Problems and Methods in Nonlinear Analysis

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: 30 June 2024 | Viewed by 2054

Special Issue Editors


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Guest Editor
Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy
Interests: nonlinear analysis; non-smooth analysis; calculus of variations; fixed point theory

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Guest Editor
Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy
Interests: nonlinear analysis; calculus of variations; regularity theory

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Guest Editor
Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy
Interests: nonlinear analysis; regularity theory; calculus of variations; harmonic analysis

Special Issue Information

Dear Colleagues,

The methods of nonlinear analysis have countless applications in ODEs, elliptic or parabolic PDEs, and fractional-type equations. The ramifications of nonlinear analysis combined with functional analysis, fixed point theory, regularity theory, and differential and algebraic geometry make it the ideal field where distant topics meet to produce fruitful results on existence, uniqueness or multiplicity, as well as qualitative properties of solutions to various integro-differential problems arising from the mathematical modelling of natural phenomena.

The aim of this Special Issue is to present new and meaningful applications of the most advanced techniques in this topic, to advertise and discuss new problems requiring genuinely innovative approaches, as well as to highlight progress in the field through review articles and/or open questions.

Prof. Salvatore A. Marano
Dr. Umberto Guarnotta
Dr. Sunra J. N. Mosconi
Guest Editors

Manuscript Submission Information

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Keywords

  • nonlinear analysis
  • partial differential equations
  • fractional operators
  • regularity theory
  • calculus of variations
  • symmetrisation
  • functional analysis
  • non-smooth analysis

Published Papers (3 papers)

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Research

13 pages, 286 KiB  
Article
On an Anisotropic Logistic Equation
by Leszek Gasiński and Nikolaos S. Papageorgiou
Mathematics 2024, 12(9), 1280; https://doi.org/10.3390/math12091280 - 24 Apr 2024
Viewed by 373
Abstract
We consider a nonlinear Dirichlet problem driven by the (p(z),q)-Laplacian and with a logistic reaction of the equidiffusive type. Under a nonlinearity condition on a quotient map, we show existence and uniqueness of positive solutions [...] Read more.
We consider a nonlinear Dirichlet problem driven by the (p(z),q)-Laplacian and with a logistic reaction of the equidiffusive type. Under a nonlinearity condition on a quotient map, we show existence and uniqueness of positive solutions and the result is global in parameter λ. If the monotonicity condition on the quotient map is not true, we can no longer guarantee uniqueness, but we can show the existence of a minimal solution uλ* and establish the monotonicity of the map λuλ* and its asymptotic behaviour as the parameter λ decreases to the critical value λ^1(q)>0 (the principal eigenvalue of (Δq,W01,q(Ω))). Full article
(This article belongs to the Special Issue Problems and Methods in Nonlinear Analysis)
27 pages, 858 KiB  
Article
A Nonlinear ODE Model for a Consumeristic Society
by Marino Badiale and Isabella Cravero
Mathematics 2024, 12(8), 1253; https://doi.org/10.3390/math12081253 - 20 Apr 2024
Viewed by 858
Abstract
In this paper, we introduce an ODE system to model the interaction between natural resources and human exploitation in a rich consumeristic society. In this model, the rate of change in population depends on wealth per capita, and the rate of consumption has [...] Read more.
In this paper, we introduce an ODE system to model the interaction between natural resources and human exploitation in a rich consumeristic society. In this model, the rate of change in population depends on wealth per capita, and the rate of consumption has a quadratic growth with respect to population and wealth. We distinguish between renewable and non-renewable resources, and we introduce a replenishment term for non-renewable resources. We first obtain some information on the asymptotic behavior of wealth and population, then we compute all system equilibria and study their stability when the resource exploitation parameter is low. Some numerical simulations confirm the theoretical results and suggest directions for future research. Full article
(This article belongs to the Special Issue Problems and Methods in Nonlinear Analysis)
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20 pages, 306 KiB  
Article
Multiplicity of Normalized Solutions for the Fractional Schrödinger Equation with Potentials
by Xue Zhang, Marco Squassina and Jianjun Zhang
Mathematics 2024, 12(5), 772; https://doi.org/10.3390/math12050772 - 5 Mar 2024
Viewed by 539
Abstract
We are concerned with the existence and multiplicity of normalized solutions to the fractional Schrödinger equation [...] Read more.
We are concerned with the existence and multiplicity of normalized solutions to the fractional Schrödinger equation (Δ)su+V(εx)u=λu+h(εx)f(u)inRN,RN|u|2dx=a,, where (Δ)s is the fractional Laplacian, s(0,1), a,ε>0, λR is an unknown parameter that appears as a Lagrange multiplier, h:RN[0,+) are bounded and continuous, and f is L2-subcritical. Under some assumptions on the potential V, we show the existence of normalized solutions depends on the global maximum points of h when ε is small enough. Full article
(This article belongs to the Special Issue Problems and Methods in Nonlinear Analysis)
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