Multiplicity of solutions for variable-order fractional Kirchhoff equations with nonstandard growth

Abstract

The aim of this paper is to study a degenerate Kirchhoff-type elliptic problem driven by the fractional Laplace operator with variable order derivative and variable exponents. More precisely, we consider{[u]s(⋅)2(θ−1)(−Δ)s(⋅)u=λa(x)|u|p(x)−2u+b(x)|u|q(x)−2uin Ω,u=0in RN∖Ω, where [u]s(⋅) is the Garliado seminorm, θ>1, N≥1, s(⋅):RN×RN→(0,1) is a continuous function, λ>0 is a parameter, 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 Laplacian, a,b∈L∞(Ω) are two positive weight functions, and p,q∈C(Ω‾) with 1<p(x)<2θ<q(x)<2N/(N−2s(x,x)). Under some suitable assumptions, we obtain that the problem admits at least two distinct nonnegative solutions by applying the Nehari manifold approach, provided λ is sufficiently small. Moreover, the existence of infinitely many solutions is also investigated by the symmetric mountain pass theorem.