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Keyword: Porous Nanofibers (3)	15 Apr 2025	Run
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articleNo Access
Extremal solutions for nonlocal dispersal equations
- Jian-Wen Sun
Analysis and Applications10 Jan 2025
Preview Abstract
This paper is devoted to the study of the extremal (global maximal or minimal) solutions for some nonlocal dispersal equations. We investigate sufficient conditions for the existence of extremal solutions. In addition, we establish the uniqueness of positive solution and obtain various uniqueness results when the nonlinearity satisfies specific monotone conditions.
articleOpen Access
Positive solutions for fractional Kirchhoff–Schrödinger–Poisson system with steep potential well
- Hui Jian,
- Qiaocheng Zhong, and
- Li Wang
Reviews in Mathematical Physics15 Jul 2023
Preview Abstract
In this paper, we deal with the following fractional Kirchhoff–Schrödinger–Poisson system:
$\{\begin{matrix} (a + b {[u]}_{s}^{2}) {(- Δ)}^{s} u + λ V (x) u + μ ϕ u = | u |^{p - 2} u & in ℝ^{3}, \\ {(- Δ)}^{t} ϕ = u^{2} & in ℝ^{3}, \end{matrix}$
where $s \in [\frac{3}{4}, 1), t \in (0, 1), 2 < p < 4$ and $a > 0$ is a constant, $b, λ, μ$ are positive parameters, $V (x)$ represents a potential well with the bottom $V^{- 1} (0)$ . By applying the truncation technique and the parameter-dependent compactness lemma, we first prove the existence of positive solutions for b small, $λ$ large and $μ$ small in the case $2 < p < 4$ . Moreover, we investigate the decay rate of positive solutions as $| x | \to \infty$ as well as their asymptotic behavior as $b \to 0, λ \to \infty$ and $μ \to 0$ , respectively.
articleNo Access
Differential equations with multiple sign changing convolution coefficients
- Christopher S. Goodrich
International Journal of Mathematics18 Jun 2021
Preview Abstract
For continuous functions $A$ and $B$ , which are allowed to change sign, we consider the nonlocal differential equation
$- A ((a * u^{q}) (1)) u^{″} (t) = λ B ((b * u^{p}) (1)) f (t, u (t)), t \in (0, 1),$
where $p$ , $q \in (1, + \infty)$ and $(a * b) (t)$ represents the finite convolution of the functions $a$ and $b$ . A model case is the equation
$- A (\int_{0}^{1} {(u (r))}^{q} d r) u^{″} (t) = λ B (\int_{0}^{1} {(u (r))}^{p} d r) f (t, u (t)), t \in (0, 1) .$
The existence of at least one positive solution to these problems subjected to a variety of boundary conditions is studied. Due to the use of a nonstandard order cone we are able to achieve our results without having to assume that the coefficients $A$ and $B$ are strictly positive.
articleNo Access
A priori bounds and existence of positive solutions to a $p$ -Kirchhoff equations
- Pengfei Li and
- Junhui Xie
International Journal of Mathematics18 Aug 2021
Preview Abstract
In this paper, we consider a $p$ -Kirchhoff problem with Dirichlet boundary problem in a bounded domain. Under suitable conditions, we get a priori estimates for positive solutions to an auxiliary problem by the well-known blow-up argument. As an application, a existence result for positive solutions is proved by the topological degree theory.
articleOpen Access
SYSTEMS OF NONLINEAR DELAY INTEGRAL EQUATIONS: POSITIVE SOLUTIONS, COMPARISON THEOREMS, AND DATA DEPENDENCE
- IBTEHAL ALAZMAN,
- MOHAMED JLELI, and
- BESSEM SAMET
Fractals01 Dec 2022
Preview Abstract
In this paper, a system of nonlinear delay integral equations related to the spread of certain infectious diseases is investigated. Namely, using Perov’s fixed-point theorem, we obtain sufficient conditions for which the system admits a unique positive solution, and provide a numerical algorithm that converges to this solution. Moreover, we establish some comparison theorems, and study the data dependence of solutions.
articleNo Access
PRESCRIBING A FOURTH ORDER CONFORMAL INVARIANT ON THE STANDARD SPHERE — PART I: A PERTURBATION RESULT
Communications in Contemporary Mathematics01 Aug 2002
Preview Abstract
In this paper we study some fourth order elliptic equation involving the critical Sobolev exponent, related to the prescription of a fourth order conformal invariant on the standard sphere. We use a topological method to prove the existence of at least a solution when the function to be prescribed is close to a constant and a finite dimensional map associated to it has non-zero degree
articleNo Access
MULTIPLE POSITIVE SOLUTIONS OF A ONE-DIMENSIONAL PRESCRIBED MEAN CURVATURE PROBLEM
- PATRICK HABETS and
- PIERPAOLO OMARI
Communications in Contemporary Mathematics01 Oct 2007
Preview Abstract
We discuss existence, non-existence and multiplicity of positive solutions of the Dirichlet problem for the one-dimensional prescribed curvature equation
in connection with the changes of concavity of the function f. The proofs are based on an upper and lower solution method, we specifically develop for this problem, combined with a careful analysis of the time-map associated with some related autonomous equations.
articleNo Access
EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR p-LAPLACIAN PROBLEMS WITH A GENERAL INDEFINITE WEIGHT
- CHAN-GYUN KIM and
- YONG-HOON LEE
Communications in Contemporary Mathematics01 Jun 2008
Preview Abstract
This paper studies the existence of positive solutions for a class of singular boundary value problems of p-Laplacian. By using Global Continuation Theorem and the fixed point index technique, criteria of the existence, multiplicity and nonexistence of positive solutions are established.
articleNo Access
NONNEGATIVE SOLUTIONS FOR INDEFINITE SUBLINEAR ELLIPTIC PROBLEMS
- FRANCISCO ODAIR DE PAIVA
Communications in Contemporary Mathematics01 Jun 2012
Preview Abstract
This paper is devoted to the study of existence, nonexistence and multiplicity of positive solutions for the semilinear elliptic problem
where Ω is a bounded domain of ℝ^N, λ ∈ ℝ and g(x, u) is a Carathéodory function. The obtained results apply to the following classes of nonlinearities: a(x)u^q + b(x)u^p and c(x)(1 + u)^p (0 ≤ q < 1 < p). The proofs rely on the sub-super solution method and the mountain pass theorem.
articleNo Access
A variational approach to nonlinear logistic equations
Communications in Contemporary Mathematics23 Mar 2015
Preview Abstract
We consider a nonlinear logistic type equation. For all big values of the parameter, we show that the problem admits nontrivial solutions of constant sign and in fact we establish the existence of extremal constant sign solutions. Using these extremal solutions, we produce a nodal (sign-changing) solution. We also investigate the uniqueness and continuous dependence on the parameter of positive solutions. Finally, we study the degenerate p-logistic equation.
articleNo Access
Positive and nodal solutions for an elliptic equation with critical growth
- Marcelo F. Furtado and
- Bruno N. Souza
Communications in Contemporary Mathematics17 Feb 2016
Preview Abstract
We consider the problem
$- div (p (x) \nabla u) = λ | u |^{q - 2} u + | u |^{2^{*} - 2} u, u \in H_{0}^{1} (Ω),$
where $Ω \subset ℝ^{N}$ is a bounded smooth domain, $N \geq 4$ , $2^{*} = 2 N / (N - 2)$ , $2 \leq q < 2^{*}$ . Under some suitable conditions on the continuous potential $p (x)$ and on the parameter $λ > 0$ , we obtain one nodal solution for $q = 2$ and one positive solution for $2 < q < 2^{*}$ .
articleNo Access
Some global results for a class of homogeneous nonlocal eigenvalue problems
- Guowei Dai
Communications in Contemporary Mathematics01 May 2019
Preview Abstract
This paper studies the global bifurcation phenomenon for the following homogeneous nonlocal eigenvalue problem
$\{\begin{matrix} - (\int_{Ω} | \nabla u |^{2} d x) Δ u = λ u^{3} + h (x, u, λ) & in Ω, \\ u = 0 & on Ω . \end{matrix}$
Under some natural hypotheses on $h$ and $Ω$ , we show that $(μ_{1}, 0)$ is a bifurcation point of the nontrivial solution set of the above problem. As application of the above result, we determine the interval of $λ$ , in which there exist positive solutions for the following Kirchhoff type problem
$\{\begin{matrix} - (\int_{Ω} | \nabla u |^{2} d x) Δ u = λ f (x, u) & in Ω, \\ u = 0 & on \partial Ω, \end{matrix}$
where $f$ is asymptotically 3-linear at zero and infinity. Our results provide a positive answer to an open problem. Moreover, we also study the spectral structure for a homogeneous nonlocal eigenvalue problem.
articleNo Access
Symmetry-breaking bifurcation for the one-dimensional Hénon equation
- Inbo Sim and
- Satoshi Tanaka
Communications in Contemporary Mathematics28 Jan 2019
Preview Abstract
We show the existence of a symmetry-breaking bifurcation point for the one-dimensional Hénon equation
$u^{″} + | x |^{l} u^{p} = 0, x \in (- 1, 1), u (- 1) = u (1) = 0,$
where $l > 0$ and $p > 1$ . Moreover, employing a variant of Rabinowitz’s global bifurcation, we obtain the unbounded connected set (the first of the alternatives about Rabinowitz’s global bifurcation), which emanates from the symmetry-breaking bifurcation point. Finally, we give an example of a bounded branch connecting two symmetry-breaking bifurcation points (the second of the alternatives about Rabinowitz’s global bifurcation) for the problem
$u^{″} + | x |^{l (λ)} u^{p} = 0, x \in (- 1, 1), u (- 1) = u (1) = 0,$
where $l$ is a specified continuous parametrization function.
articleNo Access
Choquard equations with critical nonlinearities
- Xinfu Li and
- Shiwang Ma
Communications in Contemporary Mathematics08 Apr 2019
Preview Abstract
In this paper, we study the Brezis–Nirenberg type problem for Choquard equations in $ℝ^{N}$
$- Δ u + u = (I_{α} * | u |^{p}) | u |^{p - 2} u + λ | u |^{q - 2} u in ℝ^{N},$
where $N \geq 3, α \in (0, N)$ , $λ > 0$ , $q \in (2, \frac{2 N}{N - 2}]$ , $p = \frac{N + α}{N}$ or $\frac{N + α}{N - 2}$ are the critical exponents in the sense of Hardy–Littlewood–Sobolev inequality and $I_{α}$ is the Riesz potential. Based on the results of the subcritical problems, and by using the subcritical approximation and the Pohožaev constraint method, we obtain a positive and radially nonincreasing ground-state solution in $H^{1} (ℝ^{N})$ for the problem. To the end, the regularity and the Pohožaev identity of solutions to a general Choquard equation are obtained.
articleNo Access
Standing waves with a critical frequency for a quasilinear Schrödinger equation
- Xiang-Dong Fang
Communications in Contemporary Mathematics12 Nov 2019
Preview Abstract
We consider the following quasilinear Schrödinger equation
$- 𝜀^{2} Δ w + V (x) w - 𝜀^{2} Δ (w^{2}) w = h (w), w > 0, x \in ℝ^{N},$
where $N \geq 4$ , ${lim inf}_{| x | \to \infty} V (x) > {inf}_{x \in ℝ^{N}} V (x) = 0$ , and $h$ satisfies a weaker growth condition than the Ambrosetti–Rabinowitz type condition in Byeon and Wang [Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal. 165(4) (2002) 295–316; Standing waves with a critical frequency for nonlinear Schrödinger equations, II, Calc. Var. 18(2) (2003) 207–219]. We obtain the existence of the localized bound state solutions concentrating at an isolated component of the local minimum of $V$ and whose amplitude goes to 0 as $𝜀 \to 0$ .
articleNo Access
Positive solutions of superlinear indefinite prescribed mean curvature problems
- Pierpaolo Omari and
- Elisa Sovrano
Communications in Contemporary Mathematics27 Jul 2020
Preview Abstract
This paper analyzes the superlinear indefinite prescribed mean curvature problem
$- div (\nabla u / \sqrt{1 + | \nabla u |^{2}}) = λ a (x) h (u) in Ω, u = 0 on \partial Ω,$
where $Ω$ is a bounded domain in $ℝ^{N}$ with a regular boundary $\partial Ω$ , $h \in C^{0} (ℝ)$ satisfies $h (s) \sim s^{p}$ , as $s \to 0^{+}$ , $p > 1$ being an exponent with $p < \frac{N + 2}{N - 2}$ if $N \geq 3$ , $λ > 0$ represents a parameter, and $a \in C^{0} (\bar{Ω})$ is a sign-changing function. The main result establishes the existence of positive regular solutions when $λ$ is sufficiently large, providing as well some information on the structure of the solution set. The existence of positive bounded variation solutions for $λ$ small is further discussed assuming that $h$ satisfies $h (s) \sim s^{q}$ as $s \to + \infty$ , $q > 0$ being such that $q < \frac{1}{N - 1}$ if $N \geq 2$ ; thus, in dimension $N \geq 2$ , the function $h$ is not superlinear at $+ \infty$ , although its potential $H (s) = \int_{0}^{s} h (t) d t$ is. Imposing such different degrees of homogeneity of $h$ at $0$ and at $+ \infty$ is dictated by the specific features of the mean curvature operator.
articleNo Access
Positive solutions for the Schrödinger–Poisson system with steep potential well
- Miao Du
Communications in Contemporary Mathematics14 Sep 2022
Preview Abstract
In this paper, we consider the following Schrödinger–Poisson system
$\{\begin{matrix} - Δ u + λ V (x) u + μ ϕ u = | u |^{p - 2} u & in ℝ^{3}, \\ - Δ ϕ = u^{2} & in ℝ^{3}, \end{matrix}$
where $λ, μ > 0$ are real parameters and $2 < p < 6$ . Suppose that $V (x)$ represents a potential well with the bottom $V^{- 1} (0)$ , the system has been widely studied in the case $4 \leq p < 6$ . In contrast, no existence result of solutions is available for the case $2 < p < 4$ due to the presence of the nonlocal term $ϕ u$ . With the aid of the truncation technique and the parameter-dependent compactness lemma, we first prove the existence of positive solutions for $λ$ large and $μ$ small in the case $2 < p < 4$ . Then we obtain the nonexistence of nontrivial solutions for $λ$ large and $μ$ large in the case $2 < p \leq 3$ . Finally, we explore the decay rate of the positive solutions as $| x | \to \infty$ as well as their asymptotic behavior as $λ \to \infty$ and $μ \to 0$ .
articleOpen Access
Existence and concentration of positive solutions to a fractional system with saturable term
- Menghui Li,
- Jinchun He,
- Haoyuan Xu, and
- Meihua Yang
Stochastics and Dynamics20 Jun 2022
Preview Abstract
In this paper, we consider a fractional Laplacian system with saturable nonlinearity. Under some assumptions on the parameters and potential functions, we obtain the existence and concentration behavior of the positive ground state solution by variational methods.
articleNo Access
ESTIMATES ON THE GREEN FUNCTION AND EXISTENCE OF POSITIVE SOLUTIONS FOR SOME NONLINEAR POLYHARMONIC PROBLEMS OUTSIDE THE UNIT BALL
- IMED BACHAR,
- HABIB MÂAGLI, and
- NOUREDDINE ZEDDINI
Analysis and Applications01 Apr 2008
Preview Abstract
Let be the Green function of (-Δ)^m, m ≥ 1, on the complementary D of the unit closed ball in ℝⁿ, n ≥ 2, with Dirichlet boundary conditions , 0 ≤ j ≤ m - 1. We establish some estimates on including the 3G-Inequality given by (1.3). Next, we introduce a polyharmonic Kato class of functions and we exploit the properties of this class to study the existence of positive solutions of some polyharmonic nonlinear elliptic problems.
articleNo Access
SOLVABILITY OF NEUMANN TYPE NON-HOMOGENEOUS MULTI-POINT BVP FOR SECOND ORDER DIFFERENTIAL EQUATIONS WITH p-LAPLACIAN
- YUJI LIU
Analysis and Applications01 Jul 2009
Preview Abstract
We consider the following multi-point boundary value problems
New sufficient conditions to guarantee the existence of at least one solution of the above mentioned BVP are established. Two examples are presented to illustrate the main result.

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Extremal solutions for nonlocal dispersal equations

Positive solutions for fractional Kirchhoff–Schrödinger–Poisson system with steep potential well

Differential equations with multiple sign changing convolution coefficients

A priori bounds and existence of positive solutions to a $p$ -Kirchhoff equations

SYSTEMS OF NONLINEAR DELAY INTEGRAL EQUATIONS: POSITIVE SOLUTIONS, COMPARISON THEOREMS, AND DATA DEPENDENCE

PRESCRIBING A FOURTH ORDER CONFORMAL INVARIANT ON THE STANDARD SPHERE — PART I: A PERTURBATION RESULT

MULTIPLE POSITIVE SOLUTIONS OF A ONE-DIMENSIONAL PRESCRIBED MEAN CURVATURE PROBLEM

EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR p-LAPLACIAN PROBLEMS WITH A GENERAL INDEFINITE WEIGHT

NONNEGATIVE SOLUTIONS FOR INDEFINITE SUBLINEAR ELLIPTIC PROBLEMS

A variational approach to nonlinear logistic equations

Positive and nodal solutions for an elliptic equation with critical growth

Some global results for a class of homogeneous nonlocal eigenvalue problems

Symmetry-breaking bifurcation for the one-dimensional Hénon equation

Choquard equations with critical nonlinearities

Standing waves with a critical frequency for a quasilinear Schrödinger equation

Positive solutions of superlinear indefinite prescribed mean curvature problems

Positive solutions for the Schrödinger–Poisson system with steep potential well

Existence and concentration of positive solutions to a fractional system with saturable term

ESTIMATES ON THE GREEN FUNCTION AND EXISTENCE OF POSITIVE SOLUTIONS FOR SOME NONLINEAR POLYHARMONIC PROBLEMS OUTSIDE THE UNIT BALL

SOLVABILITY OF NEUMANN TYPE NON-HOMOGENEOUS MULTI-POINT BVP FOR SECOND ORDER DIFFERENTIAL EQUATIONS WITH p-LAPLACIAN