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26 pages, 387 KiB

Open AccessFeature PaperArticle

Multiplicity Results of Solutions to the Fractional p-Laplacian Problems of the Kirchhoff–Schrödinger–Hardy Type

by Yun-Ho Kim

Mathematics 2025, 13(1), 47; https://doi.org/10.3390/math13010047 - 26 Dec 2024

Viewed by 714

This paper is devoted to establishing multiplicity results of nontrivial weak solutions to the fractional p-Laplacian equations of the Kirchhoff–Schrödinger type with Hardy potentials. The main features of the paper are the appearance of the Hardy potential and nonlocal Kirchhoff coefficients, and [...] Read more.

This paper is devoted to establishing multiplicity results of nontrivial weak solutions to the fractional p-Laplacian equations of the Kirchhoff–Schrödinger type with Hardy potentials. The main features of the paper are the appearance of the Hardy potential and nonlocal Kirchhoff coefficients, and the absence of the compactness condition of the Palais–Smale type. To demonstrate the multiplicity results, we exploit the fountain theorem and the dual fountain theorem as the main tools, respectively. Full article

(This article belongs to the Special Issue Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations)

28 pages, 374 KiB

Open AccessArticle

Global Existence, Blowup, and Asymptotic Behavior for a Kirchhoff-Type Parabolic Problem Involving the Fractional Laplacian with Logarithmic Term

by Zihao Guan and Ning Pan

Mathematics 2024, 12(1), 5; https://doi.org/10.3390/math12010005 - 19 Dec 2023

Cited by 3 | Viewed by 1228

Abstract

In this paper, we studied a class of semilinear pseudo-parabolic equations of the Kirchhoff type involving the fractional Laplacian with logarithmic nonlinearity: [...] Read more.

In this paper, we studied a class of semilinear pseudo-parabolic equations of the Kirchhoff type involving the fractional Laplacian with logarithmic nonlinearity:

\{\begin{matrix} u_{t} + M ({[u]}_{s}^{2}) {(- Δ)}^{s} u + {(- Δ)}^{s} u_{t} = {| u |}^{p - 2} u \ln | u |, & in Ω \times (0, T), \\ u (x, 0) = u_{0} (x), & in Ω, \\ u (x, t) = 0, & on \partial Ω \times (0, T), \end{matrix}

, where

{[u]}_{s}

is the Gagliardo semi-norm of u,

{(- Δ)}^{s}

is the fractional Laplacian,

s \in (0, 1)

,

2 λ < p < 2_{s}^{*} = 2 N / (N - 2 s)

,

Ω \in R^{N}

is a bounded domain with

N > 2 s

, and

u_{0}

is the initial function. To start with, we combined the potential well theory and Galerkin method to prove the existence of global solutions. Finally, we introduced the concavity method and some special inequalities to discuss the blowup and asymptotic properties of the above problem and obtained the upper and lower bounds on the blowup at the sublevel and initial level. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

16 pages, 361 KiB

Open AccessArticle

Multiple Solutions to a Non-Local Problem of Schrödinger–Kirchhoff Type in ℝ^N

by In Hyoun Kim, Yun-Ho Kim and Kisoeb Park

Fractal Fract. 2023, 7(8), 627; https://doi.org/10.3390/fractalfract7080627 - 17 Aug 2023

Cited by 4 | Viewed by 1292

Abstract

The main purpose of this paper is to show the existence of a sequence of infinitely many small energy solutions to the nonlinear elliptic equations of Kirchhoff–Schrödinger type involving the fractional p-Laplacian by employing the dual fountain theorem as a key tool. [...] Read more.

The main purpose of this paper is to show the existence of a sequence of infinitely many small energy solutions to the nonlinear elliptic equations of Kirchhoff–Schrödinger type involving the fractional p-Laplacian by employing the dual fountain theorem as a key tool. Because of the presence of a non-local Kirchhoff coefficient, under conditions on the nonlinear term given in the present paper, we cannot obtain the same results concerning the existence of solutions in similar ways as in the previous related works. For this reason, we consider a class of Kirchhoff coefficients that are different from before to provide our multiplicity result. In addition, the behavior of nonlinear terms near zero is slightly different from previous studies. Full article

(This article belongs to the Special Issue Variational Problems and Fractional Differential Equations)

34 pages, 481 KiB

Open AccessArticle

Left Riemann–Liouville Fractional Sobolev Space on Time Scales and Its Application to a Fractional Boundary Value Problem on Time Scales

by Xing Hu and Yongkun Li

Fractal Fract. 2022, 6(5), 268; https://doi.org/10.3390/fractalfract6050268 - 15 May 2022

Cited by 2 | Viewed by 2343

Abstract

First, we show the equivalence of two definitions of the left Riemann–Liouville fractional integral on time scales. Then, we establish and characterize fractional Sobolev space with the help of the notion of left Riemann–Liouville fractional derivative on time scales. At the same time, [...] Read more.

First, we show the equivalence of two definitions of the left Riemann–Liouville fractional integral on time scales. Then, we establish and characterize fractional Sobolev space with the help of the notion of left Riemann–Liouville fractional derivative on time scales. At the same time, we define weak left fractional derivatives and demonstrate that they coincide with the left Riemann–Liouville ones on time scales. Next, we prove the equivalence of two kinds of norms in the introduced space and derive its completeness, reflexivity, separability, and some embedding. Finally, as an application, by constructing an appropriate variational setting, using the mountain pass theorem and the genus properties, the existence of weak solutions for a class of Kirchhoff-type fractional p-Laplacian systems on time scales with boundary conditions is studied, and three results of the existence of weak solutions for this problem is obtained. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

16 pages, 346 KiB

Open AccessArticle

Fractional p(·)-Kirchhoff Type Problems Involving Variable Exponent Logarithmic Nonlinearity

by Jiabin Zuo, Amita Soni and Debajyoti Choudhuri

Fractal Fract. 2022, 6(2), 106; https://doi.org/10.3390/fractalfract6020106 - 12 Feb 2022

Cited by 4 | Viewed by 2125

Abstract

In this paper, we investigate a fractional

p (\cdot)

-Kirchhoff type problem involving variable exponent logarithmic nonlinearity. With the help of the Nehari manifold approach, the existence and multiplicity of nontrivial weak solutions for the above problem are obtained. The main [...] Read more.

In this paper, we investigate a fractional

p (\cdot)

-Kirchhoff type problem involving variable exponent logarithmic nonlinearity. With the help of the Nehari manifold approach, the existence and multiplicity of nontrivial weak solutions for the above problem are obtained. The main aspect and challenges of this paper are the presence of double non-local terms and logarithmic nonlinearity. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

20 pages, 372 KiB

Open AccessArticle

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

by Weichun Bu, Tianqing An, Guoju Ye and Chengwen Jiao

Mathematics 2021, 9(16), 1973; https://doi.org/10.3390/math9161973 - 18 Aug 2021

Cited by 2 | Viewed by 1954

Abstract

In this article, we study a class of Choquard–Kirchhoff type equations driven by the variable

s (x, \cdot)

-order fractional

p_{1} (x, \cdot)

and

p_{2} (x, \cdot)

-Laplacian. Assuming some reasonable conditions [...] Read more.

In this article, we study a class of Choquard–Kirchhoff type equations driven by the variable

s (x, \cdot)

-order fractional

p_{1} (x, \cdot)

and

p_{2} (x, \cdot)

-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. Full article

(This article belongs to the Special Issue Functional Differential Equations and Epidemiological Modelling)

22 pages, 369 KiB

Open AccessArticle

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

by Weichun Bu, Tianqing An, José Vanteler da C. Sousa and Yongzhen Yun

Symmetry 2021, 13(8), 1393; https://doi.org/10.3390/sym13081393 - 31 Jul 2021

Cited by 1 | Viewed by 1888

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 [...] Read more.

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 \int_{Ω \times Ω} \frac{{| ξ (x) - ξ (y) |}^{^{p}}}{{| x - y |}^{N + p s (x, y)}} d x d y)}^{p - 1} {(- Δ)}_{p}^{s (\cdot)} ξ + λ V (x) {| ξ |}^{p - 2} ξ = f (x, ξ), x \in Ω, ξ = 0, x \in \partial Ω,

where

Ω

is a bounded Lipschitz domain in

R^{N}

,

1 < p < + \infty

,

a, b > 0

are constants,

s (\cdot) : R^{N} \times R^{N} \to (0, 1)

is a continuous and symmetric function with

N > s (x, y) p

for all

(x, y) \in Ω \times Ω

,

λ > 0

is a parameter,

{(- Δ)}_{p}^{s (\cdot)}

is a fractional p-Laplace operator with variable-order,

V (x) : Ω \to R^{+}

is a potential function, and

f (x, ξ) : Ω \times R^{N} \to 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. Full article

17 pages, 334 KiB

Open AccessArticle

Existence of Solutions for Kirchhoff-Type Fractional Dirichlet Problem with p-Laplacian

by Danyang Kang, Cuiling Liu and Xingyong Zhang

Mathematics 2020, 8(1), 106; https://doi.org/10.3390/math8010106 - 8 Jan 2020

Cited by 5 | Viewed by 3112

Abstract

In this paper, we investigate the existence of solutions for a class of p-Laplacian fractional order Kirchhoff-type system with Riemann–Liouville fractional derivatives and a parameter

λ

. By mountain pass theorem, we obtain that system has at least one non-trivial weak solution [...] Read more.

In this paper, we investigate the existence of solutions for a class of p-Laplacian fractional order Kirchhoff-type system with Riemann–Liouville fractional derivatives and a parameter

λ

. By mountain pass theorem, we obtain that system has at least one non-trivial weak solution

u_{λ}

under some local conditions for each given large parameter

λ

. We get a concrete lower bound of the parameter

λ

, and then obtain two estimates of weak solutions

u_{λ}

. We also obtain that

u_{λ} \to 0

if

λ

tends to ∞. Finally, we present an example as an application of our results. Full article

(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)

21 pages, 390 KiB

Open AccessFeature PaperArticle

Multiplicity of Small or Large Energy Solutions for Kirchhoff–Schrödinger-Type Equations Involving the Fractional p-Laplacian in ℝ^N

by Jae-Myoung Kim, Yun-Ho Kim and Jongrak Lee

Symmetry 2018, 10(10), 436; https://doi.org/10.3390/sym10100436 - 26 Sep 2018

Cited by 4 | Viewed by 2963

Abstract

We herein discuss the following elliptic equations: [...] Read more.

We herein discuss the following elliptic equations:

M (\int_{R^{N}} \int_{R^{N}} \frac{{| u (x) - u (y) |}^{p}}{{| x - y |}^{N + p s}} d x d y) {(- Δ)}_{p}^{s} u + V (x) {| u |}^{p - 2} u = λ f (x, u) in R^{N},

where

{(- Δ)}_{p}^{s}

is the fractional p-Laplacian defined by

{(- Δ)}_{p}^{s} u (x) = 2 {lim}_{ε ↘ 0} \int_{R^{N} B_{ε} (x)} \frac{{| u (x) - u (y) |}^{p - 2} (u (x) - u (y))}{{| x - y |}^{N + p s}} d y, x \in R^{N} .

Here,

B_{ε} (x) : = {y \in R^{N} : | x - y | < ε}

,

V : R^{N} \to (0, \infty)

is a continuous function and

f : R^{N} \times R \to R

is the Carathéodory function. Furthermore,

M : R_{0}^{+} \to R^{+}

is a Kirchhoff-type function. This study has two aims. One is to study the existence of infinitely many large energy solutions for the above problem via the variational methods. In addition, a major point is to obtain the multiplicity results of the weak solutions for our problem under various assumptions on the Kirchhoff function

M

and the nonlinear term f. The other is to prove the existence of small energy solutions for our problem, in that the sequence of solutions converges to 0 in the

L^{\infty}

-norm. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Theory, Methods and Applications)

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