Two notions which affected nonlinear analysis (Bernard Bolzano lecture)

Drábek, Pavel

Mathematica Bohemica, Tome 139 (2014) no. 4, pp. 699-711

Abstract (VO)

General mathematical theories usually originate from the investigation of particular problems and notions which could not be handled by available tools and methods. The Fučík spectrum and the $p$-Laplacian are typical examples in the field of nonlinear analysis. The systematic study of these notions during the last four decades led to several interesting and surprising results and revealed deep relationship between the linear and the nonlinear structures. This paper does not provide a complete survey. We focus on some pioneering works and present some contributions of the author. From this point of view the list of references is by no means exhaustive.

Détail

DOI : 10.21136/MB.2014.144146

Classification : 34B15, 34B99, 35J92, 35P30
Keywords: Fučík spectrum; $p$-Laplacian

Bibliographie

[1] Anane, A.: Simplicity and isolation of first eigenvalue of the $p$-Laplacian with weight. C. R. Acad. Sci., Paris, Sér. I 305 French (1987), 725-728. | MR | Zbl

[2] Anane, A., Tsouli, N.: On the second eigenvalue of the $p$-Laplacian. Nonlinear Partial Differential Equations. Based on the International Conference on Nonlinear Analysis, Fés, Morocco, 1994 Pitman Res. Notes Math. Ser. 343 Longman, Harlow (1996), 1-9 A. Benkirane et al. | MR | Zbl

[3] Berkovits, J., Drábek, P., Leinfelder, H., Mustonen, V., Tajčová, G.: Time-periodic oscillations in suspension bridges: Existence of unique solutions. Nonlinear Anal., Real World Appl. 1 (2000), 345-362. | MR | Zbl

[4] Binding, P. A., Drábek, P., Huang, Y. X.: On the Fredholm Alternative for the $p$-Laplacian. Proc. Am. Math. Soc. 125 (1997), 3555-3559. | DOI | MR | Zbl

[5] Boccardo, L., Drábek, P., Giachetti, D., Kučera, M.: Generalization of Fredholm alternative for nonlinear differential operators. Nonlinear Anal., Theory Methods Appl. 10 (1986), 1083-1103. | DOI | MR | Zbl

[6] Dancer, E. N.: Boundary-value problems for weakly nonlinear ordinary differential equations. Bull. Aust. Math. Soc. 15 (1976), 321-328. | DOI | MR | Zbl

[7] Pino, M. del, Drábek, P., Manásevich, R.: The Fredholm alternative at the first eigenvalue for the one dimensional $p$-Laplacian. J. Differ. Equations 151 (1999), 386-419. | DOI | MR

[8] Pino, M. A. del, Manásevich, R. F.: Global bifurcation from the eigenvalues of the $p$-Laplacian. J. Differ. Equations 92 (1991), 226-251. | DOI | MR

[9] Drábek, P.: Geometry of the energy functional and the Fredholm alternative for the $p$-Laplacian in higher dimensions. Proceedings of the 2001 Luminy Conference on Quasilinear Elliptic and Parabolic Equations and System, Electron. J. Differ. Equ. (electronic only) 8 (2002), 103-120. | MR | Zbl

[10] Drábek, P.: Solvability and Bifurcations of Nonlinear Equations. Pitman Research Notes in Mathematics Series 264 Longman Scientific, Harlow; John Wiley, New York (1992). | MR | Zbl

[11] Drábek, P.: On the global bifurcation for a class of degenerate equations. Ann. Mat. Pura Appl. (4) 159 (1991), 1-16. | MR | Zbl

[12] Drábek, P.: On the resonance problem with nonlinearity which has arbitrary linear growth. J. Math. Anal. Appl. 127 (1987), 435-442. | DOI | MR | Zbl

[13] Drábek, P.: Ranges of homogeneous operators and their perturbations. Čas. Pěst. Mat. 105 (1980), 167-183. | MR | Zbl

[14] Drábek, P., Girg, P., Takáč, P., Ulm, M.: The Fredholm alternative for the $p$-Laplacian: Bifurcation from infinity, existence and multiplicity. Indiana Univ. Math. J. 53 (2004), 433-482. | DOI | MR | Zbl

[15] Drábek, P., Holubová, G.: Bifurcation of periodic solutions in symmetric models of suspension bridges. Topol. Methods Nonlinear Anal. 14 (1999), 39-58. | DOI | MR | Zbl

[16] Drábek, P., Holubová, G., Matas, A., Nečesal, P.: Nonlinear models of suspension bridges: Discussion of the results. Mathematical and computer modeling in science and engineering. Appl. Math., Praha 48 (2003), 497-514. | DOI | MR

[17] Drábek, P., Kufner, A.: Discreteness and simplicity of the spectrum of a quasilinear Sturm-Liouville-type problem on an infinite interval. Proc. Am. Math. Soc. 134 (2006), 235-242. | DOI | MR | Zbl

[18] Drábek, P., Kufner, A., Nicolosi, F.: Quasilinear Elliptic Equations with Degenerations and Singularities. De Gruyter Series in Nonlinear Analysis and Applications 5 Walter de Gruyter, Berlin (1997). | MR | Zbl

[19] Drábek, P., Kuliev, K.: Half-linear Sturm-Liouville problem with weights. Bull. Belg. Math. Soc. -- Simon Stevin 19 (2012), 107-119. | DOI | MR | Zbl

[20] Drábek, P., Robinson, S. B.: On the solvability of resonance problems with respect to the Fučík spectrum. J. Math. Anal. Appl. 418 (2014), 884-905. | DOI | MR

[21] Drábek, P., Robinson, S. B.: On the Fredholm alternative for the Fučík spectrum. Abstr. Appl. Anal. 2010 (2010), Article ID 125464, 20 pages. | MR | Zbl

[22] Drábek, P., Robinson, S. B.: On the generalization of the Courant nodal domain theorem. J. Differ. Equations 181 (2002), 58-71. | DOI | MR | Zbl

[23] Drábek, P., Robinson, S. B.: Resonance problems for the $p$-Laplacian. J. Funct. Anal. 169 (1999), Article No. jfan.1999.3501, 189-200. | DOI | MR | Zbl

[24] Elbert, A.: A half-linear second order differential equation. Qualitative Theory of Differential Equations, Vol. I, Szeged, 1979 Colloq. Math. Soc. János Bolyai 30 North-Holland, Amsterdam (1981), 153-180 M. Farkas. | MR | Zbl

[25] Fučík, S.: Solvability of Nonlinear Equations and Boundary Value Problems. Mathematics and Its Applications 4 D. Reidel Publishing, Dordrecht (1980). | MR

[26] Fučík, S.: Boundary value problems with jumping nonlinearities. Čas. Pěst. Mat. 101 (1976), 69-87. | MR | Zbl

[27] Fučík, S., Nečas, J., Souček, J., Souček, V.: Spectral Analysis of Nonlinear Operators. Lecture Notes in Mathematics 346 Springer, Berlin (1973). | DOI | MR | Zbl

[28] Kováčik, O., Rákosník, J.: On spaces $L^{p(x)}$ and $W^{k,p(x)}$. Czech. Math. J. 41 (1991), 592-618. | MR

[29] Krejčí, P.: On solvability of equations of the 4th order with jumping nonlinearities. Čas. Pěst. Mat. 108 (1983), 29-39. | MR | Zbl

[30] Landesman, E. M., Lazer, A. C.: Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. Math. Mech. 19 (1970), 609-623. | MR | Zbl

[31] Lazer, A. C., McKenna, P. J.: Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis. SIAM Rev. 32 (1990), 537-578. | DOI | MR | Zbl

[32] Růžička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics 1748 Springer, Berlin (2000). | DOI | MR | Zbl

[33] Švarc, R.: The solution of a Fučík's conjecture. Commentat. Math. Univ. Carol. 25 (1984), 483-517. | MR | Zbl