On the eigenvalues and Fu?ik spectrum of p-fractional Hardy-Sobolev operator with weight function

Abstract

In this article, we study the nonlinear eigenvalue problem of fractional Hardy–Sobolev operator

where is a bounded domain in with Lipschitz boundary containing 0, , , , and the weight function V, having nontrivial positive part, belongs to suitable integrable class and may change sign. We investigate some properties of the first eigenvalue such as simplicity and isolation. Moreover, we also study the Fu?ik spectrum of fractional Hardy-Sobolev operator, which is defined as the set such that has a non-trivial solution u. We show the existence of a first nontrivial curve of this spectrum and also we prove some properties of this curve. At the end, we study a nonresonance problem with respect to the weighted Fu?ik spectrum.