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Fractal Fract
Special Issues
Variational Problems and Fractional Differential Equations
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Variational Problems and Fractional Differential Equations

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: 31 December 2024 | Viewed by 3553

Special Issue Editors

Prof. Dr. Zhisu Liu

E-Mail Website
Guest Editor
School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China
Interests: variational problems; fractional PDEs; nonlinear elliptic DEs
Dr. Yu Su

E-Mail Website
Guest Editor
School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan 232001, China
Interests: critical point theory; fractional PDEs; calculus of variations

Special Issue Information

Dear Colleagues,

Fractional differential equations are being frequently used in physics, chemistry, biology, probability and finance modelling problems, such as, the ultrarelativistic limits of quantum mechanics, flame propagation, water waves, chemical reactions of liquids and population dynamics, etc. The Calculus of Variations provides a range of tools for the study of fractional differential equations for both mathematical theory and practical applications. The aim of this Special Issue is to present some of the recent developments on the qualitative properties of solutions for variational problems and fractional differential equations. The potential topics concerned with qualitative properties of solutions include, but are not limited to, a priori estimate, existence, non-existence, uniqueness, regularity, symmetry, stability and asymptotic behavior.

Prof. Dr. Zhisu Liu
Dr. Yu Su
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • variational problems
  • fractional differential equations
  • priori estimate
  • existence, non-existence, and uniqueness
  • regularity and symmetry
  • stability
  • asymptotic behavior

Published Papers (4 papers)

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Research

8 pages, 265 KiB  
Article
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 623
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)ps1u+(−Δ)qs2u+V(x)(|u|p−2u+|u|q−2u)=f(u)+λ|u|r−2u, where x∈RN, (−Δ)ps1 is the fractional p-Laplacian operator ((−Δ)qs2 is similar), 0<s1<s2<1<p<q<Ns2, qs2*=NqN−s2q, r≥qs2*, f is a C1 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  
Article
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 921
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  
Article
Multiple Solutions to a Non-Local Problem of Schrödinger–Kirchhoff Type in ℝN
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 1 | Viewed by 725
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  
Article
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
Viewed by 749
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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