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13 pages, 272 KiB

Open AccessFeature PaperArticle

Three Weak Solutions for a Critical Non-Local Problem with Strong Singularity in High Dimension

by Gabriel Neves Cunha, Francesca Faraci and Kaye Silva

Mathematics 2024, 12(18), 2910; https://doi.org/10.3390/math12182910 - 18 Sep 2024

Viewed by 499

Abstract

In this paper, we deal with a strongly singular problem involving a non-local operator, a critical nonlinearity, and a subcritical perturbation. We apply techniques from non-smooth analysis to the energy functional, in combination with the study of the topological properties of the sublevels [...] Read more.

In this paper, we deal with a strongly singular problem involving a non-local operator, a critical nonlinearity, and a subcritical perturbation. We apply techniques from non-smooth analysis to the energy functional, in combination with the study of the topological properties of the sublevels of its smooth part, to prove the existence of three weak solutions: two points of local minimum and a third one as a mountain pass critical point. Full article

(This article belongs to the Special Issue Problems and Methods in Nonlinear Analysis)

16 pages, 337 KiB

Open AccessFeature PaperArticle

Pairs of Positive Solutions for a Carrier p(x)-Laplacian Type Equation

by Pasquale Candito, Giuseppe Failla and Roberto Livrea

Mathematics 2024, 12(16), 2441; https://doi.org/10.3390/math12162441 - 6 Aug 2024

Viewed by 566

Abstract

The existence of multiple pairs of smooth positive solutions for a Carrier problem, driven by a

p (x)

-Laplacian operator, is studied. The approach adopted combines sub-super solutions, truncation, and variational techniques. In particular, after an explicit computation of a sub-solution, [...] Read more.

The existence of multiple pairs of smooth positive solutions for a Carrier problem, driven by a

p (x)

-Laplacian operator, is studied. The approach adopted combines sub-super solutions, truncation, and variational techniques. In particular, after an explicit computation of a sub-solution, obtained combining a monotonicity type hypothesis on the reaction term and the Giacomoni–Takáč’s version of the celebrated Díaz–Saá’s inequality, we derive a multiplicity of solution by investigating an associated one-dimensional fixed point problem. The nonlocal term involved may be a sign-changing function and permit us to obtain the existence of multiple pairs of positive solutions, one for each “positive bump” of the nonlocal term. A new result, also for a constant exponent, is established and an illustrative example is proposed. Full article

(This article belongs to the Special Issue Problems and Methods in Nonlinear Analysis)

7 pages, 233 KiB

Open AccessFeature PaperArticle

Infinitely Many Solutions for Schrödinger–Poisson Systems and Schrödinger–Kirchhoff Equations

by Shibo Liu

Mathematics 2024, 12(14), 2233; https://doi.org/10.3390/math12142233 - 17 Jul 2024

Viewed by 897

Abstract

By applying Clark’s theorem as altered by Liu and Wang and the truncation method, we obtain a sequence of solutions for a Schrödinger–Poisson system [...] Read more.

By applying Clark’s theorem as altered by Liu and Wang and the truncation method, we obtain a sequence of solutions for a Schrödinger–Poisson system

\{\begin{matrix} - Δ u + V (x) u + ϕ u = f (u) & in R^{3}, \\ - Δ ϕ = u^{2} & in R^{3} \end{matrix}

with negative energy. A similar result is also obtained for the Schrödinger-Kirchhoff equation as follows:

- (1 + \int_{R^{N}} {|\nabla u|}^{2}) Δ u + V (x) u = f (u) u \in H^{1} (R^{N})

. Full article

(This article belongs to the Special Issue Problems and Methods in Nonlinear Analysis)

11 pages, 272 KiB

Open AccessArticle

Systems of Hemivariational Inclusions with Competing Operators

by Dumitru Motreanu

Mathematics 2024, 12(11), 1766; https://doi.org/10.3390/math12111766 - 6 Jun 2024

Viewed by 533

Abstract

This paper focuses on a system of differential inclusions expressing hemivariational inequalities driven by competing operators constructed with p-Laplacians that involve two real parameters. The existence of a generalized solution is shown by means of an approximation process through approximate solutions in [...] Read more.

This paper focuses on a system of differential inclusions expressing hemivariational inequalities driven by competing operators constructed with p-Laplacians that involve two real parameters. The existence of a generalized solution is shown by means of an approximation process through approximate solutions in finite dimensional spaces. When the parameters are negative, the generalized solutions become weak solutions. The main novelty of this work is the solvability of systems of differential inclusions for which the ellipticity condition may fail. Full article

(This article belongs to the Special Issue Problems and Methods in Nonlinear Analysis)

13 pages, 286 KiB

Open AccessArticle

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 680

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

u_{λ}^{*}

and establish the monotonicity of the map

λ ⟼ u_{λ}^{*}

and its asymptotic behaviour as the parameter

λ

decreases to the critical value

{\hat{λ}}_{1} (q) > 0

(the principal eigenvalue of

(- Δ_{q}, W_{0}^{1, q} (Ω))

). Full article

(This article belongs to the Special Issue Problems and Methods in Nonlinear Analysis)

27 pages, 858 KiB

Open AccessFeature PaperArticle

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 1584

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

Open AccessFeature PaperArticle

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

Cited by 2 | Viewed by 968

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

\{\begin{matrix} {(- Δ)}^{s} u + V (ε x) u = λ u + h (ε x) f (u) & i n R^{N}, \\ \int_{R^{N}} {| u |}^{2} d x = a, \end{matrix}

, where

{(- Δ)}^{s}

is the fractional Laplacian,

s \in (0, 1)

,

a, ε > 0

,

λ \in R

is an unknown parameter that appears as a Lagrange multiplier,

h : R^{N} \to [0, + \infty)

are bounded and continuous, and f is

L^{2}

-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)

Review

Jump to: Research

17 pages, 326 KiB

Open AccessFeature PaperReview

On the Sub and Supersolution Method for Nonlinear Elliptic Equations with a Convective Term, in Orlicz Spaces

by Giuseppina Barletta

Mathematics 2024, 12(16), 2506; https://doi.org/10.3390/math12162506 - 14 Aug 2024

Viewed by 492

Abstract

In this note we provide an overview of some existence (with sign information) and regularity results for differential equations, in which the method of sub and supersolutions plays an important role. We list some classical results and then we focus on the Dirichlet [...] Read more.

In this note we provide an overview of some existence (with sign information) and regularity results for differential equations, in which the method of sub and supersolutions plays an important role. We list some classical results and then we focus on the Dirichlet problem, for problems driven by a general differential operator, depending on

(x, u, \nabla u)

, and with a convective term f. Our framework is that of Orlicz–Sobolev spaces. We also present several examples. Full article

(This article belongs to the Special Issue Problems and Methods in Nonlinear Analysis)

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