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Nontrivial solutions for some fourth order boundary value problems with parameters. (English) Zbl 1171.34006
The authors consider the fourth order boundary value problem
\[ \begin{aligned} &u^{(4)}(t)+\eta u^{(2)}(t)-\xi u(t)=f(t,u(t)), \quad 0<t<1,\\ &u(0)=u(1)=u^{(2)}(0)=u^{(2)}(1)=0 \end{aligned}\tag{BVP} \] with continuous nonlinearity \(f:[0,1] \times\mathbb R\to\mathbb R\) and fixed \(\eta ,\xi \) such that \(\frac{\xi }{\pi ^{4}}+\frac{\eta }{\pi ^{2}}<1\) , \(\xi \geq -\frac{\eta ^{2}}{4}\), \(\eta <2\pi ^{2}\). Using Green’s function the authors provide a fixed point formulation of (BVP) which they next consider by variational methods. The investigations of an equivalent variational formulation involve a square root operator of a suitable integral functional. Such an approach the authors already adopted in Y. Yang and J. Zhang [Nonlinear Anal., Theory Methods Appl. 69, No. 4 (A), 1364–1375 (2008; Zbl 1166.34012)].
In order to prove the main results the authors apply variational techniques involving Morse theory and local linking. However, contrary to the case considered in the paper mentioned previously, the Cerami compactness condition is used. Depending on the assumptions placed on the nonlinear term \(f\), the authors prove that (BVP) has at least one and next at least two nontrivial solutions. The paper is concluded with examples of functions satisfying the assumptions introduced in the paper.
The paper in fact uses topological and the variational methods which fact is very interesting.

34B15 Nonlinear boundary value problems for ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
47N20 Applications of operator theory to differential and integral equations
Full Text: DOI
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