Positive Solutions Research Articles

Existence of positive solutions of symmetric variational eigenvalue problems

Existence of positive solutions of symmetric variational eigenvalue problems

Gradient estimates for Yamabe type equations under different curvature conditions and applications

The aim of this paper is to give various aspects of gradient estimates for positive solutions to the Yamabe type equation ut=Δfu+au+buα. We study the problem in many different settings. As result, we obtain several types of gradient estimates for the solution. When the manifold is complete without boundary, we study gradient estimates of Shi type, and of Hamilton type. Moreover, we also derive gradient estimates if the manifold is evolved under the (k,m)-super Perelman-Ricci flows. Besides, we prove gradient estimates under the integral Ricci curvature bounds. Finally, when the manifold has compact boundary, we investigate gradient estimates with Dirichlet boundary conditions. Our theorems generalize and improve many previous works.

Positive Solutions to a Second-Order Sturm–Liouville Problem with Nonlocal Boundary Conditions

Positive Solutions to a Second-Order Sturm–Liouville Problem with Nonlocal Boundary Conditions

Multiple Positive Symmetric Solutions for the Fourth-Order Iterative Differential Equations Involving $\mathrm{p}$-Laplacian with Integral Boundary Conditions

Multiple Positive Symmetric Solutions for the Fourth-Order Iterative Differential Equations Involving $\mathrm{p}$-Laplacian with Integral Boundary Conditions

Symmetry and monotonicity of positive solutions for a Choquard equation involving the logarithmic Laplacian operator

Symmetry and monotonicity of positive solutions for a Choquard equation involving the logarithmic Laplacian operator

Positive solutions of nonlinear elliptic equations involving unbounded variable exponents and convection term

Positive solutions of nonlinear elliptic equations involving unbounded variable exponents and convection term

Multiple solutions to nonlocal Neumann boundary problem with sign‐changing coefficients

We consider the following Neumann boundary problem with a nonlocal Kirchhoff term where is a bounded smooth domain and and are sign‐changing coefficients which play important impact on the study. By using variational method and dividing corresponding Nehari manifold into two disjoint subsets, we study the existence of two positive solutions. To show the multiplicity of positive solutions, we find a set whose elements can be mapped to two subsets of Nehari manifold, which is also the key to proving strong convergence for minimizing sequences. Finally, the asymptotic behavior of solutions is studied as parameter tends to 0. Different from Dirichlet boundary problem, we show that the two solutions of Neumann problem tend to two different positive constants, respectively.

Stability criterion of a nonautonomous 3-species ratio-dependent diffusive predator-prey model

The global stability of a nonautonomous 3-species ratio-dependent diffusive predator-prey model is studied in this paper. Firstly, some easily verifiable sufficient conditions which guarantee the existence of the strictly positive space homogenous periodic solution (SHPS) of the ratio- dependent predator-prey model (RDPPM) with diffusive and variable coefficient are achieved by using a comparison theorem of differential equation and fixed point theorem. At the same time, some new analysis techniques are developed as a byproduct. Secondly, some sufficient conditions ensuring the globally asymptotically stability of the strictly positive SHPS of the diffusive nonautonomous predator-prey model are given by using the method of upper and lower solutions (UALS) for the parabolic partial differential equations and Lyapunov stability theory. In addition, two numerical examples are given to validate the theoretical results obtained in this paper.

Existence results for a fourth order problem with functional perturbed clamped beam boundary conditions

Abstract In this paper, we study the existence of positive solutions for a fourth order boundary value problem coupled to functional perturbed clamped beam boundary conditions. Our main ingredient is the classical fixed point index. The problem investigated is an extension of other problems studied in previous papers by covering very general nonlocal perturbed conditions on the boundary.

Multiple non-radial solutions for coupled Schrödinger equations

The paper deals with the existence of non-radial solutions for an N-coupled nonlinear elliptic system. In the repulsive regime with some structure conditions on the coupling and for each symmetric subspace of rotation symmetry, we prove the existence of an infinite sequence of non-radial positive solutions and an infinite sequence of non-radial nodal solutions.

Structure of positive solutions for a reaction-diffusion model with additional food and protection zone

The additional food and protection zone provided to predator and prey respectively are introduced into a reaction-diffusion predation model with nonlinear functional response in this paper. We mainly concentrate on the combined effects of additional food and protection zone on the existence and uniqueness of solution for the model. It is shown that, compared with the models with classical Beddington-DeAngelis functional response, the integration of additional food and protection zone into Beddington-DeAngelis functional response causes the model to produce new critical values for the existence of positive solutions. Moreover, we derive that the model permits a unique positive solution when the strength of mutual interference among predators or the quantity of additional food is large, where, the effect of quantity of additional food on the positive solution is a new finding. The results shed light on that the combination of additional food and protection zone is an effective and environmentally-friendly strategy in preserving biodiversity.

An open GNSS spoofing data repository: characterization and impact analysis with FGI-GSRx open-source software-defined receiver

Spoofing is becoming a prevalent threat to the users of Global Navigation Satellite Systems (GNSS). It is important to deepen our understanding of spoofing attacks and develop resilient techniques to effectively combat this threat. Detecting and mitigating these attacks requires thorough testing, typically conducted in a laboratory environment through the establishment of a spoofing test-bed. The complexity, cost and resource demands of creating such a test-bed underscore the necessity of utilizing openly available datasets. To address this need, this paper introduces a new GNSS spoofing data repository from Finnish Geospatial Research Institute (FGI) named hereafter as ‘FGI-SpoofRepo’. This data repository consists of raw In-phase and Quadrature (I/Q) data of live recordings of GPS L1 C/A, Galileo E1, GPS L5, and Galileo E5a signals. These datasets encompass three distinct types of spoofing characteristics (synchronous, asynchronous, and meaconing), making them very useful example candidates of open data for testing the performance of any anti-spoofing techniques (be it detection or mitigation). The inclusion of live signals in multiple GNSS frequencies and the presence of cryptographic signatures in Galileo E1 signal make these datasets potential benchmarks for assessing the resilience performance of multi-frequency multi-constellation receivers. The analysis of the datasets is carried out with an open-source MATLAB-based software-defined receiver, FGI-GSRx. An updated version of FGI-GSRx, equipped with the necessary modifications for processing and analyzing the new datasets, is released alongside the datasets. Therefore, the GNSS research community can utilize the open-source FGI-GSRx or any third-party SDR to process the publicly available raw I/Q data for implementation, testing and validation of any new anti-spoofing technique. The results show that time-synchronous spoofing seamlessly takes over positioning solution, while time-asynchronous spoofing acts as noise or in some cases, completely prevent the receiver from providing a positioning solution. Signal re-acquisition during an ongoing spoofing attack (cold start), the receiver tends to lock onto the spoofing signal with the highest peak, posing a potential threat to GNSS receivers without assisted information. Overall, this research aims to advance the understanding of complex spoofing attacks on GNSS signals, providing insight into enhancing resilience in navigation systems.

Doppler Positioning with LEO Mega-Constellation: Equation Properties and Improved Algorithm

Doppler positioning, as an early form of positioning, has regained significant research interest in the context of Low Earth Orbit (LEO) satellites.Given the LEO mega-constellation scenario, the objective function of Doppler positioning manifests significant nonlinearity, leading to ill-conditioning challenges for prevalent algorithms like iterative least squares (LS) estimation, especially in cases where inappropriate initial values are selected. In this study, we investigate the causes of ill-posed problems from two perspectives. Firstly, we analyze the linearization errors of the Doppler observation equations in relation to satellite orbital altitude and initial value errors, revealing instances where traditional algorithms may fail to converge. Secondly, from an optimization theory perspective, we demonstrate the occurrence of convergence to locally non-unique solutions for Doppler positioning. Subsequently, to address these ill-conditioning issues, we introduce Tikhonov regularization terms in the objective function to constrain algorithm divergence, with a fitted model for the regularization coefficient. Finally, we conduct comprehensive simulation experiments in both dynamic and static scenarios to validate the performance of the proposed algorithm. On the one hand, when the initial values are set to 0, our algorithm achieves high-precision positioning, whereas the iterative LS fails to converge. On the other hand, in certain simulation scenarios, the iterative LS converges to locally non-unique solutions, resulting in positioning errors exceeding 50 km in the north and east directions, several hundred kilometers in the vertical direction, and velocity errors surpassing 120 m/s. In contrast, our algorithm demonstrates typical errors of a position error of 6.8462 m, velocity error of 0.0137 m/s, and clock drift error of 8.3746 × 10−6 s/s.This work provides an effective solution to the sensitivity issue of initial points in Doppler positioning and can serve as a reference for the algorithm design of Doppler positioning receivers with LEO mega-constellations.

Dynamics of a Dengue Transmission Model with Multiple Stages and Fluctuations

A vector–host model of dengue with multiple stages and independent fluctuations is investigated in this paper. Firstly, the existence and uniqueness of the positive solution are shown by contradiction. When the death rates of aquatic mosquitoes, adult mosquitoes, and human beings respectively control the intensities of white noises, and if R0s>1, then the persistence in the mean for both infective mosquitoes and infective human beings is derived. When R0s>1 is valid, the existence of stationary distribution is derived through constructing several appropriate Lyapunov functions. If the intensities of white noises are controlled and φ<0 is valid, then the extinction for both infective mosquitoes and infective human beings is obtained by applying the comparison theorem and ergodic theorem. Further, the main findings are verified through numerical simulations by using the positive preserving truncated Euler–Maruyama method (PPTEM). Moreover, several numerical simulations on the infection scale of dengue in Fuzhou City were conducted using surveillance data. The main results indicate that the decrease in the transfer proportion from aquatic mosquitoes to adult mosquitoes reduces the infection scale of infective human beings with dengue virus, and the death rates of aquatic mosquitoes and adult mosquitoes affect the value of the critical threshold R0s. Further, the controls of the death rates of mosquitoes are the effective routes by the decision-makers of the Chinese mainland against the spread of dengue.

Triple increasing positive solutions to fractional differential equations with p$$ p $$‐Laplacian operator

Triple increasing positive solutions to fractional differential equations with p$$ p $$‐Laplacian operator

Nondegeneracy of the solutions for elliptic problem with critical exponent

This paper deals with the following nonlinear elliptic equation: −Δu=Q(|y′|,y″)uN+2N−2,u>0,inRN,u∈D1,2(RN),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ -\\Delta u=Q(|y'|,y'')u^{\\frac{N+2}{N-2}},\\,\\,u>0,\\,\\,\ ext{in}\\,{ \\mathbb{R}}^{N},\\,\\,u\\in D^{1,2}({\\mathbb{R}}^{N}), $$\\end{document} where (y′,y″)∈R2×RN−2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(y',y'')\\in {\\mathbb{R}}^{2}\ imes {\\mathbb{R}}^{N-2}$\\end{document}, N≥5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N\\geq 5$\\end{document}, Q(|y′|,y″)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$Q(|y'|,y'')$\\end{document} is a bounded nonnegative function in R2×RN−2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{2}\ imes {\\mathbb{R}}^{N-2}$\\end{document}. By using the local Pohozaev identities we prove a nondegeneracy result for the positive solutions constructed in (Peng et al. in J. Differ. Equ. 267:2503–2530, 2019).

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

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.

Existence of positive solutions for a class of p-Laplacian fractional differential equations with nonlocal boundary conditions

This article is devoted to proving the uniqueness of positive solutions for p-Laplacian equations with Caputo and Riemann-Liouville fractional derivative. The uniqueness result and the dependence of the solution on a parameter are established based on the fixed point point theorem of mixed monotone operators. In the end, a numerical simulation is given to verify the main results.

All solutions of the Diophantine equation 1/x+1/y+1/z=u/(u+2)

In the history of mathematics, many mathematical researchers have investigated the Diophantine equation in the form , where and are positive integers. Without loss of generality, we may assume that . This Diophantine equation, also known as the Egyptian fraction equation of length 3, is to write the fraction as a sum of three fractions with the numerator being one and the denominators being different positive integers. Examples of research such as, in 2021, Sandor and Atanassov studied and found that the Diophantine equation has forty-four positive integer solutions. In this paper, we will study and find the complete positive integer solutions of the Diophantine equation , by using elementary methods of number theory and computer calculations. In the process, we can see that . Then, we will consider separately the value of a positive integer in nine cases. The first case is impossible. For the second and third cases, we will separate to consider the value of . For the remaining cases, we will separate to consider the value of . The research results showed that all positive integer solutions of the Diophantine equation are eighty-seven positive integer solutions. Moreover, from the steps to find the above positive integer solutions, we expect that it can be used to find the complete positive integer solutions of the Diophantine equation , where is a positive integer with .

Dynamics of a ratio-dependent three-species food-chain system in a random environment

We consider a system of stochastic differential equations, which is established by introducing white noises into a periodic and ratio-dependent food-chain system of three-species. We investigate the asymptotic behaviors for such a system, and obtain the sufficient conditions for its extinction and persistence with the help of Itô’s formula. We discuss the existence of a nontrivial positive periodic solution based on Has’minskii theory of periodic solution. Further, we study a ratio-dependent food-chain system of three-species including not only white noise but also telephone noise. For such a model, we give the sufficient conditions to guarantee the existence of a unique ergodic stationary distribution.