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]_{s(.)}^{2(\sigma-1)}(-\Delta)^{s(.)}u =\lambda~~\frac{ a(x)}{|u|^{m (x)}}+b(x)|u|^{q(x)-2}u in Omega, u=0, on R^{n} \Omega, where Omega is a bounded domain, (-\Delta)^{s(.)} is the variable-order fractional Laplacian operator, [u]_{s(.)} is the Gagliardo seminorm and s(.,.) in C(R^{N}x{R}^{N} ,(0, 1)) and symetric function. We assume that lambda is a non-negative parameter , sigma >= 1, m,q in C(Omega) with 0<1<2\sigma<\frac{2N}{N-2s(x,x)} and N>2s(x,y) for all (x,y) in Omega x Omega). 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.