## Existence and multiplicity of solutions for fourth-order boundary value problems with three parameters.(English)Zbl 1135.34007

Summary: This paper is concerned with the existence and multiplicity of the solutions for the fourth-order boundary value problem $u^{(4)}(t)+\eta u''(t)-\zeta u(t)=\lambda f(t,u(t)),\quad 0<t<1,$
$u(0)=u(1)=u''(0)=u''(1)=0,$ where $$f:[0,1]\times \mathbb R\to\mathbb R$$ is continuous, $$\zeta,\eta$$ and $$\lambda\in\mathbb R$$ are parameters. Using the variational structure of the above boundary value problem and critical point theory, it is shown that the different locations of the pair $$(\eta,\zeta)$$ and $$\lambda\in \mathbb R$$ lead to different existence results for the above boundary value problem. More precisely, if the pair $$(\eta,\zeta)$$ is on the left side of the first eigenvalue line, then the above boundary value problem has only the trivial solution for $$\lambda\in (-\lambda,0)$$ and has infinitely many solutions for $$\lambda\in (0,\infty)$$; if $$(\eta,\zeta)$$ is on the right side of the first eigenvalue line and $$\lambda\in (-\infty,0)$$, then the above boundary value problem has two nontrivial solutions or has at least $$n_*$$ $$(n_*\in\mathbb N)$$ distinct pairs of solutions, which depends on the fact that the pair $$(\eta,\zeta)$$ is located in the second or fourth (first) quadrant.

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations 47J30 Variational methods involving nonlinear operators
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### References:

  Brezis, H.; Nirenberg, L., Remarks on finding critical points, Comm. Pure Appl. Math. XLIV, 939-963 (1999) · Zbl 0751.58006  Bai, Z.; Wang, H., On positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl., 270, 357-368 (2002) · Zbl 1006.34023  Bai, Z., The method of lower and upper solutions for a bending of a elastic beam equation, J. Math. Anal. Appl., 248, 195-202 (2000) · Zbl 1016.34010  Gupta, C. P., Existence and uniqueness results of the bending of a beam equation at resonance, J. Math. Anal. Appl., 135, 208-225 (1988) · Zbl 0655.73001  Gyulov, T.; Tersian, S., Existence of trivial and nontrivial solutions of a fourth-order ordinary differential equation, Electron. J. Differential Equations, 41, 1-14 (2004) · Zbl 1065.34016  Hao, Z. C.; Debnath, L., On eigenvalue intervals and eigenfunctions of fourth-order singular boundary value problems, Appl. Math. Lett., 18, 543-553 (2005) · Zbl 1074.34079  Li, F.; Zhang, Q.; Liang, Z., Existence and multiplicity of solutions of a kind of fourth-order boundary value problem, Nonlinear Anal., 62, 803-816 (2005) · Zbl 1076.34015  Li, Y., Positive solutions of fourth-order boundary value problems with two parameters, J. Math. Anal. Appl., 281, 477-484 (2003) · Zbl 1030.34016  Li, Y., Two-parameter nonresonance condition for the existence of fourth-order boundary value problems, J. Math. Anal. Appl., 308, 121-138 (2005) · Zbl 1071.34016  Liu, X. L.; Li, W. T., Positive solutions of the nonlinear fourth-order beam equation with three parameters, J. Math. Anal. Appl., 303, 150-163 (2005) · Zbl 1077.34027  Liu, X. L.; Li, W. T., Positive solutions of the nonlinear fourth-order beam equation with three parameters (II), Dynam. Systems Appl., 15, 415-428 (2006)  Love, A. E.H., A Treatise on the Mathematical Theory of Elasticity (1964), Noordhoff: Noordhoff Groningen · Zbl 0063.03651  Prescott, J., Applied Elasticity (1961), Dover: Dover New York · JFM 50.0554.12  Del Pino, M. A.; Manasevich, R. F., Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition, Proc. Amer. Math. Soc., 112, 1, 81-86 (1991) · Zbl 0725.34020  Peletier, L. A.; Troy, W. C.; van der Vors, R. C.A. M., Stationary solutions of a fourth order nonlinear differential equation, Differ. Equ., 31, 2, 301-314 (1995) · Zbl 0856.35029  Timoshenko, S. P., Theory of Elastic Stability (1961), McGraw-Hill: McGraw-Hill New York  Tersian, S.; Chaparora, J., Periodic and homoclinic solutions of existence Fisher-Kolmogorow equation, J. Math. Anal. Appl., 266, 490-506 (2001)  Yao, Q., On the positive solutions of a nonlinear fourth-order boundary value problem with two parameters, Appl. Anal., 83, 97-107 (2004) · Zbl 1051.34018
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