Spectrum of One-Dimensional p-Laplacian with an Indefinite Integrable Weight

TLDR: In this article, the dependence of weighted eigenvalues of the one-dimensional p-Laplacian on indefinite integrable weights was studied and a simpler explanation to the corresponding spectrum problems was given with the help of several typical techniques in nonlinear analysis such as the Frechet derivative and weak* convergence.

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 Frechet derivative and weak* convergence.