Existence and Multiplicity of Constant Sign Solutions for One-Dimensional Beam Equation
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Existence and Multiplicity of Constant Sign Solutions for One-Dimensional Beam Equation Dongliang Yan1 · Ruyun Ma1 · Zhongzi Zhao1 Received: 13 April 2020 / Revised: 1 August 2020 / Accepted: 17 August 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, we consider the nonlinear eigenvalue problems u = λh(t) f (u), 0 < t < 1, u(0) = u(1) = u (0) = u (1) = 0, where h ∈ C([0, 1], (0, ∞)); f ∈ C(R, R) and s f (s) > 0 for s = 0, and f 0 = f ∞ = 0, f 0 = lim|s|→0 f (s)/s, f ∞ = lim|s|→∞ f (s)/s. We investigate the global structure of one-sign solutions by using bifurcation techniques. Keywords Connected component · Green function · One-sign solutions · Bifurcation · Clamped beam Mathematics Subject Classification 34B27 · 34C23 · 74K10
Communicated by Shangjiang Guo. Ruyun Ma: Supported by the NSFC (No.11671322).
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Ruyun Ma [email protected] Dongliang Yan [email protected] Zhongzi Zhao [email protected]
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Department of Mathematics, Northwest Normal University, Lanzhou 730070, People’s Republic of China
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D. Yan et al.
1 Introduction The deformations of an elastic beam whose both ends clamped are described by the fourth-order problem u = λh(t) f (u), 0 < t < 1, u(0) = u(1) = u (0) = u (1) = 0,
(1.1)
where h ∈ C([0, 1], (0, ∞)); f ∈ C(R, R) and s f (s) > 0 for s = 0. Existence and multiplicity of positive solutions of (1.1) have been extensively studied by several authors, see [1–9]. Cabada and Enguiça [3] developed the method of lower and upper solutions to show the existence and multiplicity of solutions; Yao [6] and Zhai et al. [8] proved the existence and multiplicity of positive solutions via the fixed point theorem in cone. In particular, Bonanno and Bella in [10] considered the problem u = λ f (u), 0 < t < 1, u(0) = u(1) = u (0) = u (1) = 0
(1.2)
where f ∈ C(R, R) and s f (s) > 0 for s = 0, and f 0 = lim|s|→0 f (s)/s = 0, f ∞ = lim|s|→∞ f (s)/s = 0. Then for every λ > λ¯ =:
2 2 d d 8192 + 8π 2 max inf d , inf d , d>0 d 0 for |s| > 0; f 0 = 0; f ∞ = 0.
Let Y = C[0, 1] with the norm ||u||∞ = max |u(t)|. t∈[0,1]
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Existence and Multiplicity of Constant Sign Solutions for… Fig. 1 Components of one-sign solutions in Theorem 1.1
We shall use Dancer’s bifurcation theorem and some properties of superior limit of certain infinity collection of connected sets to establish the following Theorem 1.1 Let (A1), (A2), (A3) and (A4) hold. Then there exist a connected component C + ⊂ R+ × C[0, 1] of positive solutions of (1.1) and a connected component C − ⊂ R+ × C[0, 1] of negative solutions of (1.1), such that (1) C + is of ⊂-shaped and joins (+∞, 0) to (+∞, ∞); (2) For every ρ > 0, there exists ρ > 0, such that (λ, u) ∈ C + with ||u||∞ = ρ ⇒ λ > ρ ; (3) C − is of ⊂-shaped and joins (+∞, 0) to (+∞, ∞); (4) For every ρ > 0, there exists ρ > 0, such that (λ, u) ∈ C − with ||u||∞ = ρ ⇒ λ > ρ . Corollary 1.1 Let (A1), (A2), (A3) and (A4) hold. Then (1.1) with h ≡ 1 has at least two positive solutions a
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