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Nonlinear Equations Driven by Fractional Laplacian Operators

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Dr. J. Vanterler Da C. Sousa

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Guest Editor

Department of Mathematics, Aerospace Engineering, PPGEA-UEMA, DEMATI-UEMA, São Luís 65054, MA, Brazil
Interests: fractional differential equations; functional analysis; variational approach; frac-tional calculus; analysis mathematics
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Dr. Leandro Tavares

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Guest Editor

Center of Sciences and Technology, Federal University of Cariri, Juazeiro do Norte, Ceará 63048-080, Brazil
Interests: partial differential equations; mathematical analysis; equations with fractional operators

Special Issue Information

Dear Colleagues,

Fractional Differential Equations, an extension of the usual differential equations, broaden the scope of differentiation and integration to encompass arbitrary real or complex orders. Moreover, this topic has been attracting the attention of numerous researchers due to its rich applicability across several branches of science and technology. These equations play a pivotal role in describing various phenomena, including anomalous diffusion, viscoelasticity, fractional quantum mechanics, fractional dynamical systems, control theory, signal processing, and others in the fields of physics, biology, chemistry, economics, geophysics, engineering, and beyond. Unlike classical methods, problems involving fractional operators adeptly capture non-local and memory effects in complex systems, providing accurate models where traditional approaches fall short.

Researchers working on problems involving the fractional Laplacian operator are invited to contribute their original and high-quality work to this Special Issue, which is led by experienced researchers in the subject, fostering collaboration and pushing forward the boundaries of fractional equations. By doing so, they can contribute to the ongoing exploration and understanding of fractional calculus, consolidating cutting-edge research. This Special Issue aims to pave the way for innovative solutions and breakthroughs in the intricate new realm of equations driven by fractional operators, addressing real-world challenges and/or abstract mathematical problems.

Dr. J. Vanterler Da C. Sousa
Dr. Leandro Tavares
Guest Editors

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Published Papers (2 papers)

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Research

24 pages, 397 KiB

Open AccessArticle

Concentrating Solutions for Fractional Schrödinger–Poisson Systems with Critical Growth

by Liejun Shen and Marco Squassina

Fractal Fract. 2024, 8(10), 581; https://doi.org/10.3390/fractalfract8100581 - 30 Sep 2024

Viewed by 533

Abstract

In this paper, we consider a class of fractional Schrödinger–Poisson systems

{(- Δ)}^{s} u + λ V (x) u + ϕ u = f (u) + {| u |}^{2_{s}^{*} - 2} u

and [...] Read more.

In this paper, we consider a class of fractional Schrödinger–Poisson systems

{(- Δ)}^{s} u + λ V (x) u + ϕ u = f (u) + {| u |}^{2_{s}^{*} - 2} u

and

{(- Δ)}^{t} ϕ = u^{2}

R^{3}

, where

s, t \in (0, 1)

with

2 s + 2 t > 3

λ > 0

denotes a parameter,

V : R^{3} \to R

admits a potential well

Ω ≜ int V^{- 1} (0)

and

2_{s}^{*} ≜ \frac{6}{3 - 2 s}

is the fractional Sobolev critical exponent. Given some reasonable assumptions as to the potential V and the nonlinearity f, with the help of a constrained manifold argument, we conclude the existence of positive ground state solutions for some sufficiently large

λ

. Upon relaxing the restrictions on V and f, we utilize the minimax technique to show that the system has a positive mountain-pass type by introducing some analytic tricks. Moreover, we investigate the asymptotical behavior of the obtained solutions when

λ \to + \infty

. Full article

(This article belongs to the Special Issue Nonlinear Equations Driven by Fractional Laplacian Operators)

14 pages, 317 KiB

Open AccessArticle

A Fractional Magnetic System with Critical Nonlinearities

by Libo Yang, Shapour Heidarkhani and Jiabin Zuo

Fractal Fract. 2024, 8(7), 380; https://doi.org/10.3390/fractalfract8070380 - 27 Jun 2024

Viewed by 692

Abstract

In the present paper, we investigate a fractional magnetic system involving critical concave–convex nonlinearities with Laplace operators. Specifically,

{(- Δ)}_{A}^{s} u_{1} = λ_{1} {| u_{1} |}^{q - 2} u_{1}

+ [...] Read more.

In the present paper, we investigate a fractional magnetic system involving critical concave–convex nonlinearities with Laplace operators. Specifically,

{(- Δ)}_{A}^{s} u_{1} = λ_{1} {| u_{1} |}^{q - 2} u_{1}

\frac{2 α_{1}}{α_{1} + β_{1}} | u_{1} |^{α_{1} - 2} u_{1} {| u_{2} |}^{β_{1}}

Ω

{(- Δ)}_{A}^{s} u_{2} = λ_{2} | u_{2} |^{q - 2} u_{2} + \frac{2 β_{1}}{α_{1} + β_{1}} | u_{2} |^{β_{1} - 2} u_{2} {| u_{1} |}^{α_{1}}

Ω

u_{1} = u_{2} = 0

R^{n} ∖ Ω

, where

Ω

is a bounded set with Lipschitz boundary

\partial Ω

R^{n}

1 < q < 2 < \frac{n}{s}

with

s \in (0, 1)

λ_{1}, λ_{2}

are two real positive parameters,

α_{1} > 1, β_{1} > 1

α_{1} + β_{1} = 2_{s}^{*} = \frac{2 n}{n - 2 s}

2_{s}^{*}

is the fractional critical Sobolev exponent, and

{(- Δ)}_{A}^{s}

is a fractional magnetic Laplace operator. By using Lusternik–Schnirelmann’s theory, we prove the existence result of infinitely many solutions for the magnetic fractional system. Full article

(This article belongs to the Special Issue Nonlinear Equations Driven by Fractional Laplacian Operators)

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