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Three positive solutions of fourth-order problems with clamped beam boundary conditions. (English) Zbl 1476.34078

In this paper, the author investigates the existence of three positive solutions of fourth order problems \[ u^{\prime \prime \prime \prime }\left( x\right) =\lambda f\left( u\left( x\right) \right) ,x\in \left( -1,1\right) \] with \[ u\left( 1\right) =u\left( -1\right) =u^{\prime }\left( 1\right) =u^{\prime }\left( -1\right) =0, \] where \(\lambda >0\) is a parameter, \(f\in C \left[ 0,\infty \right) \) is nondecreasing, \(f\left( 0\right) =0\) and \( f\left( s\right) >0\) for all \(s>0.\)
The proof is based on the directions of a bifurcation as the author studied the existence and multiplicity of positive solutions and the shape of unbounded connected components in the positive solutions set of the considered problem.
Moreover, the author made the following assumptions:
(H1) There exist \(\alpha >0,\) \(f_{0}>0\) and \(f_{1}>0\) such that \( \lim\limits_{s\rightarrow 0^{+}}\frac{f\left( s\right) -f_{0}s}{s^{1+\alpha } }=-f_{1}.\)
(H2) There exists \(f_{\infty }:=\lim\limits_{s\rightarrow \infty }\frac{ f\left( s\right) }{s}=0.\)
(H3) There exists \(s_{0}>0\) such that \(\min\limits_{s\in \left[ s_{0},\frac{K }{m}s_{0}\right] }\frac{f\left( s\right) }{s}\geq \frac{f_{0}}{\lambda _{1}} \eta _{1},\)
where \(K\) and \(m\) are suitable chosen bounds of the related Green’s function and \(\eta _{1}\) is the first positive eigenvalue of the following linear problem:
\(\Psi ^{\prime \prime \prime \prime }\left( x\right) =\eta \Psi \left( x\right) ,x\in \left( -\frac{1}{2},\frac{1}{2}\right) \) with \(\Psi \left( \frac{1}{2}\right) =\Psi \left( -\frac{1}{2}\right) =\Psi ^{\prime }\left( \frac{1}{2}\right) =\Psi ^{\prime }\left( -\frac{1}{2}\right) =0.\)
The main result of the paper is as follows. Assume that the above conditions (H1)-(H3) hold. Then there exist \(\lambda _{\ast }\in \left( 0,\frac{\lambda _{1}}{f_{0}}\right) \) and \(\lambda ^{\ast }>\frac{\lambda _{1}}{f_{0}}\) such that
(i) the considered problem has at least one positive solution if \(\lambda =\lambda _{\ast }\);
(ii) the considered problem has at least two positive solutions if \(\lambda _{\ast }<\lambda \leq \frac{\lambda _{1}}{f_{0}}\);
(iii) the considered problem has at least three positive solutions if \(\frac{ \lambda _{1}}{f_{0}}<\lambda <\lambda ^{\ast }\);
(iv) the considered problem has at least two positive solutions if \(\lambda =\lambda ^{\ast }\);
(v) the considered problem has at least one positive solution if \(\lambda >\lambda ^{\ast }\).

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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References:

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