×

Multiplicity of solutions for gradient systems using Landesman-Lazer conditions. (English) Zbl 1207.35126

Summary: We establish existence and multiplicity of solutions for an elliptic system which presents resonance at infinity of Landesman-Lazer type. In order to describe the resonance, we use an eigenvalue problem with indefinite weights. In all results, we use variational methods, Morse theory and critical groups.

MSC:

35J47 Second-order elliptic systems
35J50 Variational methods for elliptic systems
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B40 Asymptotic behavior of solutions to PDEs
35P05 General topics in linear spectral theory for PDEs
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, NY, USA, 1992. · Zbl 0777.35001
[2] J. Smoller, Shock Waves and Reaction-Diffusion Equations, vol. 258 of Fundamental Principles of Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 1994. · Zbl 0807.35002
[3] K. Chang, “An extension of the Hess-Kato theorem to elliptic systems and its applications to multiple solution problems,” Acta Mathematica Sinica, vol. 15, no. 4, pp. 439-454, 1999. · Zbl 0932.35067 · doi:10.1007/s10114-999-0078-0
[4] D. G. de Figueiredo, “Positive solutions of semilinear elliptic problems,” in Differential Equations (Sao Paulo, 1981), vol. 957 of Lecture Notes in Mathematics, pp. 34-87, Springer, Berlin, Germany, 1982. · Zbl 0506.35038
[5] M. F. Furtado and F. O. V. de Paiva, “Multiplicity of solutions for resonant elliptic systems,” Journal of Mathematical Analysis and Applications, vol. 319, no. 2, pp. 435-449, 2006. · Zbl 1108.35046 · doi:10.1016/j.jmaa.2005.06.038
[6] E. M. Landesman and A. C. Lazer, “Nonlinear perturbations of linear elliptic boundary value problems at resonance,” vol. 19, pp. 609-623, 1969-1970. · Zbl 0193.39203
[7] E. A. B. Silva, “Linking theorems and applications to semilinear elliptic problems at resonance,” Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no. 5, pp. 455-477, 1991. · Zbl 0731.35042 · doi:10.1016/0362-546X(91)90070-H
[8] T. Bartsch and S. Li, “Critical point theory for asymptotically quadratic functionals and applications to problems with resonance,” Nonlinear Analysis: Theory, Methods & Applications, vol. 28, no. 3, pp. 419-441, 1997. · Zbl 0872.58018 · doi:10.1016/0362-546X(95)00167-T
[9] T. Bartsch, K.-C. Chang, and Z.-Q. Wang, “On the Morse indices of sign changing solutions of nonlinear elliptic problems,” Mathematische Zeitschrift, vol. 233, no. 4, pp. 655-677, 2000. · Zbl 0946.35023 · doi:10.1007/s002090050492
[10] K.-C. Chang, “Principal eigenvalue for weight matrix in elliptic systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 46, no. 3, pp. 419-433, 2001. · Zbl 1194.35135 · doi:10.1016/S0362-546X(00)00140-1
[11] H.-M. Suo and C.-L. Tang, “Multiplicity results for some elliptic systems near resonance with a nonprincipal eigenvalue,” Nonlinear Analysis: Theory, Methods and Applications, vol. 73, no. 7, pp. 1909-1920. · Zbl 1194.35140 · doi:10.1016/j.na.2009.11.004
[12] E. D. da Silva, “Multiplicity of solutions for gradient systems with strong resonance at higher eigenvalues,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 11, pp. 3918-3928, 2010. · Zbl 1187.35061 · doi:10.1016/j.na.2010.01.010
[13] Z.-Q. Ou and C.-L. Tang, “Existence and multiplicity results for some elliptic systems at resonance,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2660-2666, 2009. · Zbl 1172.35382 · doi:10.1016/j.na.2009.01.106
[14] S. Ahmad, A. C. Lazer, and J. L. Paul, “Elementary critical point theory and perturbations of elliptic boundary value problems at resonance,” Indiana University Mathematics Journal, vol. 25, no. 10, pp. 933-944, 1976. · Zbl 0351.35036 · doi:10.1512/iumj.1976.25.25074
[15] K. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, Mass, USA, 1993. · Zbl 0779.58005
[16] D. G. de Figueiredo and J.-P. Gossez, “Strict monotonicity of eigenvalues and unique continuation,” Communications in Partial Differential Equations, vol. 17, no. 1-2, pp. 339-346, 1992. · Zbl 0777.35042 · doi:10.1080/03605309208820844
[17] L. Homander, Linear Partial Differential Equations, Springer, Berlin, Germany, 1969.
[18] L. Hörmander, “Uniqueness theorems for second order elliptic differential equations,” Communications in Partial Differential Equations, vol. 8, no. 1, pp. 21-64, 1983. · Zbl 0546.35023 · doi:10.1080/03605308308820262
[19] O. Lopes, “Radial symmetry of minimizers for some translation and rotation invariant functionals,” Journal of Differential Equations, vol. 124, no. 2, pp. 378-388, 1996. · Zbl 0842.49004 · doi:10.1006/jdeq.1996.0015
[20] R. Regbaoui, “Strong uniqueness for second order differential operators,” Journal of Differential Equations, vol. 141, no. 2, pp. 201-217, 1997. · Zbl 0887.35040 · doi:10.1006/jdeq.1997.3327
[21] M. Sanada, “Strong unique continuation property for some second order elliptic systems,” Japan Academy Proceedings. Series A, vol. 83, no. 7, pp. 119-122, 2007. · Zbl 1180.35213 · doi:10.3792/pjaa.83.119
[22] S. Li and J. Q. Liu, “Some existence theorems on multiple critical points and their applications,” Kexue Tongbao, vol. 17, pp. 1025-1027, 1984.
[23] J. Su, “Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues,” Nonlinear Analysis: Theory, Methods & Applications, vol. 48, no. 6, pp. 881-895, 2002. · Zbl 1018.35037 · doi:10.1016/S0362-546X(00)00221-2
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.