Abstract
Motivated by extremal problems of weighted Dirichlet or Neumann eigenvalues, we will establish two fundamental results on the dependence of weighted eigenvalues of the one-dimensional p-Laplacian on indefinite integrable weights. One is the continuous differentiability of eigenvalues in weights in the Lebesgue spaces L γ with the usual norms. Another is the continuity of eigenvalues in weights with respect to the weak topologies in L γ spaces. Here 1 ≤ γ ≤ ∞. In doing so, we will give a simpler explanation to the corresponding spectrum problems, with the help of several typical techniques in nonlinear analysis such as the Fréchet derivative and weak* convergence.
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References
Anane A., Chakrone O., Monssa M. (2002) Spectrum of one dimensional p- Laplacian with indefinite weight. Electr. J. Qualitative Theory Differential Equations 2002(17): 11
Arias M., Campos J., Cuesta M., Gossez J.-P. (2008) An asymmetric Neumann problem with weights. Ann. Inst. H. Poincaré Anal. Non Linéarie 25: 267–280
Binding P.L., Dràbek P. (2003) Sturm-Liouville theory for the p-Laplacian. Stud. Sci. Math. Hungar. 40: 375–396
Binding P.L., Rynne B.P. (2007) The spectrum of the periodic p-Laplacian. J. Differential Equations 235: 199–218
Brown K.J., Lin S.S. (1980) On the existence of positive eigenvalue problem with indefinite weight function. J. Math. Anal. 75: 112–120
Cuesta M. (2001) Eigenvalue problems for the p-Laplacian with indefinite weights. Electr. J. Differential Equations 2001(33): 9
Eberhard W., Elbert Á. (2000) On the eigenvalues of half-linear boundary value problems. Math. Nachr. 213: 57–76
Kajikiya R., Lee Y.-H., Sim I. (2008) One-dimensional p-Laplacian with a strong singular indefinite weight, I, Eigenvalues. J. Differential Equations 244: 1985–2019
Karaa S. (1998) Sharp estimates for the eigenvalues of some differential equations. SIAM J. Math. Anal. 29: 1279–1300
Kong Q., Zettl A. (1996) Eigenvalues of regular Sturm-Liouville problems. J. Differential Equations 131: 1–19
Krein M.G. (1955) On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability. Amer. Math. Soc. Transl. Ser. 2(1): 163–187
Lou Y., Yanagida E. (2006) Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics. Japan J. Indust. Appl. Math. 23: 275–292
Megginson R.E. (1998) An Introduction to Banach Space Theory, Graduate Texts Math., Vol. 183. Springer-Verlag, New York
G. Meng and M. Zhang, Continuity in weak topology: first order linear systems of ODE, Preprint, 2008. http://faculty.math.tsinghua.edu.cn/~mzhang/
Möller M., Zettl A. (1996) Differentiable dependence of eigenvalues of operators in Banach spaces. J. Operator Theory 36: 335–355
Pöschel J., Trubowitz E. (1987) The Inverse Spectral Theory. Academic Press, New York
Senn S., Hess P. (1982) On positive solutions of a linear boundary value problem with Neumann boundary conditions. Math. Ann. 258: 459–470
Taira K. (2008) Degenerate elliptic eigenvalue problems with indefinite weights. Mediterr. J. Math., 5: 133–162
P. Yan and M. Zhang, Continuity in weak topology and extremal problems of eigenvalues of the p-Laplacian, Trans. Amer. Math. Soc., in press.
Zhang M. (2000) Nonuniform nonresonance of semilinear differential equations. J. Differential Equations 166: 33–50
Zhang M. (2001) The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials. J. London Math. Soc. (2) 64: 125–143
Zhang M. (2008) Continuity in weak topology: higher order linear systems of ODE. Sci. China Ser. A 51: 1036–1058
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The third author is supported by the National Basic Research Program of China (Grant no. 2006CB805903), the National Natural Science Foundation of China (Grant no. 10531010) and the 111 Project of China (2007).
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Meng, G., Yan, P. & Zhang, M. Spectrum of One-Dimensional p-Laplacian with an Indefinite Integrable Weight. Mediterr. J. Math. 7, 225–248 (2010). https://doi.org/10.1007/s00009-010-0040-5
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DOI: https://doi.org/10.1007/s00009-010-0040-5