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Fixed Point Theory and Applications 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 Article type 1687-1812 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) For information about publishing your research in Fixed Point Theory and Applications go to http://www.fixedpointtheoryandapplications.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2012 Lee et al ; licensee Springer This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Global bifurcation results for general Laplacian problems Eun Kyoung Lee1 , Yong-Hoon Lee2∗ and Byungjae Son2 Department of Mathematics Education, Pusan National University, Busan 609-735, Korea 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 for the following general Laplacian problem,      −(ϕ(u′ (t)))′ = λψ(u(t)) + f (t, u, λ), t ∈ (0, 1), (P )    u(0) = u(1) = 0,  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 Introduction In this article, we consider the following general Laplacian problem,    −(ϕ(u′ (t)))′ = λψ(u(t)) + f (t, u, λ), t ∈ (0, 1),  (P )    u(0) = u(1) = 0, where f : [0, 1] × R × R → R is continuous with f (t, u, 0) = and ϕ, ψ : R → R are odd increasing homeomorphisms of R with ϕ(0) = ψ(0) = We consider the following conditions; (Φ1 ) limt→0 ϕ(σt) ψ(t) (Φ2 ) lim|t|→∞ = σ p−1 , for all σ ∈ R+ , for some p > ϕ(σt) ψ(t) = σ q−1 , for all σ ∈ R+ , for some q > (F1 ) f (t, u, λ) = ◦(|ψ(u)|) near zero, uniformly for t and λ in bounded intervals (F2 ) f (t, u, λ) = ◦(|ψ(u)|) near infinity, uniformly for t and λ in bounded intervals (F3 ) uf (t, u, λ) ≥ We note that ϕr (t) = |t|r−2 t, r > are special cases of ϕ and ψ We first prove following global bifurcation result Theorem 1.1 Assume (Φ1 ), (Φ2 ), (F1 ), (F2 ) and (F3 ) Then for any j ∈ N, there exists a connected component Cj of the set of nontrivial solutions for (P ) connecting (0, λj (p)) to (∞, λj (q)) such that (u, λ) ∈ Cj implies that u has exactly j − simple zeros in (0, 1), where λj (r) is the j-th eigenvalue of (ϕr (u′ (t)))′ + λϕr (u(t)) = and u(0) = u(1) = 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, which satisfy (Φ1 ) and (Φ2 ) with ϕ = ψ Also ug(t, u) ≥ and there exist positive integers k, n with k ≤ n such that µ = lims→0 g(t,s) ϕ(s) < λk (p) ≤ λn (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 − simple zeros in (0, 1) Thus, (A) possesses at least n − k + nontrivial solutions In [1], the authors studied the existence of solutions and global bifurcation results for    −(tN −1 ϕ(u′ (t)))′ = tN −1 λψ(u(t)) + tN −1 f (t, u, λ), t ∈ (0, R),    ′  u (0) = u(R) = The main purpose of this article is to derive the same result for N = case with Dirichlet boundary condition which was not considered in [1] For p-Laplacian problems, i.e., ϕ = ψ = ϕp , 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 introduce an existence result as an application of the previous result and give some examples Degree estimate Let us consider problem (P ) with f ≡ 0, i.e.,    −(ϕ(u′ (t)))′ = λψ(u(t)), t ∈ (0, 1),  (P )    u(0) = u(1) = 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),      (AP ) u(0) = u(1) = 0, where h ∈ L1 (0, 1) Here, a function u is called a solution of (AP ) if u ∈ C0 [0, 1] with ϕ(u′ ) absolutely continuous which satisfies (AP ) We note that (AP ) is equivalently written as ∫ u(t) = G(h)(t) = t ϕ −1 ∫ ( a(h) + s ) h(ξ)dξ ds, where a : L1 (0, 1) → R is a continuous function which sends bounded sets of L1 into bounded sets of R and satisfying ∫ −1 ϕ ∫ ( a(h) + s ) h(ξ)dξ ds = (1) It is known that G : L1 (0, 1) → C0 [0, 1] is continuous and maps equi-integrable sets of a L1 (0, 1) into relatively compact sets of C0 [0, 1] One may refer Man´sevich-Mawhin [4, 3] λ and Garcia-Huidobro-Man´sevich-Ward [7] for more details If we define the operator Tϕψ : a 1 C0 [0, 1] → C0 [0, 1] by ∫ λ Tϕψ (u)(t) = G(−λψ(u))(t) = t −1 ∫ ( ϕ s a(−λψ(u)) + ) −λψ(u(ξ))dξ ds, (2) λ then (P ) is equivalently written as u = Tϕψ (u) Now let us consider p-Laplacian problem     −(ϕp (u′ (t)))′ = λϕp (u(t)), t ∈ (0, 1), (Ep )    u(0) = u(1) = By the similar argument, we can also get the equivalent integral operator of problem (Ep ), λ which is known by Garcia-Huidobro-Man´sevich-Schmitt [1] Let us define Tp : C0 [0, 1] → a C0 [0, 1] by ∫ λ Tp (u)(t) t = ϕ−1 p ∫ ( ap (−λϕp (u)) + s ) −λϕp (u(ξ))dξ ds, (3) where ap : L1 (0, 1) → R is a continuous function which sends bounded sets of L1 into bounded sets of R and satisfying ∫ ϕ−1 p ∫ ( s ap (h) + ) h(ξ)dξ ds = 0, for all h ∈ L1 (0, 1) Note that ap has homogineity property, i.e., ap (λt) = λap (t) Problem (Ep ) can be equivaλ λ λ lently written as u = Tp (u) Obviously, Tϕψ and Tp are completely continuous λ The main purpose of this section is to compute the Leray-Schauder degree of I − Tϕψ Following Lemma is for the property of ϕ and ψ with asymptotic homogeneity condition (Φ1 ) and (Φ2 ), 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 (Φ1 ) and (Φ2 ) Then, we have ϕ−1 (σt) = ϕ−1 (σ), f or all σ ∈ R+ , f or some p > 1, p t→0 ψ −1 (t) (4) ϕ−1 (σt) = ϕ−1 (σ), f or all σ ∈ R+ , f or some q > q |t|→∞ ψ −1 (t) (5) (i) lim and (ii) lim 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 (Ep ), p > and r > 0, then λ deg(I − Tp , B(0, r), 0) =     1 if λ < λ1 (p), (6)    (−1)k if λ ∈ (λk (p), λk+1 (p)) λ Now, let us compute deg(I − Tϕψ , B(0, r), 0) when λ is not an eigenvalue of (Ep ) Theorem 2.3 Assume that ϕ, ψ are odd increasing homeomorphisms of R which satisfy (Φ1 ) and (Φ2 ) Then, λ (i) The Leray-Schauder degree of I − Tϕψ is defined for B(0, ε), for all sufficiently small ε Moreover, we have deg(I −    1  λ Tϕψ , B(0, ε), 0) = if λ < λ1 (p), (7)    (−1)m if λ ∈ (λm (p), λm+1 (p)) λ (ii) The Leray-Schauder degree of I − Tϕψ is defined for B(0, M ), for all sufficiently large M , and deg(I − λ Tϕψ , B(0, M ), 0)    1  =     if λ < λ1 (q), (8) if λ ∈ (λl (q), λl+1 (q)) (−1)l Proof: We give the proof for assertion (i) Proof for the latter case is similar Define 1 λ λ T λ : C0 [0, 1] × [0, 1] → C0 [0, 1] by T λ (u, τ ) = τ Tϕψ (u) + (1 − τ )Tp (u) We claim that the Leray-Schauder degree for I − T λ (·, τ ) is defined for B(0, ε) in C0 [0, 1] for all small ε Indeed, suppose there exist sequences {un }, {τn } and {εn } with εn → and ∥un ∥0 = εn such that un = T λ (un , τn ), i.e., ∫ t un (t) = τn ϕ −1 ∫ ( s a(−λψ(un )) + ∫ t + (1 − τn ) ϕ−1 p ) −λψ(un (ξ))dξ ds ( ∫ un (t) , εn ) −λϕp (un (ξ))dξ ds 0 Setting (t) = s ap (−λϕp (un )) + we have ∥vn ∥0 = 1, τn (t) = εn ∫ t ϕ −1 ∫ ( a(−λψ(un )) + ∫ + (1 − τn ) t ϕ−1 p s ) −λψ(un (ξ))dξ ds ( ∫ ap (−λϕp (vn )) + 0 s ) −λϕp (vn (ξ))dξ ds, and ′ (t) ∫ t ) τn −1 ( = ϕ a(−λψ(un )) + −λψ(un (ξ))dξ εn ∫ t ( ) −1 + (1 − τn )ϕp ap (−λϕp (vn )) + −λϕp (vn (ξ))dξ ′ Now, we show that {vn } is uniformly bounded Since ∥vn ∥0 = 1, ∫t −λϕp (vn (ξ))dξ ≤ λ Moreover, there exists C1 such that ap (−λϕp (vn )) ≤ C1 These results imply the uniform ( ) ∫t boundedness of ϕ−1 ap (−λϕp (vn )) + −λϕp (vn (ξ))dξ Let p ( qn (t) = ϕ−1 a(−λψ(un )) + εn and ∫ dn (t) = ∫ t ) −λψ(un (ξ))dξ , t λψ(un (ξ))dξ Then dn ∈ C[0, 1], and ∫ ∫ t ∥dn ∥0 = maxt∈[0,1] | Since ∫1 ϕ −1 ( λψ(un (ξ))dξ| ≤ λψ(∥un ∥0 )dξ ≤ λψ(εn ) ) a(−λψ(un )) − dn (s) ds = 0, we have |a(−λψ(un ))| ≤ λψ(εn ) Otherwise, ∫1 ( ) ϕ−1 a(−λψ(un ))−dn (s) ds < (or > 0) Now, we show that is bounded Indeed, suppose that it is not true, i.e., for arbitrary A > 0, there exists N0 ∈ N such that implies that 2λ ≥ ϕ(Aεn ) ψ(εn ) −1 ϕ εn for all n > N0 However, −1 ϕ εn ( ( −1 ϕ εn ( ) 2λψ(εn ) ) 2λψ(εn ) → ∞ as n → ∞ Then, ) 2λψ(εn ) ≥ A, for all n > N0 This ϕ(Aεn ) ψ(εn ) → ϕp (A) as n → ∞ This is a satisfies un = F(un , λn ) for each n ∈ N Equivalently, (un , λn ) satisfies ∫ t un (t) = ϕ −1 ( ∫ a(−λn ψ(un ) − f (·, un , λn )) + −λn ψ(un (ξ)) − f (ξ, un , λn )dξ ds with ∫1 ) s ( ) ∫s ϕ−1 a(−λn ψ(un ) − f (·, un , λn )) + −λn ψ(un (ξ)) − f (ξ, un , λn )dξ ds = Let εn = ∥un ∥0 and (t) = (t) = εn ∫ t −1 ϕ ( un (t) εn Then ∫ s a(−λn ψ(un ) − f (·, un , λn )) + ) −λn ψ(un (ξ)) − f (ξ, un , λn )dξ ds, 0 and ′ (t) −1 ( = ϕ a(−λn ψ(un ) − f (·, un , λn )) + εn Now, define dn (t) = ∫t ∫ t −λn ψ(un (ξ)) − f (ξ, un , λn )dξ −λn ψ(un (ξ)) − f (ξ, un , λn )dξ Since f (t, u, λ) = ◦(|ψ(u)|) near zero, uniformly for t and λ, for some constants K1 and K2 ∫ ∥dn ∥0 = maxt∈[0,1] | t ≤ maxt∈[0,1] ∫ t λn ψ(un (ξ)) + f (ξ, un , λn )dξ| ∫ |λn ψ(un (ξ))| + |f (ξ, un , λn )|dξ λn ψ(∥un ∥0 ) + K1 ψ(∥un ∥0 )dξ ≤ ≤ K2 ψ(εn ) Since ∫1 ) ( ) ϕ−1 a(−λn ψ(un ) − f (·, un , λn )) − dn (s) ds = 0, we have |a(−λn ψ(un ) − f (·, un , λn ))| ≤ K2 ψ(εn ) 13 ∫1 Otherwise, that −1 ϕ εn ( ( ) ϕ−1 a(−λn ψ(un ) − f (·, un , λn )) − dn (s) ds > or < Now, let us verify ) 2K2 ψ(εn ) is bounded If −1 ϕ εn ( ) 2K2 ψ(εn ) → ∞ as n → ∞, then for arbitrary A > 0, there exists N0 ∈ N such that ) −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 ( ϕ a(−λn ψ(un ) − f (·, un , λn )) + εn ∫ t ) −λn ψ(un (ξ)) − f (ξ, un , λn )dξ ≤ K3 ′ Consequently, {vn } is uniformly bounded and by the Arzela-Ascoli Theorem, {vn } has a uniformly convergent subsequence in C[0, 1] Let → v in C[0, 1] Now claim that ∫ v(t) = t ϕ−1 p ( ∫ s ˆ ap (−λϕp (v)) + ) ˆ p (v(ξ))dξ ds −λϕ Clearly, ′ (t) −1 ( = ϕ a(−λn ψ(un ) − f (·, un , λn )) + εn ( ) ϕ−1 hn (t)ψ(εn ) ( ) , = ψ −1 ψ(εn ) where hn (t) = Since a(−λn ψ(un )−f (·,un ,λn )) ψ(εn ) a(−λn ψ(un )−f (·,un ,λn )) ψ(εn ) + ∫t ∫ t −λn ψ(un (ξ)) − f (ξ, un , λn )dξ ) (ξ))ϕ(εn −λn ψ(unn )ψ(εn ) ) − ϕ(ε f (ξ,un ,λn ) dξ ψ(εn ) is bounded, considering a subsequence if necessary, we may assume 14 n )−f that sequence { a(−λn ψ(uψ(εn )(·,un ,λn )) } converges to d as n → ∞ This implies that ′ (t) → ϕ−1 p ∫ ( d+ t ) ˆ p (v(ξ))dξ as n → ∞, −λϕ and thus v(t) = ∫t ( ) ∫s ˆ ˆ ϕ−1 d + −λϕp (v(ξ))dξ ds Since (1) = 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 is a neighborhood of (0, µ) containing no nontrivial solutions of (P ) In particular, we may choose an ε-ball Bε such that there are no solutions of (P ) on B ì [à − ε, µ + ε] and µ is the only eigenvalue of (Ep ) on [µ − ε, µ + ε] Let Φ(u, λ) = u − F(u, λ) Then deg(Φ(·, λ), B(0, ε), 0) is well-defined for λ with |λ − µ| ≤ ε Moreover, from the homotopy invariance theorem, deg(Φ(·, λ), B(0, ε), 0) ≡ constant, for all λ with |λ − µ| ≤ ε Now, we claim that deg(Φ(·, µ − ε), B(0, ε), 0) = deg(Φp (·, µ − ε), B(0, ε), 0), 15 µ−ε 1 where Φp (u, µ − ε) = u − Tp (u) Dene H : C0 [0, 1] ì [0, 1] → C0 [0, 1] by µ−ε H µ−ε (u, τ )(t) = τ F(u, µ − ε)(t) + (1 − τ )Tp (u)(t) µ−ε We know that F(·, µ−ε) and Tp are completely continuous To apply the homotopy invari- ance theorem, we need to show that ∈ u − H µ−ε (u, τ )(∂Bε ) to guarantee well-definedness / of deg(I −H µ−ε (·, τ ), B(0, ε), 0) Suppose that this is not the case, then there exist sequences {un }, {τn } and {εn } with εn → and ∥un ∥0 = εn such that un = H µ−ε (un , τn ), i.e., ∫ t un (t) = τn −1 ( ϕ ∫ s + a(−(µ − ε)ψ(un ) − f (·, un , µ − ε)) ) −(µ − ε)ψ(un (ξ)) − f (ξ, un , µ − ε)dξ ds ∫ t + (1 − τn ) ϕ−1 p ( ∫ Setting (t) = un (t) , εn τn (t) = εn ∫ ∫ ) −(µ − ε)ϕp (un (ξ))dξ ds we have that ∥vn ∥0 = and t s + s ap (−(µ − ε)ϕp (un )) + ( ϕ−1 a(−(µ − ε)ψ(un ) − f (·, un , µ − ε)) ) −(µ − ε)ψ(un (ξ)) − f (ξ, un , µ − ε)dξ ds ∫ + (1 − τn ) t ϕ−1 p ( ∫ ap (−(µ − ε)ϕp (vn )) + 0 16 s ) −(µ − ε)ϕp (vn (ξ))dξ ds Hence, we obtain that ′ (t) = τn −1 ( ϕ a(−(µ − ε)ψ(un ) − f (·, un , µ − ε)) εn ∫ t ) + −(µ − ε)ψ(un (ξ)) − f (ξ, un , µ − ε)dξ + (1 − τn )ϕ−1 p ( ∫ ) t ap (−(µ − ε)ϕp (vn )) + −(µ − ε)ϕp (vn (ξ))dξ , ′ 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 → v Moreover, using the fact that ∫ t ) −1 ( ϕ a(−(µ − ε)ψ(un ) − f (·, un , µ − ε)) + −(µ − ε)ψ(un (ξ)) − f (ξ, un , µ − ε)dξ εn ∫ t ( ) → ϕ−1 ap (−(µ − ε)ϕp (v)) + −(µ − ε)ϕp (v(ξ))dξ , p we can obtain that ∫ v(t) = t ϕ−1 p ( ∫ s ap (−(µ − ε)ϕp (v)) + ) −(µ − ε)ϕp (v(ξ))dξ ds This implies v ≡ and this is a contradiction Consequently, deg(I − H µ−ε (·, τ ), B(0, ε), 0) is well defined Therefore, by the homotopy invariance theorem, deg(Φ(·, µ − ε), B(0, ε), 0) = deg(Φp (·, µ − ε), B(0, ε), 0) Similarly, deg(Φ(·, µ + ε), B(0, ε), 0) = deg(Φp (·, µ + ε), B(0, ε), 0) 17 Let µ is k-th eigenvalue of (Ep ) Then by Lemma 2.2, we get deg(Φ(·, µ − ε), B(0, ε), 0) = (−1)k−1 and deg(Φ(·, µ + ε), B(0, ε), 0) = (−1)k This is a contradiction to the fact deg(Φ(·, µ − ε), B(0, ε), 0) = deg(Φ(·, µ + ε), B(0, ε), 0) Thus (0, µ) is a bifurcation point of (P ) Now, we shall adopt Rabinowitz’s standard arguement [11] Let S denote the closure of + the set of nontrivial solutions of (P ) and Sk denote the set of u ∈ C0 [0, 1] such that u has exactly k − simple zeros in (0,1), u > near 0, and all zeros of u in [0,1] are simple Let − + + − + − Sk = −Sk and Sk = Sk ∪ Sk We note that the sets Sk , Sk and Sk are open in C0 [0, 1] Moreover, let Ck denote the component of S which meets (0, µk ), where µk = λk (p) By the similar argument of Theorem 1.10 in [11], we can show the existence of two types of components C emanating from (0, µ) contained in S, when µ is an eigenvalue of (Ep ); either it is unbounded or it contains (0, µ), where µ(̸= µ) is an eigenvalue of (Ep ) The existence ˆ ˆ of a neighborhood Ok of (0, µk ) such that (u, λ) ∈ S ∩ Ok and u ̸≡ imply u ∈ Sk is also proved in [11] Actually, only the first alternative is possible as shall be shown next Lemma 3.5 Assume (Φ1 ), (Φ2 ), and (F1 ) Then, Ck is unbounded in Sk × R Proof: Suppose Ck (Sk ì R) {(0, àk )} Then since Sk ∩ Sj = ∅ for j ̸= k, it follows from the above facts, Ck must be unbounded in Sk ×R Hence, Lemma 3.5 will be established once we show Ck (Sk ìR){(0, àk )} is impossible It is clear that Ck Ok (Sk ìR){(0, àk )} Hence if Ck (Sk ìR){(0, àk )}, then there exists (u, λ) ∈ Ck ∩(∂Sk ×R) with (u, λ) ̸= (0, µk ) and (u, λ) = limn→∞ (un , λn ), un ∈ Sk If u ∈ ∂Sk , u ≡ 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 true, then there exists a sequence {(un , λn )} ⊂ Ck such that λn → ∞ Let ρjn be the jth zero of un Then there exists j ∈ {1, , k − 1} such that |ρ(j+1)n − ρjn | ≥ k Thus for each n, there exists σjn ∈ (ρjn , ρ(j+1)n ) such that u′n (σjn ) = Let un (t) > for all t ∈ (ρjn , ρ(j+1)n ) Suppose σjn ∈ (ρjn , ρjn +3ρ(j+1)n ] Then by integrating the equation in (P ) from σjn to t ∈ [σjn , ρ(j+1)n ], we see that un satisfies ∫ ρ(j+1)n un (t) = −1 (∫ t For t ∈ [ s ϕ ) λn ψ(un (ξ)) + f (ξ, un , λn )dξ ds σj n ρjn +4ρ(j+1)n ρjn +5ρ(j+1)n , ], ∫ un (t) ≥ ∫ ≥ ρ(j+1)n ρjn +5ρ(j+1) n ϕ ρ(j+1)n ρjn +5ρ(j+1) n ϕ −1 −1 (∫ (∫ t ) λn ψ(un (t))dξ ds σjn ρjn +4ρ(j+1) n ρjn +3ρ(j+1) n ) λn ψ(un (t))dξ ds ) ρ(j+1)n − ρjn −1 ( ρ(j+1)n − ρjn ϕ λn ψ(un (t)) 20 ( ) −1 ϕ λn ψ(un (t)) ≥ 6k 20k = Thus λn ϕ(6kun (t)) ≥ ψ(un (t)) 20k (11) The left side of (11) is bounded and independent on n, but the right side goes to ∞ as 19 n → ∞ This is impossible Now, if σjn ∈ ( ρjn +3ρ(j+1)n , ρ(j+1)n ), then by integrating the equation in (P ) from t ∈ [ρjn , σjn ] to σjn , we see that un satisfies ∫ t un (t) = ϕ −1 (∫ ρjn For t ∈ [ σj n ) λn ψ(un (t)) + f (ξ, un , λn )dξ ds σj n ) λn ψ(un (t))dξ ds s ρjn +ρ(j+1)n ρjn +2ρ(j+1)n , ], ∫ t un (t) ≥ ϕ −1 (∫ ρjn ∫ ≥ t ρjn +ρ(j+1) n ϕ σjn −1 (∫ ) ρjn +3ρ(j+1) n ρjn +2ρ(j+1) n λn ψ(un (t))dξ ds ) ρ(j+1)n − ρjn −1 ( ρ(j+1)n − ρjn ϕ λn ψ(un (t)) 12 ( ) −1 ≥ ϕ λn ψ(un (t)) 2k 12k = From the above argument, we get ϕ(2kun (t)) λn ≥ ψ(un (t)) 12k (12) This is impossible because the left side is bounded and independent on n, but the right side goes to infinity as n goes to infinity We can get similar results when un (t) < Indeed, if σjn ∈ (ρjn , ρjn +3ρ(j+1)n ], then we have λn ϕ(6k|un (t)|) ≥ ψ(|un (t)|) 20k 20 (13) Also if σjn ∈ ( ρjn +3ρ(j+1)n , ρ(j+1)n ), then we have λn ϕ(2k|un (t)|) ≥ ψ(|un (t)|) 12k (14) Since both (13) and (14) are impossible, there is no sequence {(un , λn )} ⊂ Ck satisfying λn → ∞ Consequently, there exists an Mk ∈ (0, ∞) such that λ ≤ Mk Proof of Theorem 1.1 By Lemmas 3.3, 3.4, and 3.5, for any j ∈ N, there exists an unbounded connected component Cj of the set of nontrivial solutions emanating from (0, λj (p)) such that (u, λ) ∈ Cj implies u has exactly j − simple zeros in (0,1) From Lemma 3.6, there is an Mj such that (u, λ) ∈ Cj implies that λ ≤ Mj , and there are no nontrivial solutions of (P ) for λ = 0, it follows that for any M > 0, there is (u, λ) ∈ Cj such that ∥u∥1 > M Hence, we can choose subsequence ˆ ˆ {(un , λn )} ⊂ Cj such that λn → λ and ∥un ∥1 → ∞ Thus, (∞, λ) is a bifurcation point and ˆ λ = λj (q) 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) = 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 in (Ag ) can be equivalently changed into the following equation    −(ϕ(u′ (t)))′ = (λ + µ)ϕ(u(t)) + f (t, u, λ), t ∈ (0, 1),  (Af )    u(0) = u(1) = By the similar argument in the proof of Theorem 1.1, for each k ≤ j ≤ n, there is a connected branch Cj of solutions to (Af ) emanating from (0, λj (p) − µ) which is unbounded in C0 [0, 1] × R and such that (u, λ) ∈ Cj implies that u has exactly j − simple zeros in (0,1) From the fact ug(t, u) ≥ 0, it can be proved that there is an Mj > such that (u, λ) ∈ Cj implies that λ ≤ Mj , by the same argument as in the proof of Lemma 3.6 Since there is a constant Kg > such that g(t, s) ≤ Kg ϕ(s) for all (t, s) ∈ [0, 1] × R, if (u, λ) ∈ Cj , then λ > −Kg Hence Cj will bifurcate from infinity also, which can only happen for λ = λj (q)−ν Since λj (q) − ν < < λj (p) − µ and Cj is connected, there exists u ̸= such that (u, 0) ∈ Cj This u is a solution of (A) Since this is true for every such j, (A) has at least n − k + nontrivial solutions Finally, we illustrate several examples of Theorems 1.1 and 1.2 Example 4.1 Define ϕ, ψ, f by ϕ(u) =   u + u2 + u3 , if u ≥ 0,   u − u2 + u3 , if u < 0,  ψ(u) = 22   u + 2u2 + u3 , if u ≥ 0,   u − 2u2 + u3 , if u < 0,  f (t, u, λ) =    λu2 , if u ≥ 0,   −λu2 , if u <  Then ϕ and ψ are odd increasing homeomorphisms of R and ϕ(σu) = σ = ϕ2 (σ), u→0 ψ(u) lim ϕ(σu) = σ = ϕ4 (σ) |u|→∞ ψ(u) lim Moreover, f satisfies f (t, u, λ) = ◦(|ψ(u)|) near zero and infinity, uniformly in t and λ, and uf (t, u, λ) ≥ Therefore, all hypotheses of Theorem 1.1 are satisfied Example 4.2 Define ϕ, g by ϕ(u) =   u + u2 , if u ≥ 0,   u − u2 , if u < 0,  g(t, u) =   πu + π u2 , if u ≥ 0,   πu − π u2 , if u <  Then ϕ is odd increasing homeomorphism of R and ϕ(σu) = σ = ϕ2 (σ), u→0 ϕ(u) lim ϕ(σu) = σ = ϕ3 (σ) |u|→∞ ϕ(u) lim Moreover, ug(t, u) ≥ and g(t, u) = π, u→0 ϕ(u) lim g(t, u) = π3 |u|→∞ ϕ(u) lim Thus we can check on the fact that ( 4√3π )3 g(t, u) g(t, u) < λ1 (2) = π < λ1 (3) = < lim lim u→0 ϕ(u) |u|→∞ ϕ(u) 23 All hypotheses of Theorem 1.2 for k = n = are satisfied so that (A) possesses at least one nontrivial solution Example 4.3 Define ϕ, g by     ϕ(u) = u2 ln(u + 1), if u ≥ 0,  −u ln(−u + 1), if u < 0,  g(t, u) =   10  u ln(u + 1) tan−1 u, if u ≥ 0,   10 2 u ln(−u + 1) tan−1 u, if u <  Then ϕ is odd increasing homeomorphism of R and ϕ(σu) = σ = ϕ4 (σ), u→0 ϕ(u) lim ϕ(σu) = σ = ϕ3 (σ) |u|→∞ ϕ(u) lim Moreover, ug(t, u) ≥ and g(t, u) = 0, u→0 ϕ(u) lim g(t, u) = 29 π |u|→∞ ϕ(u) lim Thus we can check on the fact that ( √2π )4 ( 4√3π )3 g(t, u) g(t, u) lim < λ1 (4) = < λ3 (3) = < lim u→0 ϕ(u) |u|→∞ ϕ(u) All hypotheses of Theorem 1.2 for k = and n = are satisfied so that (A) possesses at least three nontrivial solutions Competing interests The authors declare that they have no competing interests 24 Author’s contributions All authors have equally contributed in obtaining new results in this article and also read and approved the final manuscript Acknowledgment YH LEE was supported by the Mid-career Researcher Program through NRF grant funded by the MEST (No 2010-0000377) References [1] Garc´ ia-Huidobro, M, Man´sevich, R, Schmitt, K: Some bifurcation results for a class of a p-Laplacian like operators Diff Integral Eqns 10, 5166 (1997) [2] Agarwal, RP, Lă, 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 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Analysis Springer, Berlin (1985) [11] Rabinowitz, PH: Some aspect of nonlinear eigenvalue problems Rocky Mountain J Math 3, 161–202 (1973) Figure 1: A connected component Cj of (P ) 26 Figure .. .Global bifurcation results for general Laplacian problems Eun Kyoung Lee1 , Yong-Hoon Lee2∗ and Byungjae Son2 Department... ψ(t) (Φ2 ) lim|t|→∞ = σ p−1 , for all σ ∈ R+ , for some p > ϕ(σt) ψ(t) = σ q−1 , for all σ ∈ R+ , for some q > (F1 ) f (t, u, λ) = ◦(|ψ(u)|) near zero, uniformly for t and λ in bounded intervals... 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: