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Existence of solutions for some fourth-order boundary value problems with parameters. (English) Zbl 1166.34012
The authors consider the boundary value problem (BVP)
\[ u^{( 4) }( t) +\eta u^{( 2) }( t) -\xi u( t) =\lambda f( t,u( t)),\quad 0<t<1, \]
\[ u(0) =u(1) =u^{(2)}(0) =u^{(2)}(1) =0 \] with continuous nonlinearity \(f:\left[ 0,1\right] \times \mathbb{R}\rightarrow \mathbb{R}\) and fixed \(\eta ,\xi \) such that
\[ \frac{\xi}{\pi^{4}}+\frac{\eta}{\pi^{2}}<1,\;\xi \geq -\frac{\eta ^{2}}{4},\;\eta <2\pi ^{2} \] and where \(\lambda \in \mathbb{R}^{+}\) is a parameter. Using Green’s function the authors provide a fixed point formulation of (BVP) for which they find an action functional and apply variational methods. The investigations of the equivalent variational formulation involve a square root operator of a suitable integral functional.
Depending on the assumptions on the nonlinear term \(f\), they further obtain the values of \(\lambda \) for which (BVP) has at least one and at least two nontrivial solutions.
The proofs are based on variational techniques involving Morse theory and local linking. The paper is interesting since it shows the interplay between the topological and the variational methods.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
58E30 Variational principles in infinite-dimensional spaces
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References:
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