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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.

MSC:
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
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[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.; 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] Han, G.D.; Xu, Z.B., Multiple solutions of some nonlinear fourth-order beam equations, Nonlinear anal., 68, 3646-3656, (2008) · Zbl 1145.34008
[4] 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
[5] Yang, Y.; Zhang, J.H., Existence of solutions for some fourth order boundary value problems with parameters, Nonlinear anal., 69, 1364-1375, (2008) · Zbl 1166.34012
[6] Chang, K.C., Solutions of asymptotically linear operator equations via Morse theory, Comm. pure. appl. math., 34, 693-712, (1981) · Zbl 0444.58008
[7] Taylor, A.E.; Lay, D.C., Introduction to functional analysis, (1980), Wiley New York
[8] Rabinowitz, Y.H., ()
[9] Chang, K.C., Infinite dimensional Morse theory and multiple solution problems, (1993), Birkhäuser Boston
[10] Su, J.B.; Zhao, L.G., An elliptic resonance problem with multiple solutions, J. math. anal. appl., 319, 604-616, (2006) · Zbl 1155.35358
[11] Furtado, M.F.; Paiva, F.O.V., Multiplicity of solutions for resonant elliptic systems, J. math. anal. appl., 319, 435-449, (2006) · Zbl 1108.35046
[12] Amrouss, A.R.E., Nontrivial solutions of semilinear equations at resonance, J. math. anal. appl., 325, 19-35, (2007) · Zbl 1167.35365
[13] Liu, J.Q., The Morse index of a saddle point, Systems. sci. math. sci., 2, 32-39, (1989) · Zbl 0732.58011
[14] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer-Verlag New York · Zbl 0676.58017
[15] 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
[16] 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
[17] Cerami, G., Un criterio de esistenza per i punti critic su varietà ilimitade, Istit. lombardo accad. sci. lett. dend., A 112, 332-336, (1978) · Zbl 0436.58006
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