Fractional Kirchhoff-Hardy Problem: Breaking of Resonance and General Existence Results

In this paper, we study a fractional Kirchhoff problem with a Hardy-type singular potential and general nonlinearities depending on the solution and its gradient: M∫∫RN×RN|u(x)−u(y)|q|x−y|N+qsdxdy(−Δ)su=λu|x|2s+f(x,u,∇u)inΩ, where Ω⊂RN is a bounded domain containing the origin, s∈(0,1), q∈(1,2] with N>2s, λ>0, and f is a measurable non-negative function satisfying suitable hypotheses. The main objective is to establish the existence of positive solutions for the largest possible class of nonlinearities f without imposing restrictions on λ. Two main cases areconsidered: (I)−f(x,u,∇u)=up+μ,and(II)−f(x,u,∇u)=|∇u|p+μg. Existence is proved under suitable hypotheses on q,p and the data g,μ. The results are new, including for the local case s=1.