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

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.