Non-trivial solutions for two-point boundary-value problems of fourth-order Sturm-Liouville type equations. (English) Zbl 1252.34035
The author proves the existence of at least one non-trivial solution for the two-point boundary value problem of fourth-order Sturm-Liouville type
\[
(p_i(x) u_i''(x))'' - (q_i(x) u_i'(x))' + r_i(x) u_i(x) =\lambda F_{u_i}(x,u_1,\dots,u_n), \quad x \in (0,1);
\]
\[ u_{i}(0)=u_{i}(1)=u_{i}''(0)=u_{i}''(1)=0, \quad 1\leq i \leq n, \] where \(\lambda\) is a parameter and the functions \(p_i, q_i, r_i, F\) satisfy some conditions. The proof is based on a recent result of G. Bonanno [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 5, 2992–3007 (2012; Zbl 1239.58011)].
\[ u_{i}(0)=u_{i}(1)=u_{i}''(0)=u_{i}''(1)=0, \quad 1\leq i \leq n, \] where \(\lambda\) is a parameter and the functions \(p_i, q_i, r_i, F\) satisfy some conditions. The proof is based on a recent result of G. Bonanno [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 5, 2992–3007 (2012; Zbl 1239.58011)].
Reviewer: Zhiqing Han (Dalian)
MSC:
| 34B24 | Sturm-Liouville theory |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 47J10 | Nonlinear spectral theory, nonlinear eigenvalue problems |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
