Fractal and Fractional

Research

21 pages, 384 KiB

Open AccessArticle

The Existence of a Solution to a Class of Fractional Double Phase Problems

by Maoji Ri and Yongkun Li

Fractal Fract. 2024, 8(11), 621; https://doi.org/10.3390/fractalfract8110621 - 24 Oct 2024

Viewed by 512

Abstract

This paper focuses on the study of a class of fractional

p & q

-Laplacian problems with unbalanced growth, which includes vanishing potential and a supercritical growth exponent. By employing the mountain pass theorem alongside the Truncation method, penalization method, and Moser iteration [...] Read more.

This paper focuses on the study of a class of fractional

p & q

-Laplacian problems with unbalanced growth, which includes vanishing potential and a supercritical growth exponent. By employing the mountain pass theorem alongside the Truncation method, penalization method, and Moser iteration method, the main result establishes the existence of a nontrivial solution under conditions of low perturbations of supercritical nonlinearity. Furthermore, we derive

L^{\infty} (R^{N})

estimates and the interior Hölder regularity of weak solutions in the context of supercritical growth. Full article

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

17 pages, 333 KiB

Open AccessArticle

Solutions for a Logarithmic Fractional Schrödinger-Poisson System with Asymptotic Potential

by Lifeng Guo, Yuan Li and Sihua Liang

Fractal Fract. 2024, 8(9), 528; https://doi.org/10.3390/fractalfract8090528 - 10 Sep 2024

Viewed by 585

Abstract

In this paper, we consider a logarithmic fractional Schrödinger-Poisson system where the potential is a sign-changing function. When the potential is coercive, we get the existence of infinitely many solutions for the system. When the potential is bounded, we get the existence of [...] Read more.

In this paper, we consider a logarithmic fractional Schrödinger-Poisson system where the potential is a sign-changing function. When the potential is coercive, we get the existence of infinitely many solutions for the system. When the potential is bounded, we get the existence of a ground state solution for the system. Full article

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

18 pages, 324 KiB

Open AccessArticle

Multiplicity of Normalized Solutions to a Fractional Logarithmic Schrödinger Equation

by Yan-Cheng Lv and Gui-Dong Li

Fractal Fract. 2024, 8(7), 391; https://doi.org/10.3390/fractalfract8070391 - 29 Jun 2024

Viewed by 640

Abstract

We study the existence and multiplicity of normalized solutions to the fractional logarithmic Schrödinger equation

{(- Δ)}^{s} u + V (ϵ x) u = λ u + u log u^{2} in R^{N},

under the mass [...] Read more.

We study the existence and multiplicity of normalized solutions to the fractional logarithmic Schrödinger equation

{(- Δ)}^{s} u + V (ϵ x) u = λ u + u log u^{2} in R^{N},

under the mass constraint

\int_{R^{N}} {| u |}^{2} d x = a .

Here,

N \geq 2

,

a, ϵ > 0

,

λ \in R

is an unknown parameter,

{(- Δ)}^{s}

is the fractional Laplacian and

s \in (0, 1)

. We introduce a function space where the energy functional associated with the problem is of class

C^{1}

. Then, under some assumptions on the potential V and using the Lusternik–Schnirelmann category, we show that the number of normalized solutions depends on the topology of the set for which the potential V reaches its minimum. Full article

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

27 pages, 412 KiB

Open AccessArticle

Multiple Normalized Solutions to a Choquard Equation Involving Fractional p-Laplacian in ℝ^N

by Xin Zhang and Sihua Liang

Fractal Fract. 2024, 8(6), 310; https://doi.org/10.3390/fractalfract8060310 - 23 May 2024

Viewed by 1130

Abstract

In this paper, we study the existence of multiple normalized solutions for a Choquard equation involving fractional p-Laplacian in

R^{N}

. With the help of variational methods, minimization techniques, and the Lusternik–Schnirelmann category, the existence of multiple normalized solutions is obtained [...] Read more.

In this paper, we study the existence of multiple normalized solutions for a Choquard equation involving fractional p-Laplacian in

R^{N}

. With the help of variational methods, minimization techniques, and the Lusternik–Schnirelmann category, the existence of multiple normalized solutions is obtained for the above problem. Full article

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

8 pages, 265 KiB

Open AccessArticle

The Sign-Changing Solution for Fractional (p,q)-Laplacian Problems Involving Supercritical Exponent

by Jianwen Zhou, Chengwen Gong and Wenbo Wang

Fractal Fract. 2024, 8(4), 186; https://doi.org/10.3390/fractalfract8040186 - 25 Mar 2024

Viewed by 1041

Abstract

In this article, we consider the following fractional

(p, q)

-Laplacian problem [...] Read more.

In this article, we consider the following fractional

(p, q)

-Laplacian problem

{(- Δ)}_{p}^{s_{1}} u + {(- Δ)}_{q}^{s_{2}} u + {V (x) (| u |}^{p - 2} u + {| u |}^{q - 2} {u) = f (u) + λ | u |}^{r - 2} u,

where

x \in R^{N}

,

{(- Δ)}_{p}^{s_{1}}

is the fractional p-Laplacian operator (

{(- Δ)}_{q}^{s_{2}}

is similar),

0 < s_{1} < s_{2} < 1 < p < q < \frac{N}{s_{2}}

,

q_{s_{2}}^{*} = \frac{N q}{N - s_{2} q}

,

r \geq q_{s_{2}}^{*}

, f is a

C^{1}

real function and V is a coercive function. By using variational methods, we prove that the above problem admits a sign-changing solution if

λ > 0

is small. Full article

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

20 pages, 330 KiB

Open AccessArticle

Existence of Ground State Solutions for a Class of Non-Autonomous Fractional Kirchhoff Equations

by Guangze Gu, Changyang Mu and Zhipeng Yang

Fractal Fract. 2024, 8(2), 113; https://doi.org/10.3390/fractalfract8020113 - 14 Feb 2024

Viewed by 1372

Abstract

We take a look at the fractional Kirchhoff problem in this paper. Using a variational approach, we show that there exists a ground state solution for this problem. Furthermore, using the approach developed by Szulkin and Weth, we also find that positive ground [...] Read more.

We take a look at the fractional Kirchhoff problem in this paper. Using a variational approach, we show that there exists a ground state solution for this problem. Furthermore, using the approach developed by Szulkin and Weth, we also find that positive ground state solutions exist for the fractional Kirchhoff equation with

p = 4

. Full article

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

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 2 | Viewed by 1046

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)

23 pages, 402 KiB

Open AccessArticle

Ground State Solutions of Fractional Choquard Problems with Critical Growth

by Jie Yang and Hongxia Shi

Fractal Fract. 2023, 7(7), 555; https://doi.org/10.3390/fractalfract7070555 - 17 Jul 2023

Cited by 1 | Viewed by 1200

Abstract

In this article, we investigate a class of fractional Choquard equation with critical Sobolev exponent. By exploiting a monotonicity technique and global compactness lemma, the existence of ground state solutions for this equation is obtained. In addition, we demonstrate the existence of ground [...] Read more.

In this article, we investigate a class of fractional Choquard equation with critical Sobolev exponent. By exploiting a monotonicity technique and global compactness lemma, the existence of ground state solutions for this equation is obtained. In addition, we demonstrate the existence of ground state solutions for the corresponding limit problem. Full article

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

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