Multiplicity results for variable-order fractional Laplacian equations with variable growth

In this paper, we study the multiplicity of solutions for an elliptic type problem driven by the variable-order fractional Laplace operator involving variable exponents. More precisely, we consider (−Δ)s(⋅)u+λV(x)u=α|u|p(x)−2u+β|u|q(x)−2uinΩ,u=0inRN∖Ω,where N≥1, s(⋅):RN×RN→(0,1) is a continuous function, Ω is a bounded domain in RN with N>2s(x,y) for all (x,y)∈Ω×Ω, (−Δ)s(⋅) is the variable-order fractional Laplace operator, λ>0 is a parameter, V:Ω→[0,∞) is a continuous function, α,β>0 are two parameters and p,q∈C(Ω). Under some suitable assumptions, we show that the above problem admits at least two distinct solutions by applying the mountain pass theorem and Ekeland's variational principle. Then we prove that these two solutions converge to two solutions of a limit problem as λ→∞. Moreover, we obtain the existence of infinitely many solutions for the limit problem.