Abstract
Given a bounded domain \(\Omega \) in \(\mathbb {R}^N\), \(N\ge 1\) we study the homogenization of the weighted Fučík spectrum with Dirichlet boundary conditions. In the case of periodic weight functions, precise asymptotic rates of the curves are obtained.
Article PDF
Similar content being viewed by others
References
Alif, M., Gossez, J.-P.: On the Fučík spectrum with indefinite weights. Differ. Integral Equ. 14(12), 1511–1530 (2001)
Arias, M., Campos, J.: Fučik spectrum of a singular Sturm–Liouville problem. Nonlinear Anal. 27(6), 679–697 (1996)
Arias, M., Campos, J., Cuesta, M., Gossez, J.-P.: An asymmetric Neumann problem with weights. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(2), 267–280 (2008)
Azorero, J.G., Peral Alonso, I.: Comportement asymptotique des valeurs propres du \(p\)-Laplacien. C. R. Acad. Sci Paris Sér. I Math. 307(2), 75–78 (1988)
Brown, B.M., Reichel, W.: Computing eigenvalues and Fučík-spectrum of the radially symmetric \(p\)-Laplacian. J. Comput. Appl. Math. 148(1), 183–211 (2002) (on the occasion of the 65th birthday of Professor Michael Eastham)
Conti, M., Terracini, S., Verzini, G.: On a class of optimal partition problems related to the Fučík spectrum and to the monotonicity formulae. Calc. Var. Partial Differ. Equ. 22(1), 45–72 (2005)
Dancer, E.N.: On the Dirichlet problem for weakly non-linear elliptic partial differential equations. Proc. R. Soc. Edinb. Sect. A 76(4), 283–300 (1976/77)
de Figueiredo, D.G., Gossez, J.-P.: On the first curve of the Fučík spectrum of an elliptic operator. Differ. Integral Equ. 7(5), 1285–1302 (1994)
del Pino, M., Drábek, P., Manásevich, R.: The Fredholm alternative at the first eigenvalue for the one-dimensional \(p\)-Laplacian. J. Differ. Equ. 151(2), 386–419 (1999)
Drábek, P.: Solvability and bifurcations of nonlinear equations. In: Pitman Research Notes in Mathematics Series, vol. 264, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York (1992)
Fernández Bonder, J., Pinasco, J.P., Salort, A.M.: Eigenvalue homogenization for quasilinear elliptic equations with various boundary conditions. Electron J. Differ. Equ. 2016(30), 1–15 (2016)
Fernández Bonder, J., Pinasco, J.P., Salort, A.M.: Homogenization of Fučík eigencurves Mediterr. J. Math. 14, 90. doi:10.1007/s00009-017-0890-1
Fučík, S., Kufner, A.: Nonlinear Differential Equations, Studies in Applied Mechanics, vol. 2. Elsevier Scientific Publishing Co., Amsterdam-New York (1980)
Leadi, L., Marcos, A.: On the first curve in the Fučik spectrum with weights for a mixed \(p\)-Laplacian. Int. J. Math. Math. Sci. (2007) Art. ID 57607, 13
Li, W., Yan, P.: Various half-eigenvalues of scalar \(p\)-Laplacian with indefinite integrable weights. Abstr. Appl. Anal. (2009) Art. ID 109757, 27
Pinasco, J.P., Commun, A.M.: Salort, Asymptotic behavior of the curves in the fucik spectrum. Contemp. Math. (2015) (to appear)
Pinasco, J.P., Salort, A.M.: Quasilinear eigenvalues. Rev. Un. Mat. Argent. 56(1), 1–25 (2015)
Pinasco, J.P., Salort, A.M.: Eigenvalue homogenisation problem with indefinite weights. Bull. Aust. Math. Soc. 93(1), 113–127 (2016)
Reichel, W., Walter, W.: Radial solutions of equations and inequalities involving the \(p\)-Laplacian. J. Inequal. Appl. 1(1), 47–71 (1997)
Rynne, B.P.: The Fučík spectrum of general Sturm–Liouville problems. J. Differ. Equ. 161(1), 87–109 (2000)
Salort, A.M.: Convergence rates in a weighted Fuc̆ik problem. Adv. Nonlinear Stud. 14(2), 427–443 (2014)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Salort, A.M. Precise homogenization rates for the Fučík spectrum. Nonlinear Differ. Equ. Appl. 24, 31 (2017). https://doi.org/10.1007/s00030-017-0452-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-017-0452-z