Global bifurcation and nodal solutions for fourth-order problems with sign-changing weight. (English) Zbl 1294.34017

Summary: In this paper, we shall establish unilateral global bifurcation result for a class of fourth-order eigenvalue problems with sign-changing weight. Under some natural hypotheses on the perturbation function, we show that \((\mu_k^{\nu},0)\) is a bifurcation point of the above problems and there are two distinct unbounded continua, \((\mathcal C_k^{\nu})^+\) and \((\mathcal C_k^{\nu})^-\), consisting of the bifurcation branch \(\mathcal C_k^{\nu}\) from \(({\mu}_k^{\nu},0)\), where \(\mu_k^{\nu}\) is the \(k\)th positive or negative eigenvalue of the linear problem corresponding to the above problems, \(\nu\in \{+,-\}\). As applications of the above result, we study the existence of nodal solutions for a class of fourth-order eigenvalue problems with sign-changing weight. Moreover, we also establish the Sturm type comparison theorem for fourth-order problems with sign-changing weight.


34B09 Boundary eigenvalue problems for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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[1] Brezis, H., Operateurs maximaux monotone et semigroup de contractions dans LES espase de Hilbert, Math. Stud., vol. 5, (1973), North-Holland Amsterdam
[2] Bailey, P. B.; Shampine, L. F.; Waltman, P. E., Nonlinear two-point boundary value problems, (1968), Academic Press New York · Zbl 0169.10502
[3] Bai, Z. B.; Wang, H. Y., On positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl., 270, 357-368, (2002) · Zbl 1006.34023
[4] Cantrell, R. S.; Cosner, C., Spatial ecology via reaction-diffusion equations, (2003), Wiley Chichester · Zbl 1059.92051
[5] Dai, G.; Ma, R., Unilateral global bifurcation phenomena and nodal solutions for p-Laplacian, J. Differ. Equ., 252, 2448-2468, (2012) · Zbl 1242.34068
[6] G. Dai, R. Ma, Spectrum of Navier p-biharmonic problem with sign-changing weight, arXiv:1207.7159v1 [math.CA].
[7] Dancer, E. N., On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J., 23, 1069-1076, (1974) · Zbl 0276.47051
[8] Drábek, P.; Ôtani, M., Global bifurcation result for the p-biharmonic operator, Electron. J. Differ. Equ., 2001, 1-19, (2001) · Zbl 0983.35099
[9] Ferrero, A.; Warnault, G., On a solutions of second and fourth order elliptic with power type nonlinearities, Nonlinear Anal., 70, 2889-2902, (2009) · Zbl 1171.35374
[10] Liu, Y.; O’Regan, D., Bifurcation techniques for fourth order m-point boundary value problems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 18, 215-234, (2011) · Zbl 1216.34018
[11] Ma, R.; Gao, C.; Han, X., On linear and nonlinear fourth-order eigenvalue problems with indefinite weight, Nonlinear Anal., 74, 4186-4191, (2011) · Zbl 1222.39008
[12] Ma, R.; Wang, H., On the existence of positive solutions of fourth-order ordinary differential equations, Appl. Anal., 59, 225-231, (1995) · Zbl 0841.34019
[13] Ma, R. Y.; Xu, J., Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem, Nonlinear Anal., 72, 1, 113-122, (2010) · Zbl 1200.34023
[14] Myers, T. G., Thin films with high surface tension, SIAM Rev., 40, 3, 441-462, (1998) · Zbl 0908.35057
[15] Rabinowitz, P. H., Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7, 487-513, (1971) · Zbl 0212.16504
[16] Yao, Q. L., Existence, multiplicity and infinite solvability of positive solutions to a nonlinear fourth-order periodic boundary value problem, Nonlinear Anal., 63, 237-246, (2005) · Zbl 1082.34025
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