Existence and multiplicity of solutions for fractional p(x)-Kirchhoff-type problems

In this paper, we deal with the existence and multiplicity of solutions for fractional p(x)-Kirchhoff-type problems as follows:{ M(Q1p(x,y) vertical bar v(x)-v(y)vertical bar p(x,y)|x-y|d+sp(x,y) dxdy)(-p(x))sv(x) =|v(x)|r(x)-v(x),inO,v=0,inRd\O, where (-p(x))s is the fractional p(x)-Laplacian. Different from the previous ones which have recently appeared, we weaken the condition of M and obtain the existence and multiplicity of solutions via the symmetric mountain pass theorem and the theory of the fractional Sobolev space with variable exponents.