×

On sign-changing solution for a fourth-order asymptotically linear elliptic problem. (English) Zbl 1184.35137

Summary: We deal with a fourth-order elliptic problem whose nonlinear term is asymptotically linear at both zero and infinity. By using the variational method, we obtain an existence result of sign-changing solutions as well as positive and negative solutions.

MSC:

35J61 Semilinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J20 Variational methods for second-order elliptic equations
35B09 Positive solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lazer, A. C.; McKenna, P. J., Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Rev., 32, 537-578 (1990) · Zbl 0725.73057
[2] Tarantello, G., A note on a semilinear elliptic problem, Differential Integral Equations, 5, 3, 561-565 (1992) · Zbl 0786.35060
[3] Lazer, A. C.; McKenna, P. J., Global bifurcation and a theorem of Tarantello, J. Math. Anal. Appl., 181, 648-655 (1994) · Zbl 0797.34021
[4] Micheletti, A. M.; Pistoia, A., Nontrivial solutions for some fourth order semilinear elliptic problems, Nonlinear Anal., 34, 509-523 (1998) · Zbl 0929.35053
[5] Micheletti, A. M.; Pistoia, A., Multiplicity results for a fourth-order semilinear elliptic problem, Nonlinear Anal., 31, 895-903 (1998) · Zbl 0898.35032
[6] Zhang, J. H., Existence results for some fourth-order nonlinear elliptic problems, Nonlinear Anal., 45, 29-36 (2001) · Zbl 0981.35016
[7] Xu, G. X.; Zhang, J. H., Existence results for some fourth-order nonlinear elliptic problems of local superlinearity and sublinearity, J. Math. Anal. Appl., 281, 633-640 (2003) · Zbl 1146.35362
[8] Zhang, J. H.; Li, S. J., Multiple nontrivial solutions for some fourth-order semilinear elliptic problems, Nonlinear Anal., 60, 221-230 (2005) · Zbl 1103.35027
[9] Zhou, J. W.; Wu, X., Sign-changing solutions for some fourth-order nonlinear elliptic problems, J. Math. Anal. Appl., 342, 542-558 (2008) · Zbl 1138.35335
[10] Qian, A. X.; Li, S. J., Multiple solutions for a fourth-order asymptotically linear elliptic problem, Acta Math. Sin., 22, 4, 1121-1126 (2006) · Zbl 1274.35140
[11] An, Y. K.; Liu, R. Y., Existence of nontrivial solutions of an asymptotically linear fourth-order elliptic equation, Nonlinear Anal., 68, 3325-3331 (2008) · Zbl 1158.35041
[12] Zhou, H. S., Existence of asymptotically linear Dirichlet problem, Nonlinear Anal., 44, 909-918 (2001) · Zbl 1194.35188
[13] Liu, Z. L.; Sun, J. X., Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations, 172, 257-299 (2001) · Zbl 0995.58006
[14] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Eqiuations of Second Order (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0691.35001
[15] Dahlberg, Björn E. J., A note on Sobolev spaces, (Proceedings of Symposia in Pure Mathematics, vol. XXXV 1 (1979)), 183-185 · Zbl 0421.46027
[16] Ziemer, William P., (Weakly Differentiable Functions. Weakly Differentiable Functions, GTM 120 (1989), Springer-Verlag) · Zbl 0692.46022
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.