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.
Highlights
Introduction and the Main ResultsIn this article, we investigate the following Schrödinger–Kirchhoff type equations (Pv) : a+b dxdy ξ = 0, x ∈ ∂Ω, p−1(−∆)sp(·)ξ + λV(x)|ξ|p−2ξ = f (x, ξ), x ∈ Ω, where Ω is a bounded Lipschitz domain in RN and 1 < p < +∞, s(·) : RN × RN → (0, 1) is a continuous symmetric function with N > s(x, y)p for all (x, y) ∈ Ω × Ω, (−∆)sp(·) is the fractional p-Laplace operator with variable-order, defined as (−∆)sp(·)ξ(x) := P.V
Motivated by the above cited works, we find that there are some papers on Kirchhoff equations or Schrödinger equations involving the fractional p-Laplace operator; there are no results for Schrödinger–Kirchhoff type equations driven by the fractional p-Laplace operator with variable-order
We recall some preliminary knowledge of generalized Lebesgue spaces with variable exponent
Summary
When p = 2, the fractional Laplace operator with variable-order was studied by Xiang et al in [11], they investigated the following Laplacian equations Ξ = 0, x ∈ R\Ω, where (−∆)s(·) is the fractional Laplacian operator First of all, they proved the embedding theorem of variable-order fractional Sobolev space, and they obtained a multiplicity result for a Schrödinger equation via variational methods. The fractional Kirchhoff type equation regarding non-local integro-differential operator was first introduced in [19] by Fiscella et al, and they studied the non-negative solutions for this kind of equation as follows. We are devoted to investigating the existence of infinitely many solutions for Schrödinger–Kirchhoff type equations involving a variableorder fractional p-Laplace operator by applying the fountain theorem and symmetric mountain pass theorem, respectively. When θ(x) ≡ constant, the results of Lemmas 1 and 2 still hold
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
