Infinitely Many Solutions for Fractional p-Laplacian Schrödinger–Kirchhoff Type Equations with Symmetric Variable-Order

In this article, we first obtain an embedding result for the Sobolev spaces with variable-order, and then consider following Schrödinger–Kirchhoff type equations a+b∫Ω×Ω|ξ(x)−ξ(y)|p|x−y|N+ps(x,y)dxdyp−1(−Δ)ps(·)ξ+λV(x)|ξ|p−2ξ=f(x,ξ),x∈Ω,ξ=0,x∈∂Ω, where Ω is a bounded Lipschitz domain in RN, 1<p<+∞, a,b>0 are constants, s(·):RN×RN→(0,1) continuous symmetric function N>s(x,y)p all (x,y)∈Ω×Ω, λ>0 parameter, (−Δ)ps(·) fractional p-Laplace operator V(x):Ω→R+ potential function, f(x,ξ):Ω×RN→R nonlinearity function. Assuming that V f satisfy some reasonable hypotheses, existence of infinitely many solutions above problem by using fountain theorem mountain pass without Ambrosetti–Rabinowitz ((AR) short) condition.