Abstract
In this paper, we discuss the existence of solutions for a 2nth-order differential equation with Sturm–Liouville operator. Using critical point theorems due to Bonanno and Candito, we prove that there exist at least three classical solutions for the boundary value problem.
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Agarwal, R.P., Henderson, J.: Positive solutions and nonlinear eigenvalue problems for third-order difference equations. Comput. Math. Appl. 36, 347–355 (1998)
Bonanno, G., Candito, P.: Non-difierentiable functionals and applications to elliptic problems with discontinuous nonlinearities. J. Difier. Equ. 244, 3031–3059 (2008)
Graef, G.R., Heidarkhani, S., Kong, L.: Multiple solutions for systems of multi-point boundary value problems. Opuscula Math. 33, 293–306 (2013)
Graef, J.R., Qian, C., Yang, B.: A three point boundary value problem for nonlinear fourth order differential equations. J. Math. Anal. Appl. 287, 217–233 (2003)
Graef, J.R., Heidarkhani, S., Kong, L.: Multiple solutions for systems of Sturm–Liouville boundary value problems. Mediterr. J. Math. 13, 1625–1640 (2016)
Graef, J.R., Heidarkhani, S., Kong, L.: Infinitely many solutions for systems of Sturm–Liouville boundary value problems. Results Math. 66, 327–341 (2014)
Graef, J.R., Heidarkhani, S., Kong, L.: Nontrivial solutions for systems of Sturm–Liouville boundary value problems. Differ. Equ. Appl. 6, 255–265 (2014)
Heidarkhani, S.: Existence of solutions for a two-point boundary-value problem of a fourth order Sturm–Liouville type. Electron. J. Differ. Equ. 84, 1–15 (2012)
Kong, L.: Eigenvalues for a fourth-order elliptic problem. Proc. Am. Math. Soc. 143, 249–258 (2015)
Saavedra, L., Tersian, S.: Existence of solutions for 2nth-order nonlinear p-Laplacian differential equations. Nonlinear Anal. Real World Appl. 34, 505–519 (2017)
Liu, X., Tian, Y.: On positive solutions of Sturm–Liouville boundary-value problem for fourth-order impulsive differential equations. Dyn. Syst. Appl. 23, 189–201 (2014)
Tian, Y., Ge, W.: Multiple solutions for a second-order Sturm–Liouville boundary value problem. Taiwanese. J. Math. 11, 975–988 (2007)
Tian, Y., Ge, W.: Second-order Sturm–Liouville boundary value problem involving the one-dimensional p-Laplacian. Rocky Mt. J. Math. 38, 309–327 (2008)
Wu, H.: Positive solutions to fourth-order three-point nonlinear eigenvalue problem. Ann. Appl. Math. 31, 96–104 (2015)
Zhou, Y., Yang, Y., Liu, L.: Multiple positive solutions for a fourth-order nonlinear eigenvalue problem. Ann. Appl. Math. 32, 418–428 (2016)
Zeidler, E.: Nonlinear functional analysis and its applications, vol. 2. Springer, Berlin (1990)
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This work is supported by the National Natural Science Foundation of China (Grant nos 61377067, 11375033) and Beijing University of Posts and Telecommunications in 2017 teaching research and reform project 500517183.
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Tian, Y., Zhang, M. & Shang, S. Existence Results for a 2nth-Order Differential Equation via Variational Methods. Mediterr. J. Math. 15, 10 (2018). https://doi.org/10.1007/s00009-017-1055-y
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DOI: https://doi.org/10.1007/s00009-017-1055-y