We consider the problem of attainability of the best constant in the following critical fractional Hardy-Sobolev inequality:, the latter being the best fractional Hardy constant on R n .
We consider linear and non-linear boundary value problems associated to the fractional Hardy-Schrödinger operator Lγ,α := (−∆) α 2 − γ |x| α on domains of R n containing the singularity 0, where 0 < α < 2 and 0 ≤ γ < γ H (α), the latter being the best constant in the fractional Hardy inequality on R n . We tackle the existence of least-energy solutions for the borderline boundaryn−α is the critical fractional Sobolev exponent. We show that if γ is below a certain threshold γ crit , then such solutions exist for all 0 < λ < λ 1 (Lγ,α), the latter being the first eigenvalue of Lγ,α. On the other hand, for γ crit < γ < γ H (α), we prove existence of such solutions only for those λ in (0, λ 1 (Lγ,α)) for which the domain Ω has a positive fractional Hardy-Schrödinger mass m γ,λ (Ω). This latter notion is introduced by way of an invariant of the linear equation (Lγ,α − λI)u = 0 on Ω.
In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities:where Ω ⊂ R n is a smooth bounded domain in R n containing 0 in its interior, and f, g ∈ C(Ω) with f + , g + ≡ 0 which may change sign in Ω. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for λ sufficiently small. The variational approach requires that 0 < α < 2, 0 < s < α < n, 1 < q < 2 < p ≤ 2 * α (s) := 2(n−s) n−α , and γ < γH (α), the latter being the best fractional Hardy constant on R n .
We consider the Hardy-Schrödinger operator Lγ := −∆ B n − γV 2 on the Poincaré ball model of the Hyperbolic space B n (n ≥ 3). Here V 2 is a well chosen radially symmetric potential, which behaves like the Hardy potential around its singularity at 0, i.e., V 2 (r) ∼ 1 r 2 . Just like in the Euclidean setting, the operator Lγ is positive definite whenever γ < (n−2) 2 4 , in which case we exhibit explicit solutions for the Sobolev critical equation Lγ u = V 2 * (s) u 2 * (s)−1 in B n , where 0 ≤ s < 2, 2 * (s) = 2(n−s) n−2 , and V 2 * (s) is a weight that behaves like 1 r s around 0. In dimensions n ≥ 5, the above equation in a domain Ω of B n containing 0 and away from the boundary, has a ground state solution, whenever 0 < γ ≤ n(n−4) 4 , and provided Lγ is replaced by a linear perturbation Lγ − λu, where λ > n − 2 n − 4 n(n − 4) 4 − γ . On the other hand, in dimensions 3and 4, the existence of solutions depends on whether the domain has a postive "hyperbolic mass," a notion that we introduce and analyze therein.
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