Properties of eigenvalues and some regularities on fractional $p$-Laplacian with singular weights

Ky Ho; Inbo Sim

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arXiv · 1809.07020

Properties of eigenvalues and some regularities on fractional $p$-Laplacian with singular weights

Ky Ho·Inbo Sim

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Abstract

We provide fundamental properties of the first eigenpair for fractional $p$-Laplacian eigenvalue problems under singular weights, which is related to Hardy type inequality, and also show that the second eigenvalue is well-defined. We obtain a-priori bounds and the continuity of solutions to problems with such singular weights with some additional assumptions. Moreover, applying the above results, we show a global bifurcation emenating from the first eigenvalue, the Fredholm alternative for non-resonant problems, and obtain the existence of infinitely many solutions for some nonlinear problems involving singular weights. These are new results, even for (fractional) Laplacian.

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Ky Ho, Inbo Sim. 2018-09-19. Properties of eigenvalues and some regularities on fractional $p$-Laplacian with singular weights. https://arxiv.org/abs/1809.07020

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