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

Abstract

In this article, we first obtain an embedding result for the Sobolev spaces with variable-order, and then we consider the 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) is a continuous and symmetric function with N>s(x,y)p for all (x,y)∈Ω×Ω, λ>0 is a parameter, (−Δ)ps(·) is a fractional p-Laplace operator with variable-order, V(x):Ω→R+ is a potential function, and f(x,ξ):Ω×RN→R is a continuous nonlinearity function. Assuming that V and f satisfy some reasonable hypotheses, we obtain the existence of infinitely many solutions for the above problem by using the fountain theorem and symmetric mountain pass theorem without the Ambrosetti–Rabinowitz ((AR) for short) condition.

Fractional-order diffusion coupled with integer-order diffusion for multiplicative noise removal

“…In this section, we state some necessary preliminaries of variable-order fractional Sobolev spaces that will be used below. We refer to [38,39] and the references therein for some detailed proof. Henceforth, we will always assume that…”

“…The established variational model is built on the basis of the Stinga-Torrea extension, instead of directly defining (−∆) s(x) , because there is no clear way to define (−∆) s(x) , actually. The well-defined variable-order fractional Laplacian (−∆) s(•,•) (see [38]) and the variable-order fractional p-Laplacian (−∆) s(•,•) p (see [39]) have not been applied to image processing yet, and there are rare theoretical results on their evolution problems.…”