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This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Global Bifurcation Results for General Laplacian Problems Fixed Point Theory and Applications 2012, 2012:7 doi:10.1186/1687-1812-2012-7 Eun Kyoung Lee (eunkyoung165@gmail.com) Yong-Hoon Lee (yhlee@pusan.ac.kr) Byungjae Son (mylife1882@hanmail.net) ISSN 1687-1812 Article type Research Submission date 23 December 2010 Acceptance date 18 January 2012 Publication date 18 January 2012 Article URL http://www.fixedpointtheoryandapplications.com/content/2012/1/7 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). 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Global bifurcation results for general Laplacian problems Eun Kyoung Lee 1 , Yong-Hoon Lee 2∗ and Byungjae Son 2 1 Department of Mathematics Education, Pusan National University, Busan 609-735, Korea 2 Department of Mathematics, Pusan National University, Busan 609-735, Korea *Corresponding author: yhlee@pusan.ac.kr Email addresses: EK LEE: eunkyoung165@gmail.com B SON: mylife1882@hanmail.net Abstract In this article, we consider the global bifurcation result and existence of solutions 1 for the following general Laplacian problem,            −(ϕ(u ′ (t))) ′ = λψ(u(t)) + f(t, u, λ), t ∈ (0, 1), u(0) = u(1) = 0, (P ) where f : [0, 1] × R × R → R is continuous and ϕ, ψ : R → R are odd increasing homeomorphisms of R, when ϕ, ψ satisfy the asymptotic homogeneity conditions. 1 Introduction In this article, we consider the following general Laplacian problem,          −(ϕ(u ′ (t))) ′ = λψ(u(t)) + f(t, u, λ), t ∈ (0, 1), u(0) = u(1) = 0, (P ) where f : [0, 1] × R × R → R is continuous with f(t, u, 0) = 0 and ϕ, ψ : R → R are odd increasing homeomorphisms of R with ϕ(0) = ψ(0) = 0. We consider the following conditions; (Φ 1 ) lim t→0 ϕ(σt) ψ(t) = σ p−1 , for all σ ∈ R + , for some p > 1. (Φ 2 ) lim |t|→∞ ϕ(σt) ψ(t) = σ q−1 , for all σ ∈ R + , for some q > 1. (F 1 ) f(t, u, λ) = ◦(|ψ(u)|) near zero, uniformly for t and λ in bounded intervals. (F 2 ) f(t, u, λ) = ◦(|ψ(u)|) near infinity, uniformly for t and λ in bounded intervals. (F 3 ) uf(t, u, λ) ≥ 0. 2 We note that ϕ r (t) = |t| r−2 t, r > 1 are special cases of ϕ and ψ. We first prove following global bifurcation result. Theorem 1.1. Assume (Φ 1 ), (Φ 2 ), (F 1 ), (F 2 ) and (F 3 ). Then for any j ∈ N, there exists a connected component C j of the set of nontrivial solutions for (P ) connecting (0, λ j (p)) to (∞, λ j (q)) such that (u, λ) ∈ C j implies that u has exactly j −1 simple zeros in (0, 1), where λ j (r) is the j-th eigenvalue of (ϕ r (u ′ (t))) ′ + λϕ r (u(t)) = 0 and u(0) = u(1) = 0. By the aid of this theorem, we can prove the following existence result of solutions. Theorem 1.2. Consider problem          −(ϕ(u ′ (t))) ′ = g(t, u), t ∈ (0, 1), u(0) = u(1) = 0, (A) where g : [0, 1] × R × R → R is continuous and ϕ is odd increasing homeomorphism of R, 3 which satisfy (Φ 1 ) and (Φ 2 ) with ϕ = ψ. Also ug(t, u) ≥ 0 and there exist positive integers k, n with k ≤ n such that µ = lim s→0 g(t,s) ϕ(s) < λ k (p) ≤ λ n (q) < lim |s|→∞ g(t,s) ϕ(s) = ν uniformly in t ∈ [0, 1]. Then for each integer j with k ≤ j ≤ n, problem (A) has a solution with exactly j −1 simple zeros in (0, 1). Thus, (A) possesses at least n −k + 1 nontrivial solutions. In [1], the authors studied the existence of solutions and global bifurcation results for          −(t N−1 ϕ(u ′ (t))) ′ = t N−1 λψ(u(t)) + t N−1 f(t, u, λ), t ∈ (0, R), u ′ (0) = u(R) = 0. The main purpose of this article is to derive the same result for N = 1 case with Dirichlet boundary condition which was not considered in [1]. For p-Laplacian problems, i.e., ϕ = ψ = ϕ p , many authors have studied for the existence and multiplicity of nontrivial solutions [2–6]. In [2, 5, 6], the authors used fixed point theory or topological degree argument. Also global bifurcation theory was mainly employed in [3, 4]. Moreover, there are some studies related to general Laplacian problems [3, 7, 8], but most of them are about ϕ = ψ case. In [3], the authors proved some results under several kinds of boundary conditions and in [7], the authors considered a system of general Laplacian problems. In [8], the author studied global continuation result for the singular problem. In this paper, we mainly study the global bifurcation phenomenon for general Laplacian problem (P ) and prove the existence and multiplicity result for (A). This article is organized as follows: In Section 2, we set up the equivalent integral operator of (P ) and compute the degree of this operator. In Section 3, we verify the existence of global bifurcation having bifurcation points at zero and infinity simultaneously. In Section 4, we 4 introduce an existence result as an application of the previous result and give some examples. 2 Degree estimate Let us consider problem (P ) with f ≡ 0, i.e.,          −(ϕ(u ′ (t))) ′ = λψ(u(t)), t ∈ (0, 1), u(0) = u(1) = 0. (P ) We introduce the equivalent integral operator of problem (P ). For this, we consider the following problem          (ϕ(u ′ (t))) ′ = h(t), a.e., t ∈ (0, 1), u(0) = u(1) = 0, (AP ) where h ∈ L 1 (0, 1). Here, a function u is called a solution of (AP ) if u ∈ C 1 0 [0, 1] with ϕ(u ′ ) absolutely continuous which satisfies (AP ). We note that (AP ) is equivalently written as u(t) = G(h)(t) =  t 0 ϕ −1  a(h) +  s 0 h(ξ)dξ  ds, where a : L 1 (0, 1) → R is a continuous function which sends bounded sets of L 1 into bounded sets of R and satisfying  1 0 ϕ −1  a(h) +  s 0 h(ξ)dξ  ds = 0. (1) It is known that G : L 1 (0, 1) → C 1 0 [0, 1] is continuous and maps equi-integrable sets of L 1 (0, 1) into relatively compact sets of C 1 0 [0, 1]. One may refer Man´asevich-Mawhin [4, 3] 5 and Garcia-Huidobro-Man´asevich-Ward [7] for more details. If we define the operator T λ ϕψ : C 1 0 [0, 1] → C 1 0 [0, 1] by T λ ϕψ (u)(t) = G(−λψ(u))(t) =  t 0 ϕ −1  a(−λψ(u)) +  s 0 −λψ(u(ξ))dξ  ds, (2) then (P ) is equivalently written as u = T λ ϕψ (u). Now let us consider p-Laplacian problem          −(ϕ p (u ′ (t))) ′ = λϕ p (u(t)), t ∈ (0, 1), u(0) = u(1) = 0. (E p ) By the similar argument, we can also get the equivalent integral operator of problem (E p ), which is known by Garcia-Huidobro-Man´asevich-Schmitt [1]. Let us define T λ p : C 1 0 [0, 1] → C 1 0 [0, 1] by T λ p (u)(t) =  t 0 ϕ −1 p  a p (−λϕ p (u)) +  s 0 −λϕ p (u(ξ))dξ  ds, (3) where a p : L 1 (0, 1) → R is a continuous function which sends bounded sets of L 1 into bounded sets of R and satisfying  1 0 ϕ −1 p  a p (h) +  s 0 h(ξ)dξ  ds = 0, for all h ∈ L 1 (0, 1). Note that a p has homogineity property, i.e., a p (λt) = λa p (t). Problem (E p ) can be equiva- lently written as u = T λ p (u). Obviously, T λ ϕψ and T λ p are completely continuous. The main purpose of this section is to compute the Leray-Schauder degree of I − T λ ϕψ . Following Lemma is for the property of ϕ and ψ with asymptotic homogeneity condition (Φ 1 ) 6 and (Φ 2 ), which is very useful for our analysis. The proof can be modified from Proposition 4.1 in [9]. Lemma 2.1. Assume that ϕ, ψ are odd increasing homeomorphisms of R which satisfy (Φ 1 ) and (Φ 2 ). Then, we have (i) lim t→0 ϕ −1 (σt) ψ −1 (t) = ϕ −1 p (σ), for all σ ∈ R + , for some p > 1, (4) and (ii) lim |t|→∞ ϕ −1 (σt) ψ −1 (t) = ϕ −1 q (σ), for all σ ∈ R + , for some q > 1. (5) To compute the degree, we will make use of the following well-known fact [10]. Lemma 2.2. If λ is not an eigenvalue of (E p ), p > 1 and r > 0, then deg(I − T λ p , B(0, r), 0) =          1 if λ < λ 1 (p), (−1) k if λ ∈ (λ k (p), λ k+1 (p)). (6) Now, let us compute deg(I − T λ ϕψ , B(0, r), 0) when λ is not an eigenvalue of (E p ). Theorem 2.3. Assume that ϕ, ψ are odd increasing homeomorphisms of R which satisfy (Φ 1 ) and (Φ 2 ). Then, (i) The Leray-Schauder degree of I − T λ ϕψ is defined for B(0, ε), for all sufficiently small ε. 7 Moreover, we have deg(I − T λ ϕψ , B(0, ε), 0) =          1 if λ < λ 1 (p), (−1) m if λ ∈ (λ m (p), λ m+1 (p)). (7) (ii) The Leray-Schauder degree of I − T λ ϕψ is defined for B(0, M), for all sufficiently large M, and deg(I − T λ ϕψ , B(0, M), 0) =          1 if λ < λ 1 (q), (−1) l if λ ∈ (λ l (q), λ l+1 (q)). (8) Proof: We give the proof for assertion (i). Proof for the latter case is similar. Define T λ : C 1 0 [0, 1] × [0, 1] → C 1 0 [0, 1] by T λ (u, τ ) = τT λ ϕψ (u) + (1 − τ )T λ p (u). We claim that the Leray-Schauder degree for I −T λ (·, τ ) is defined for B(0, ε) in C 1 0 [0, 1] for all small ε. Indeed, suppose there exist sequences {u n }, {τ n } and {ε n } with ε n → 0 and ∥u n ∥ 0 = ε n such that u n = T λ (u n , τ n ), i.e., u n (t) = τ n  t 0 ϕ −1  a(−λψ(u n )) +  s 0 −λψ(u n (ξ))dξ  ds + (1 −τ n )  t 0 ϕ −1 p  a p (−λϕ p (u n )) +  s 0 −λϕ p (u n (ξ))dξ  ds. Setting v n (t) = u n (t) ε n , we have ∥v n ∥ 0 = 1, v n (t) = τ n ε n  t 0 ϕ −1  a(−λψ(u n )) +  s 0 −λψ(u n (ξ))dξ  ds + (1 −τ n )  t 0 ϕ −1 p  a p (−λϕ p (v n )) +  s 0 −λϕ p (v n (ξ))dξ  ds, 8 and v ′ n (t) = τ n ε n ϕ −1  a(−λψ(u n )) +  t 0 −λψ(u n (ξ))dξ  + (1 −τ n )ϕ −1 p  a p (−λϕ p (v n )) +  t 0 −λϕ p (v n (ξ))dξ  . Now, we show that {v ′ n } is uniformly bounded. Since ∥v n ∥ 0 = 1,  t 0 −λϕ p (v n (ξ))dξ ≤ λ. Moreover, there exists C 1 such that a p (−λϕ p (v n )) ≤ C 1 . These results imply the uniform boundedness of ϕ −1 p  a p (−λϕ p (v n )) +  t 0 −λϕ p (v n (ξ))dξ  . Let q n (t) = 1 ε n ϕ −1  a(−λψ(u n )) +  t 0 −λψ(u n (ξ))dξ  , and d n (t) =  t 0 λψ(u n (ξ))dξ. Then d n ∈ C[0, 1], and ∥d n ∥ 0 = max t∈[0,1] |  t 0 λψ(u n (ξ))dξ| ≤  1 0 λψ(∥u n ∥ 0 )dξ ≤ λψ(ε n ). Since  1 0 ϕ −1  a(−λψ(u n )) − d n (s)  ds = 0, we have |a(−λψ(u n ))| ≤ λψ(ε n ). Otherwise,  1 0 ϕ −1  a(−λψ(u n ))−d n (s)  ds < 0 (or > 0). Now, we show that 1 ε n ϕ −1  2λψ(ε n )  is bounded. Indeed, suppose that it is not true, i.e., 1 ε n ϕ −1  2λψ(ε n )  → ∞ as n → ∞. Then, for arbitrary A > 0, there exists N 0 ∈ N such that 1 ε n ϕ −1  2λψ(ε n )  ≥ A, for all n > N 0 . This implies that 2λ ≥ ϕ(Aε n ) ψ(ε n ) for all n > N 0 . However, ϕ(Aε n ) ψ(ε n ) → ϕ p (A) as n → ∞. This is a 9 [...]... ia-Huidobro, M, Man´sevich, R, Schmitt, K: Some bifurcation results for a class of a p -Laplacian like operators Diff Integral Eqns 10, 51–66 (1997) [2] Agarwal, RP, L¨, J, O’Regan, D: Eigenvalues and the one-dimensional p -Laplacian J u Math Anal Appl 266, 383–400 (2002) [3] Man´sevich, R, Mawhin, J: Boundary value problems for nonlinear perturbations of a vector p -Laplacian- like-operators J Korean Math Soc... λ) is a bifurcation point of (P ), then λ = λn (p) for some p ∈ N ˆ ˆ (ii) Assume (Φ2 ) and (F2 ) If (∞, λ) is a bifurcation point of (P ), then λ = λn (q) for some q ∈ N ˆ Proof: We prove assertion (i) Suppose that (0, λ) is a bifurcation point of (P ) Then there 1 ˆ exists a sequence {(un , λn )} in C0 [0, 1] × R with (un , λn ) → (0, λ) and such that (un , λn ) 12 satisfies un = F(un , λn ) for each... → ∞ as n → ∞, then for arbitrary A > 0, there exists N0 ∈ N such that ) 1 −1 ( ϕ 2K2 ψ(εn ) ≥ A, for all n ≥ N0 εn This implies that 2K2 ≥ ϕ(Aεn ) , ψ(εn ) for all n ≥ N0 This is impossible Thus 1 −1 ( ϕ a(−λn ψ(un ) − f (·, un , λn )) + εn ∫ t ) −λn ψ(un (ξ)) − f (ξ, un , λn )dξ ≤ K3 0 ′ Consequently, {vn } is uniformly bounded and by the Arzela-Ascoli Theorem, {vn } has a uniformly convergent subsequence... 0 −λϕp (v(ξ))dξ ds Since vn (1) = 0 for all n, d = ap (−λϕp (v)) p ˆ and v is a solution of (Ep ) Consequently, λ must be an eigenvalue of the p -Laplacian operator The converse of first part of Theorem 3.3 is true in our problem Lemma 3.4 Assume (Φ1 ) and (F1 ) If µ is an eigenvalue of (Ep ), then (0, µ) is a bifurcation point Proof: Suppose that (0, µ) is not a bifurcation point of (P ) Then there... ε)ψ(un (ξ)) − f (ξ, un , µ − ε)dξ 0 + (1 − τn )ϕ−1 p ( ∫ ) t ap (−(µ − ε)ϕp (vn )) + −(µ − ε)ϕp (vn (ξ))dξ , 0 ′ and we see that {vn } is uniformly bounded Therefore, by the Arzela-Ascoli Theorem, {vn } has a uniformly convergent subsequence in C[0, 1] Without loss of generality, let vn → v Moreover, using the fact that ∫ t ) 1 −1 ( ϕ a(−(µ − ε)ψ(un ) − f (·, un , µ − ε)) + −(µ − ε)ψ(un (ξ)) − f (ξ, un... ∈ ∂Sk , u ≡ 0 because u dose not have double zero Henceforth λ = µj , j ̸= k But then, (un , λn ) ∈ (Sk × R) ∩ Oj for large n which is 18 impossible by the fact that (un , λn ) ∈ S ∩ Oj implies un ∈ Sj The proof is complete Lemma 3.6 Assume (Φ1 ), (Φ2 ), (F1 ), and (F3 ) Then for each k ∈ N, there exists a constant Mk ∈ (0, ∞) such that λ ≤ Mk for every λ with (u, λ) ∈ Ck Proof: Suppose it is not... that λn → λ and ∥un ∥1 → ∞ Thus, (∞, λ) is a bifurcation point and ˆ λ = λj (q) 4 Application and some examples Proof of Theorem 1.2 Let us consider the bifurcation problem    −(ϕ(u′ (t)))′ = λϕ(u(t)) + g(t, u), t ∈ (0, 1),  (Ag )    u(0) = u(1) = 0 21 Put f (t, u, λ) = −µϕ(u) + g(t, u) We can easily see that f (t, u, λ) = ◦(|ϕ(u)|) near zero uniformly for t and λ in bounded intervals The equation...contradiction Thus by the above inequality, we get 1 −1 ( ϕ a(−λψ(un )) + εn ∫ 0 t ) ) 1 −1 ( −λψ(un (ξ))dξ ≤ ϕ 2λψ(εn ) ≤ C2 , εn ′ for some C2 > 0 Therefore, {vn } is uniformly bounded By the Arzela-Ascoli Theorem, {vn } has a uniformly convergent subsequence in C[0, 1] relabeled as the original sequence so let limn→∞ vn = v Now, we claim that qn (t) → q(t), where q(t) = ϕ−1 p ∫... Mawhin, J: Periodic solutions of nonlinear sysetems with p-Laplaciana like operators J Diff Eqns 145, 367–393 (1998) [5] S´nchez, J: Multiple positive solutions of singular eigenvalue type problems involving a the one-dimensional p -Laplacian J Math Anal Appl 292, 401–414 (2004) 25 [6] Wang, J: The existence of positive solutions for the one-dimensional p -Laplacian Proc Am Math Soc 125, 2275–2283 (1997) [7]... Man´sevich, R, Ward, JR: A homotopy along p for systems with a a vector p-Laplace operator Adv Diff Eqns 8, 337–356 (2003) [8] Sim, I: On the existence of nodal solutions for singular one-dimensional φ -Laplacian problem with asymptotic condition Commun Pure Appl Math 7, 905–923 (2008) [9] Garc´ ia-Huidobro, M, Man´sevich, R, Ubilla, P: Existence of positive solutions for some a Dirichlet problems with an . article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Global Bifurcation Results for General Laplacian Problems Fixed Point Theory and. and reproduction in any medium, provided the original work is properly cited. Global bifurcation results for general Laplacian problems Eun Kyoung Lee 1 , Yong-Hoon Lee 2∗ and Byungjae Son 2 1 Department. lim t→0 ϕ(σt) ψ(t) = σ p−1 , for all σ ∈ R + , for some p > 1. (Φ 2 ) lim |t|→∞ ϕ(σt) ψ(t) = σ q−1 , for all σ ∈ R + , for some q > 1. (F 1 ) f(t, u, λ) = ◦(|ψ(u)|) near zero, uniformly for t and λ in