Multiplicity of solutions for variable-order fractional Kirchhoff problem with singular term

Abstract

In this paper, we consider a class of singular variable-order fractional Kirchhoff problem of the form:

[u]2(σ−1)/s(.) (−Δ)^s(.)u = λ a(x)/|u|m(x) + b(x)|u|q(x)−2u in Ω,
u = 0, on Rⁿ⧹Ω,

where Ω ⊂ R^N, (N ≥ 2), is a bounded domain, (−Δ)^s(.) is the variable-order fractional Laplacian operator, [u]_s(.) is the Gagliardo seminorm and s(.) : R^N×R^N →(0, 1) is a continuous and symmetric function. We assume that λ is a non-negative parameter, σ ≥ 1, m, q ∈ C(Ω) with 0 < m(x) <  1  <  2σ < q(x) < 2N /N−2s(x,x) and N > 2s (x, y) for all (x, y) ∈ (Ω × Ω).

We combine some variational techniques with a truncation argument in order to show the existence and the multiplicity of positive solutions to the above problem.