Yang, Yang; Zhang, Jihui Existence of solutions for some fourth-order boundary value problems with parameters. (English) Zbl 1166.34012 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 4, 1364-1375 (2008). 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. Reviewer: Marek Galewski (Łódź) Cited in 1 ReviewCited in 21 Documents 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 Keywords:boundary value problem; homological nontrivial critical point; Morse theory; local linking PDF BibTeX XML Cite \textit{Y. Yang} and \textit{J. Zhang}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 4, 1364--1375 (2008; Zbl 1166.34012) Full Text: DOI References: [1] Liu, X.L.; Li, W.T., Existence and multiplicity of solutions for fourth order boundary value problems with parameters, J. math. anal. appl., 327, 362-375, (2007) · Zbl 1109.34015 [2] Li, F.Y.; Li, Y.H.; Liang, Z.P., Existence of solutions to nonlinear Hammerstein integral equations and applications, J. math. anal. appl., 323, 209-227, (2006) · Zbl 1104.45003 [3] Li, F.Y.; Liang, Z.P.; Zhang, Q., Existence and multiplicity of solutions of a kind of fourth-order boundary value problem, Nonlinear. anal., 62, 803-816, (2005) · Zbl 1076.34015 [4] Taylor, A.E.; Lay, D.C., Introduction to functional analysis, (1980), Wiley New York [5] Bartsch, T.; Li, S.J., Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear. anal., 28, 419-441, (1997) · Zbl 0872.58018 [6] Liu, J.Q., The Morse index of a saddle point, Systems. sci. math. sci., 2, 32-39, (1989) · Zbl 0732.58011 [7] Liu, J.Q.; Su, J.B., Remarks on multiple nontrivial solutions for quasi-linear resonant problems, J. math. anal. appl., 258, 209-222, (2001) · Zbl 1050.35025 [8] Zhang, J.H.; Li, S.J., Multiple nontrivial solutions for some fourth order semilinear elliptic problems, Nonlinear anal., 60, 221-230, (2005) · Zbl 1103.35027 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.