×

Multiple sign-changing solutions to semilinear elliptic resonant problems. (English) Zbl 1185.35096

Summary: We use the topological degree theory and the critical groups to investigate the elliptic equation \(-\Delta u = f(x,u)\) in \(\Omega \) and subject to \(u = 0\) on \(\partial \Omega \), and establish a multiple solutions theorem which guarantees that this problem has at least six nontrivial solutions under some resonant conditions. If this problem has only finitely many solutions then, among them, there are two positive solutions, two negative solutions and two sign-changing solutions.

MSC:

35J61 Semilinear elliptic equations
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35B09 Positive solutions to PDEs
35J20 Variational methods for second-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Chang, K.-C.; Li, S.; Liu, J., Remarks on multiple solutions for asymptotically linear elliptic boundary value problems, Topol. methods nonlinear anal., 3, 179-187, (1994) · Zbl 0812.35031
[2] Dancer, E.; Du, Y., Multiple solutions of some semilinear elliptic equations via the generalized Conley index, J. math. anal. appl., 189, 848-871, (1995) · Zbl 0834.35049
[3] Bartsch, T.; Wang, Z.-Q., On the existence of sign-changing solutions for semilinear Dirichlet problems, Topol. methods nonlinear anal., 7, 115-131, (1996) · Zbl 0903.58004
[4] Dancer, E.; Du, Y., The generalized Conley index and multiple solutions of some semilinear elliptic problems, Abstract appl. anal., 1, 103-135, (1996) · Zbl 0933.35069
[5] Bartsch, T.; Chang, K.-C.; Wang, Z.-Q., On the Morse indices of sign changing solutions of nonlinear elliptic problems, Math. Z., 233, 655-677, (2000) · Zbl 0946.35023
[6] Li, S.; Wang, Z.-Q., Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems, J. anal. math., 81, 373-396, (2000) · Zbl 0962.35065
[7] Li, S.; Wang, Z.-Q., Ljusternik – schnirelman theory in partially ordered Hilbert spaces, Trans. amer. math. soc., 354, 3207-3227, (2002) · Zbl 1219.35067
[8] Liu, Z.; Sun, J., An elliptic problem with jumping nonlinearities, Nonlinear anal., 63, 1070-1082, (2005) · Zbl 1161.35395
[9] Zou, W., On finding sign-changing solutions, J. funct. anal., 234, 364-419, (2006) · Zbl 1095.35047
[10] Hofer, H., Variational and topological methods in partially order Hilbert spaces, Math. ann., 261, 493-514, (1982) · Zbl 0488.47034
[11] Pang, C.; Dong, W.; Wei, Z., Multiple solutions for fourth-order boundary value problems, J. math. anal. appl., 314, 464-476, (2006) · Zbl 1094.34012
[12] Chang, K.-C., Infinite dimensional Morse theory and multiple solution problems, (1993), Birkhäuser Boston
[13] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer New York · Zbl 0676.58017
[14] Bartsch, T.; Li, S., Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear anal., 28, 419-441, (1997) · Zbl 0872.58018
[15] Liu, Z., On dancer’s conjecture and multiple solutions of elliptic differential equations, Northeast. math. J., 9, 388-394, (1993) · Zbl 0818.35031
[16] Xu, X., Multiple sign-changing solutions for some \(m\)-point boundary value problems, Electron. J. differential equations, 89, 1-14, (2004)
[17] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag Berlin Heidelberg · Zbl 0559.47040
[18] Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM rev., 18, 620-709, (1976) · Zbl 0345.47044
[19] Li, S.; Liu, J., Computations of critical groups at degenerate critical point and applications to nonlinear differential equations with resonance, Houston J. math., 25, 563-582, (1999) · Zbl 0981.58011
[20] Zou, W.; Li, S.; Liu, J., Nontrivial solutions for resonant cooperative elliptic systems via computations of critical groups, Nonlinear anal., 38, 229-247, (1999) · Zbl 0940.35074
[21] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press New York · Zbl 0661.47045
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.