satisfies < w t , for all t ∈ 0, , and w t −1/ p t −1 ∈ L1 0, ; < η1 < · · · < ηm−2 < 1, < ξ1 < · · · < ξm−2 < 1; αi ≥ 0, βi ≥ i 1, , m − , and < m−2 αi < 1, < m−2 βi < 1; e0 , e1 ∈ RN ; δ is a positive parameter i i Journal of Inequalities and Applications The study of differential equations and variational problems with variable exponent growth conditions is a new and interesting topic Many results have been obtained on these problems, for example, 1–14 We refer to 2, 15, 16 the applied background on these problems If w t ≡ and p t ≡ p a constant , −Δp t ,w t is the well-known p-Laplacian If p t is a general function, −Δp t ,w t represents a nonhomogeneity and possesses more nonlinearity, thus −Δp t ,w t is more complicated than −Δp We have the following examples If Ω ⊂ RN is a bounded domain, the Rayleigh quotient λp x 1,p x u∈W0 1/p x |∇u|p x dx Ω inf Ω \{0} Ω 1/p x |u|p x dx 1.3 is zero in general, and only under some special conditions λp x > see , but the fact that λp > is very important in the study of p-Laplacian problems If w t ≡ and p t ≡ p a constant and −Δp u > 0, then u is concave, this property is used extensively in the study of one-dimensional p-Laplacian problems, but it is invalid for −Δp t ,1 It is another difference on −Δp and −Δp t ,1 On the existence of solutions of the following typical −Δp x ,1 problem: − u p x −2 |u|q x −2 u u u C, x ∈ Ω ⊂ RN , 1.4 on ∂Ω, because of the nonhomogeneity of −Δp x ,1 , if maxx∈Ω q x < minx∈Ω p x , then the corresponding functional is coercive; if maxx∈Ω p x < minx∈Ω q x , then the corresponding functional satisfies Palais-Smale condition see 4, 7, 12 If minx∈Ω p x ≤ q x ≤ maxx∈Ω p x , we can see that the corresponding functional is neither coercive nor satisfying Palais-Smale conditions, the results on this case are rare There are many results on the existence of solutions for p-Laplacian equation with multipoint boundary value conditions see 17–20 On the existence of solutions for p x Laplacian systems boundary value problems, we refer to 5, 7, 10, 11 But results on the existence of solutions for weighted p t -Laplacian systems with multipoint boundary value conditions are rare In this paper, when p t is a general function, we investigate the existence of solutions for weighted p t -Laplacian systems with multipoint boundary value conditions Moreover, the case of mint∈ 0,1 p t ≤ q t ≤ maxt∈ 0,1 p t has been discussed Let N ≥ and I 0, , the function f f , , f N : I × RN × RN → RN is assumed to be Caratheodory, by this we mean the following: i for almost every t ∈ I the function f t, ·, · is continuous; ii for each x, y ∈ RN × RN the function f ·, x, y is measurable on I; iii for each R > there is a βR ∈ L1 I, R , such that for almost every t ∈ I and every x, y ∈ RN × RN with |x| ≤ R, |y| ≤ R, one has f t, x, y ≤ βR t 1.5 Qihu Zhang et al Throughout the paper, we denote w u w u p −2 p −2 lim w r u u p r −2 r →0 lim− w r u u p r −2 r →1 u r , 1.6 u r The inner product in RN will be denoted by ·, · , |·| will denote the absolute value C I, RN , C1 {u ∈ C | u ∈ and the Euclidean norm on RN For N ≥ 1, we set C p t −2 p t −2 N u t , and limt → 1− w t |u | u t exist} For any u t C 0, , R , limt → w t |u | N i 1/2 supt∈ 0,1 |ui t |, u , and u u1 t , , uN t , we denote |ui |0 i |u |0 w t 1/ p t −1 u Spaces C and C1 will be equipped with the norm · and · , u respectively Then C, · and C1 , · are Banach spaces We say a function u : I → RN is a solution of 1.1 if u ∈ C1 with w t |u |p t −2 u absolutely continuous on 0, , which satisfies 1.1 a.e on I In this paper, we always use Ci to denote positive constants, if it cannot lead to confusion Denote z− z t , t∈I z max z t , t∈I for any z ∈ C I, R 1.7 We say f satisfies sub-p− − growth condition, if f satisfies lim |u| |v| → ∞ f t, u, v |u| |v| q t −1 0, for t ∈ I uniformly, 1.8 where q t ∈ C I, R and < q− ≤ q < p− We say that f satisfies general growth condition, if we not know whether f satisfies sub-p− − growth condition or not We will discuss the existence of solutions of 1.1 - 1.2 in the following two cases: i f satisfies sub-p− − growth condition; ii f satisfies general growth condition This paper is divided into four sections In the second section, we will some preparation In the third section, we will discuss the existence of solutions of 1.1 - 1.2 , when f satisfies sub-p− − growth condition Finally, in Section 4, we will discuss the existence of solutions of 1.1 - 1.2 , when f satisfies general growth condition Preliminary For any t, x ∈ I × RN , denote ϕ t, x |x|p t −2 x Obviously, ϕ has the following properties Lemma 2.1 see ϕ is a continuous function and satisfies the following: i for any t ∈ 0, , ϕ t, · is strictly monotone, that is, ϕ t, x1 − ϕ t, x2 , x1 − x2 > 0, for any x1 , x2 ∈ RN , x1 / x2 ; 2.1 Journal of Inequalities and Applications ii there exists a function ρ : 0, ∞ → 0, ∞ , ρ s → ∞ as s → ∞, such that ϕ t, x , x ≥ ρ |x| |x|, ∀x ∈ RN 2.2 It is well known that ϕ t, · is a homeomorphism from RN to RN for any fixed t ∈ 0, For any t ∈ I, denote by ϕ−1 t, · the inverse operator of ϕ t, · , then ϕ−1 t, x |x| 2−p t / p t −1 for x ∈ RN \ {0}, ϕ−1 t, x, 2.3 It is clear that ϕ−1 t, · is continuous and sends bounded sets to bounded sets Let us now consider the following problem with boundary value condition 1.2 : w t ϕ t, u t g t , 2.4 where g ∈ L1 If u is a solution of 2.4 with 1.2 , by integrating 2.4 from to t, we find that t w t ϕ t, u t w ϕ 0, u g s ds 2.5 Denote a w ϕ 0, u It is easy to see that a is dependent on g t Define operator t g s ds By solving for u in 2.5 and integrating, we find F : L → C as F g t u t m−2 i βi u From u u ξi −1 m−2 i αi ϕ t, w t 1− m−2 i αi u ξi t, w t a F g t 2.6 e0 , we have ηi −1 m−2 i βi ϕ u From u ηi −1 F ϕ−1 t, w t u −1 a F g t dt e0 m−2 i βi 2.7 e1 , we obtain −1 a dt − F g t 1− −1 ϕ t, w t −1 a F g t dt e1 m−2 i αi 2.8 From 2.7 and 2.8 , we have ηi −1 m−2 i βi ϕ t, w t −1 a 1− ξi −1 m−2 i αi ϕ t, w t F g t dt e0 m−2 i βi −1 a F g t 1− dt − −1 ϕ m−2 i αi t, w t −1 a F g t dt e1 2.9 Qihu Zhang et al For fixed h ∈ C, we denote ηi −1 m−2 i βi ϕ Λh a −1 t, w t 1− − ξi −1 m−2 i αi ϕ a h t dt e0 m−2 i βi −1 t, w t a −1 ϕ dt − h t t, w t −1 a h t dt e1 m−2 i αi 1− 2.10 Throughout the paper, we denote E −1/ p t −1 w t dt Lemma 2.2 The function Λh · has the following properties: i for any fixed h ∈ C, the equation Λh a 2.11 has a unique solution a h ∈ RN ; ii the function a : C → RN , defined in (i), is continuous and sends bounded sets to bounded sets Moreover, a h ≤ 3N E 1− m−2 i βi E 1− E p −1 m−2 i αi · h E 2N p e0 e1 p# −1 , 2.12 where the notation Mp # −1 means Mp # −1 ⎧ ⎨Mp ⎩Mp −1 − , M>1 −1 , M ≤ 2.13 Proof i It is easy to see that Λh a ηi −1 m−2 i βi ϕ −1 m−2 i αi ξi ϕ t, w t 1− ϕ−1 t, w t −1 a h t dt e0 m−2 i βi 1− −1 t, w t a −1 a h t m−2 i αi h t dt dt − e1 2.14 Journal of Inequalities and Applications From Lemma 2.1, it is immediate that Λh a1 − Λh a2 , a1 − a2 > 0, for a1 / a2 , 2.15 and hence, if 2.11 has a solution, then it is unique Let t0 3N E E m−2 i βi 1− m−2 i αi 1− E p −1 1 E · h 2N p e1 e0 p# −1 2.16 If |a| ≥ t0 , since w t −1/ p t −1 ∈ L1 0, and h ∈ C, it is easy to see that there exists an i ∈ {1, , N} such that the ith component of a satisfies ≥ E 1− E m−2 i βi m−2 i αi 1− E p −1 1 E · h 2N p e1 e0 p# −1 2.17 Thus ai hi t ≥2 hi t keeps sign on I and ≥ − h E 1− E m−2 i βi E 1− p −1 m−2 i αi E · h 2N p e1 e0 p# −1 , ∀t ∈ I, 2.18 then hi t 1/ p t −1 ≥ 21/ p E −1 1− m−2 i βi E E 1− m−2 i αi E e0 e1 , ∀t ∈ I 2.19 have Thus, when |a| is large enough, the ith component Λi a of Λh a is nonzero, then we h Λh a / 2.20 Let us consider the equation λΛh a 1−λ a 0, λ ∈ 0, 2.21 Qihu Zhang et al {x ∈ RN | |x| < t0 } So, It is easy to see that all the solutions of 2.21 belong to b t0 we have dB Λh a , b t0 , dB I, b t0 , / 0, 2.22 it means the existence of solutions of Λh a In this way, we define a function a h : C 0, → RN , which satisfies Λh a h 2.23 ii By the proof of i , we also obtain that a sends bounded sets to bounded sets, and a h E ≤ 3N E m−2 i βi E p −1 1 m−2 i αi E · h p# −1 e1 e0 2.24 It only remains to prove the continuity of a Let {un } be a convergent sequence in C and un → u as n → ∞ Since {a un } is a bounded sequence, then it contains a convergent 0, letting j → ∞, subsequence {a unj } Let a unj → a0 as j → ∞ Since Λunj a unj we have Λu a0 From i , we get a0 a u , it means that a is continuous This completes the proof Now, we define a : L1 → RN as a u a F u 2.25 It is clear that a · is continuous and sends bounded sets of L1 to bounded sets of RN , and hence it is a complete continuous mapping If u is a solution of 2.4 with 1.2 , then u t u F ϕ−1 t, w t −1 a g F g t t , ∀t ∈ 0, 2.26 The boundary condition 1.2 implies that u ηi −1 m−2 i βi ϕ −1 t, w t 1− a g F g t m−2 i βi dt e0 2.27 We denote that K1 h t : K1 ◦ h t F ϕ−1 t, w t −1 a h F h t , ∀t ∈ 0, 2.28 Lemma 2.3 The operator K1 is continuous and sends equi-integrable sets in L1 to relatively compact sets in C1 Journal of Inequalities and Applications −1/ p t −1 Proof It is easy to check that K1 h t ∈ C1 Since w t K1 h −1 ϕ−1 t, w t t a h F h ∈ L1 and ∀t ∈ 0, , , 2.29 it is easy to check that K1 is a continuous operator from L1 to C1 Let now U be an equi-integrable set in L1 , then there exists ρ∗ ∈ L1 , such that ≤ ρ∗ t u t un a.e in I, for any u ∈ U 2.30 We want to show that K1 U ⊂ C1 is a compact set Let {un } be a sequence in K1 U , then there exists a sequence {hn } ∈ U such that K1 hn For any t1 , t2 ∈ I, we have F hn t1 − F hn t1 t2 t2 hn t dt − t2 hn t dt 0 hn t dt ≤ t1 t2 ρ∗ t dt t1 2.31 Hence the sequence {F hn } is uniformly bounded and equicontinuous By AscoliArzela theorem, there exists a subsequence of {F hn } which we rename the same convergent in C According to the bounded continuous operator a, we can choose a F hn } which we still denote {a hn F hn } which is convergent subsequence of {a hn a hn F hn is convergent in C in C, then w t ϕ t, K1 hn t Since K1 hn t F ϕ−1 t, w t −1 a hn F hn t , ∀t ∈ 0, , 2.32 according to the continuity of ϕ−1 and the integrability of w t −1/ p t −1 in L1 , we can see that K1 hn is convergent in C Thus {un } is convergent in C1 This completes the proof Let us define P : C1 → C1 as P h m−2 i βi K1 ◦ h ηi 1− e0 m−2 i βi 2.33 It is easy to see that P is compact continuous We denote Nf u : 0, × C1 → L1 the Nemytski operator associated to f defined by Nf u t f t, u t , w t 1/ p t −1 u t , a.e on I 2.34 Lemma 2.4 u is a solution of 1.1 - 1.2 if and only if u is a solution of the following abstract equation: u P δNf u K1 δNf u 2.35 Qihu Zhang et al Proof If u is a solution of 1.1 - 1.2 , by integrating 1.1 from to t, we find that w t ϕ t, u t a δNf u F δNf u t 2.36 From 2.36 , we have u F ϕ−1 r, w r −1 βi u F ϕ−1 r, w r −1 u t m−2 u a δNf u F δNf u t , a δNf u F δNf u ηi 2.37 e0 , i then we have −1 ϕ−1 r, w r m−2 i βi F u a δNf u 1− m−2 i βi K1 δNf u ηi e0 ηi e0 2.38 P δNf u m−2 i βi 1− F δNf u m−2 i βi So we have u P δNf u K1 δNf u 2.39 Conversely, if u is a solution of 2.35 , it is easy to see that m−2 i βi K1 P δNf u u δNf u 1− m−2 u ηi e0 m−2 i βi , m−2 βi u K1 δNf u ηi βi u ηi e0 i 2.40 e0 , i P δNf u u K1 δNf u By the condition of the mapping a, m−2 i βi K1 u δNf u 1− ηi e0 m−2 i αi K1 δNf u m−2 i βi 1− ξi − K1 δNf u m−2 i αi e1 , 2.41 then we have u m−2 i αi K1 δNf u 1− ξi − K1 δNf u m−2 i αi e1 K1 δNf u 1, 2.42 10 Journal of Inequalities and Applications thus m−2 αi u − K1 δNf u u 1 K1 δNf u ξi e1 i m−2 2.43 m−2 αi P δNf u K1 δNf u ξi αi u ξi e1 i e1 , i from 2.40 and 2.43 , we obtain 1.2 From 2.35 , we have ϕ−1 t, w t u t −1 a F δNf u , 2.44 δNf u t w t ϕ t, u Hence u is a solution of 1.1 - 1.2 This completes the proof Lemma 2.5 If u is a solution of 1.1 - 1.2 , then for any j that uj ≤ C∗ : ςj 1, , N, there exists a ςj ∈ 0, , such e0 e1 m−2 i βi 1− m−2 i αi 1− Proof For any j 1, , N, if there exists ςj ∈ 0, such that uj If it is false, then uj is strictly monotone i If uj is strictly decreasing in 0, , then uj > uj ξi > uj , uj > uj ηi > uj , ςj 2.45 0, then 2.45 is valid 1, , m − i 2.46 Thus m−2 uj βi uj ηi j m−2 i e0 , i m−2 uj j βi uj e0 < m−2 αi uj ξi i j 2.47 j αi uj e1 > e1 , i it means that j e0 1− m−2 i βi j > uj > u j > e1 1− m−2 i αi , 2.48 Qihu Zhang et al 11 then there exists a tj ∈ 0, such that j > uj j e1 uj − uj >− 1−0 − m−2 αi i tj e0 m−2 i βi 1− 2.49 ii If uj is strictly increasing in 0, , then uj < uj ξi < uj , uj < uj ηi < uj , 1, , m − i 2.50 Thus m−2 m−2 j βi uj ηi uj j βi uj e0 > i j j e1 , i m−2 e0 , m−2 j u j e1 j αi u ξi i < 2.51 αi u i it means that j j e0 1− m−2 i βi < uj < u j < e1 1− m−2 i αi , 2.52 then there exists a r j ∈ 0, such that j j < uj rj e1 uj − uj < 1−0 − m−2 αi i e0 1− m−2 i βi 2.53 Combining 2.49 and 2.53 , then we obtain 2.45 This completes the proof f satisfies sub-p− − growth condition In this section, we will apply Leray-Schauder’s degree to deal with the existence of solutions for 1.1 - 1.2 , when f satisfies sub-p− − growth condition Theorem 3.1 If f satisfies sub-p− − growth condition, then for any fixed parameter δ, problem 1.1 - 1.2 has at least one solution K1 λδNf u , where Nf u is defined in 2.34 We Proof Denote Ψf u, λ : P λδNf u know that 1.1 - 1.2 has the same solution of u when λ Ψf u, λ , 3.1 12 Journal of Inequalities and Applications It is easy to see that the operator P is compact continuous According to Lemmas 2.2 and 2.3, then we can see that Ψf ·, λ is compact continuous from C1 to C1 for any λ ∈ 0, We claim that all the solutions of 3.1 are uniformly bounded for λ ∈ 0, In fact, if it is false, we can find a sequence of solutions { un , λn } for 3.1 such that un → ∞ as n → ∞, and un > for any n 1, 2, Let tn ∈ 0, such that sup w t un t t∈ 0,1 p t −1 p tn −1 ≤ w t n un t n , n 1, 2, 3.2 1, 2, , there exists an in ∈ {1, , N} such that For any fixed n uin n tn ≥ u tn N n 3.3 Thus, {uin } becomes a sequence with respect to n n Since un , λn are solutions of 3.1 , according to Lemma 2.5, for any n i i exists ξnn ∈ 0, such that | uin ξnn | ≤ C∗ , then n w t un p t −2 uin n i i w ξnn un ξnn t t i ξnn i p ξnn −2 q r −1 λn δf un uin n in 1, 2, , there i ξnn r, un , w r 1/ p r −1 un q r −1 un ∀t ∈ 0, dr, 3.4 For any t ∈ 0, , we have w t un p t −2 uin n t i i ≤ w ξnn un ξnn t i ξnn un i p ξnn −2 q r −1 λn δf uin n in i ξnn r, un , w r un 1/ p r −1 un q r −1 3.5 dr i Without loss of generality, we assume that |un ξnn | > i 1◦ If p ξnn − ≤ 0, then i i w ξnn |un ξnn | i p ξnn −2 | uin n i ξnn | i w ξnn | uin n i ξnn | i |un ξnn | i w ξnn | uin n i 2−p ξnn i ξnn | i ≤ w ξnn i p ξnn −1 | uin n | uin n i ξnn | i ξnn | i p ξnn −1 i ≤ w ξnn C∗ i 2−p ξnn , 3.6 where C∗ is defined in 2.45 Qihu Zhang et al 13 Combining 3.2 , 3.3 , 3.5 , and 3.6 , we have sup w t un 2N t∈ 0,1 p t −1 ≤ w t n un t n N p tn −1 tn i p ξnn −1 i ≤ w ξnn C∗ ≤ C1 q r −1 λn δf un i ξnn C2 un p tn −2 ≤ w t n un t n in uin n r, un , w r tn 1/ p r −1 q r −1 un un dr q −1 3.7 Then we have p t −1 w t un t ≤ 2N C1 C2 un q −1 , ∀t ∈ 0, 3.8 q − / p− − , we have Denoting α sup 1/ p t −1 w t t∈ 0,1 ≤ C3 un α 3.9 ≤ NC3 un α 3.10 un t Thus 1/ p t −1 w t i 2◦ If p ξnn − > 0, since | uin n i p ξnn −2 i i w ξnn un ξnn w ≤ C4 i ξnn uin n i 1/ p ξnn −1 i ξnn | ≤ C∗ , we have i ξnn uin n sup w t un t un t i ξnn p t −1 w i ξnn , i p ξnn −1 i un ξnn i i p ξnn −2 / p ξnn −1 3.11 p −2 p −1 where t∈ 0,1 According to 3.2 , 3.3 , 3.5 , and 3.11 , we have sup w t un t 2N t∈ 0,1 p t −1 ≤ w tn un t n ≤ C4 p tn −2 sup w t un t t∈ 0,1 uin n p t −1 tn 3.12 ε q −1 C2 un 14 Journal of Inequalities and Applications Since < is a positive constant, 3.12 means that p t −1 sup w t un t ≤ C5 un t∈ 0,1 q −1 3.13 Thus 1/ p t −1 w t un t ≤ NC6 un α 3.14 Summarizing this argument, we have 1/ p t −1 w t Since |a h | ≤ C8 F h ≤ C8 a δNf |e0 | un t p# −1 |e1 | F Nf ≤ C7 un α 3.15 , then we have e1 e0 0 p# −1 ≤ C9 u q −1 3.16 Thus a δNf un ≤ a δNf un F δNf un F δNf un ≤ C10 un q −1 3.17 Combining 2.38 and 3.17 , we have ≤ C11 un un For any j j un t α , where α q −1 p− − 3.18 1, , N, since t j j un un r dr t j ≤ un −1/ p r −1 w r α ≤ un C11 sup w t 1/ p t −1 j un t dr 3.19 t∈ 0,1 C7 E , we have j un ≤ C12 un α , j 1, , N; n 1, 2, 3.20 Thus un ≤ NC12 un α , n 1, 2, Combining 3.15 and 3.21 , then we obtain that { un } is bounded 3.21 Qihu Zhang et al 15 Thus, there exists a large enough R0 > such that all the solutions of 3.1 belong to {u ∈ C1 | u < R0 }, then the Leray-Schauder degree dLS I − Ψf ·, λ , B R0 , is B R0 well defined for λ ∈ 0, , and dLS I − Ψf ·, , B R0 , dLS I − Ψf ·, , B R0 , 3.22 Let u0 ηi −1 m−2 i βi ϕ t, w t −1 a dt r e0 m−2 i βi 1− ϕ−1 t, w t −1 a dt, 3.23 where a is defined in 2.25 , thus u0 is the unique solution of u Ψf u, It is easy to see that u is a solution of u Ψf u, if and only if u is a solution of the following: ⎧ ⎪−Δp t ,w t u ⎪ ⎨ I ⎪u ⎪ ⎩ t ∈ 0, , 0, m−2 3.24 m−2 βi u ηi e0 , u αi u ξi i e1 i Obviously, system I possesses only one solution u0 Since u0 ∈ B R0 , thus the LeraySchauder degree dLS I − Ψf ·, , B R0 , dLS I − Ψf ·, , B R0 , / 0, 3.25 therefore, we obtain that 1.1 - 1.2 has at least one solution This completes the proof f satisfies general growth condition In the following, we will deal with the existence of solutions for p t -Laplacian ordinary system, when f satisfies general growth condition Denote Ωε u ∈ C1 | max ui 1≤i≤N w t 1/ p t −1 ui 0, then u is concave, this property is used extensively in the study of one-dimensional p- Laplacian problems,