## 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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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