  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.
