Abstract The aim of this work is to investigate the spectrum of a singular eigenvalue boundary problems involving the fractional p (x) p (x) -Laplacian operator. We first establish a new (p (x) ; q (x) ) (p (x) ;q (x) ) -Hardy inequality related to this operator. Then, using the theory of the fractional Sobolev space with variable exponent W s, p (x, y) (Ω) W^{s, p (x, y) () } and the variational technique based on the Ljusternik–Schnirelmann theory on C 1 C^{1} -manifolds, we prove the existence of at least one unbounded, nondecreasing sequence of nonnegative eigenvalues (λ k) k ≥ 1 (₊) ₊ ₁. Finally, we provide sufficient conditions that guarantee the positivity of the infimum of the set Λ of eigenvalues.
Aboubker et al. (Thu,) studied this question.