## Nontrivial solutions of singular fourth-order Sturm-Liouville boundary value problems with a sign-changing nonlinear term.(English)Zbl 1213.34045

Summary: This paper concerns the existence of nontrivial solutions for the following singular boundary value problem with a sign-changing nonlinear term:
$\begin{cases} u^{(4)}(t) =h(t)f(t,u(t),u''(t)),\quad 0<t<1,\\ \alpha_1u(0)-\beta_1u'(0)=\delta_1u(1)+\gamma_1u'(1)=0,\\ \alpha_2u''(0)-\beta_2u'''(0)=\delta_2u''(1)+\gamma_2u'''(1)=0,\end{cases}$
where $$h(t)$$ is allowed to be singular at $$t = 0$$ and/or $$t=1$$. Moreover, $$f(t,x,y):[0,1]\times \mathbb R^2\to\mathbb R$$ is a sign-changing continuous function and may be unbounded from below with respect to $$x$$ and $$y$$. By applying the topological degree of a completely continuous field and eigenvalue, some new results on the existence of nontrivial solutions for the above boundary value problem are obtained.

### MSC:

 34B24 Sturm-Liouville theory 34B16 Singular nonlinear boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations
Full Text:

### References:

 [1] O’Regan, D., Theory of singular boundary value problems, (1994), World Science Singapore · Zbl 0808.34022 [2] Liu, L.; Kang, P.; Wu, Y.; Wiwatanapataphee, B., Positive solutions of singular boundary value problems for systems of nonlinear fourth order differential equations, Nonlinear anal., 68, 485-498, (2008) · Zbl 1134.34015 [3] Liu, L.; Zhang, X.; Wu, Y., Positive solutions of fourth-order nonlinear singular sturm – liouville eigenvalue problems, J. math. anal. appl., 326, 1212-1224, (2007) · Zbl 1113.34022 [4] Zhang, X.; Liu, L.; Zou, H., Eigenvalues of fourth-order singular sturm – liouville boundary value problems, Nonlinear anal., 68, 384-392, (2008) · Zbl 1131.34022 [5] Webb, J.R.L.; Infante, G.; Franco, D., Positive solutions of nonlinear fourth-order boundary value problems with local and non-local boundary conditions, Proc. roy. soc. Edinburgh, 138A, 427-446, (2008) · Zbl 1167.34004 [6] Zhang, X.; Liu, L.; Jiang, J., Positive solutions of fourth-order sturm – liouville boundary value problems with changing sign nonlinearity, Nonlinear anal., 69, 4764-4774, (2008) · Zbl 1159.34318 [7] Sun, J.; Zhang, G., Nontrivial solutions of singular super-linear sturm – liouville problem, J. math. anal. appl., 313, 518-536, (2006) · Zbl 1100.34019 [8] Cui, Y.; Zou, Y., Nontrivial solutions of singular superlinear boundary value problems, Appl. math. comput., 187, 1256-1264, (2007) · Zbl 1121.34030 [9] Han, G.; Wu, Y., Nontrivial solutions of singular two-point boundary value problems with sign-changing nonlinear terms, J. math. anal. appl., 325, 1327-1338, (2007) · Zbl 1111.34019 [10] Sun, Y.; Liu, L., Solvability for nonlinear second-order three-point boundary value problem, J. math. anal. appl., 296, 265-275, (2004) · Zbl 1069.34018 [11] Agarwal, R.P.; Staněk, S., Nonnegative solutions of singular boundary value problems with sign changing nonlinearities, Comput. math. appl., 46, 1827-1837, (2003) · Zbl 1156.34310 [12] Guo, D., Nonlinear functional analysis, (2001), Shangdong Science and Technology press Jinan, (in Chinese) [13] Guo, D.; Sun, J., Nonlinear integral equations, (1987), Shangdong Science and Technology press Jinan, (in Chinese) [14] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag Berlin · Zbl 0559.47040
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.