Variable order nonlocal Choquard problem with variable exponents

This paper is concerned with the existence and multiplicity of solutions for the following variable s(·)‐order fractional p(·)‐Kirchhoff type problem M()∬ℝ2N1pfalse(x,yfalse)false|vfalse(xfalse)−vfalse(yfalse)false|pfalse(x,yfalse)false|x−yfalse|N+pfalse(x,yfalse)sfalse(x,yfalse)dxdyfalse(−normalΔfalse)pfalse(·false)sfalse(·false)vfalse(xfalse)+false|vfalse(xfalse)false|truep‾false(xfalse)−2vfalse(xfalse)=μgfalse(x,vfalse)0.1em0.1emin0.3emℝN,v∈Wsfalse(·false),pfalse(·false)false(ℝNfalse), where N > p(x, y)s(x, y) for any false(x,yfalse)∈ℝN×ℝN, false(−normalΔfalse)pfalse(·false)sfalse(·false) is a variable s(·)‐order p(·)‐fractional Laplace operator with sfalse(·false):ℝ2N→false(0,1false) and pfalse(·false):ℝ2N→false(1,∞false), truep‾false(xfalse)=pfalse(x,xfalse) for x∈ℝN, and M is a continuous Kirchhoff‐type function, g(x, v) is a Carathéodory function, and μ > 0 is a parameter. Under the weaker conditions, we obtain that there are at least two distinct solutions for the above problem by applying the generalized abstract critical point theorem. Moreover, we also show the existence of one solution and infinitely many solutions by using the mountain pass lemma and fountain theorem, respectively. In particular, the new compact embedding result of the space Wsfalse(·false),pfalse(·false)false(ℝNfalse) into Lafalse(xfalse)qfalse(·false)false(ℝNfalse) will be used to overcome the lack of compactness in ℝN. The main feature and difficulty of this paper is the presence of a double non‐local term involving two variable parameters.

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“…The space X is a separable reflexive Banach space, see previous studies 17,50 . We define the convex modular function

ϱ_{p false(\cdot false)}^{s false(\cdot false)} : X \to ℝ

ϱ_{p (\cdot)}^{s (\cdot)} (v) = \iint_{ℝ^{2 N}} \frac{| v (x) - v (y) |^{p (x, y)}}{| x - y |^{N + p (x, y) s (x, y)}} d x d y + \int_{ℝ^{N}} | v (x) |^{\overline{p} (x)} d x,

whose associated norm define by

‖ v ‖ = ‖ v ‖_{ϱ_{p (\cdot)}^{s (\cdot)}} = \inf \{γ > 0 : ϱ_{p (\cdot)}^{s (\cdot)} \{\frac{v}{γ}\} \leq 1\},

which is equivalent to the norm ‖ · ‖ X .…”

Section: Functional Analytic Setup and Preliminaries Resultsmentioning

confidence: 99%

A variable‐order fractional p(·)‐Kirchhoff type problem in ℝN

Zuo

Yang

Liang

2020

Math Methods in App Sciences

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“…For classical Sobolev space theory such as constant order and constant exponential, see [43][44][45]. And for variable order, variable exponent cases, see [21,41].…”

Section: Preliminariesmentioning

confidence: 99%

Existence and Uniqueness of Weak Solutions to Variable-Order Fractional Laplacian Equations with Variable Exponents

Guo

2021

Journal of Function Spaces

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In this paper, the variable-order fractional Laplacian equations with variable exponents and the Kirchhoff-type problem driven by p · -fractional Laplace with variable exponents were studied. By using variational method, the authors obtain the existence and uniqueness results.

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“…In the framework of variable exponents involving fractional p(x, •)-Laplace operator with variable s(x, •)-order, such as Kirchhoff equations, Choquard equations, etc., there have been some papers on this topic-see [19,41,[47][48][49][50][51]. We point out that very recently in [47], Biswas et al firstly proved a embedding theorem for variable exponential Sobolev spaces and Hardy-Littlewood-Sobolev type result, and then they studied the existence of solutions for Choquard equations as follows…”

Section: Introductionmentioning

confidence: 99%

Existence Results for p1(x,·) and p2(x,·) Fractional Choquard–Kirchhoff Type Equations with Variable s(x,·)-Order

et al. 2021

Mathematics

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In this article, we study a class of Choquard–Kirchhoff type equations driven by the variable s(x,·)-order fractional p1(x,·) and p2(x,·)-Laplacian. Assuming some reasonable conditions and with the help of variational methods, we reach a positive energy solution and a negative energy solution in an appropriate space of functions. The main difficulties and innovations are the Choquard nonlinearities and Kirchhoff functions with the presence of double Laplace operators involving two variable parameters.

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Variable order nonlocal Choquard problem with variable exponents

Abstract: In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents

Cited by 33 publications

References 35 publications

A variable‐order fractional p(·)‐Kirchhoff type problem in ℝN

A variable‐order fractional p(·)‐Kirchhoff type problem in ℝN

Existence and Uniqueness of Weak Solutions to Variable-Order Fractional Laplacian Equations with Variable Exponents

Existence Results for p1(x,·) and p2(x,·) Fractional Choquard–Kirchhoff Type Equations with Variable s(x,·)-Order

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