Nonzero Solutions Research Articles

Periodic motion for FPU lattice dynamical systems with the strongly indefinite case

Sun and Ma (2013, 2019) proved the existence of nonzero T-periodic solution for the Fermi–Pasta–Ulam (FPU) lattice dynamical system. In this paper, we prove the existence of ground state solutions, i.e., the least energy of nonzero T-periodic solution for FPU lattice dynamical systems via Non-Nehari method developed recently by Tang et al. (2020) without strict monotonous condition.

A Peer Search on Integer Solutions to Quadratic Diophantine Equation with Three Unknowns (Equation)

The non- homogeneous ternary quadratic diophantine (Equation) is analyzed for its patterns of non-zero distinct integral solutions. Various interesting relations between the solutions and special numbers namely polygonal, Pronic and Gnomonic numbers are exhibited.

Ground states for infinite lattices with nearest neighbor interaction

Sun and Ma (J. Differ. Equ. 255:2534–2563, 2013) proved the existence of a nonzero T-periodic solution for a class of one-dimensional lattice dynamical systems, qi¨=Φi−1′(qi−1−qi)−Φi′(qi−qi+1),i∈Z,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\ddot{q_{i}}=\\varPhi _{i-1}'(q_{i-1}-q_{i})- \\varPhi _{i}'(q_{i}-q_{i+1}),\\quad i\\in \\mathbb{Z}, \\end{aligned}$$ \\end{document} where q_{i} denotes the co-ordinate of the ith particle and varPhi _{i} denotes the potential of the interaction between the ith and the (i+1)th particle. We extend their results to the case of the least energy of nonzero T-periodic solution under general conditions. Of particular interest is a new and quite general approach. To the best of our knowledge, there is no result for the ground states for one-dimensional lattice dynamical systems.

An elementary proof of Fermat’s last theorem for all even exponents

AbstractAn elementary proof that the equation x2n + y2n = z2n can not have any non-zero positive integer solutions when n is an integer ≥ 2 is presented. To prove that the equation has no integer solutions it is first hypothesized that the equation has integer solutions. The absence of any integer solutions of the equation is justified by contradicting the hypothesis.

On the Positive Pell Equation

The binary quadratic equation () representing the hyperbola is studied for its non-zero distinct integer solutions. A few interesting properties among the solutions are presented. Employing the integer solutions of the equation under consideration, integer solutions for special straight lines, hyperbolas and parabolas are exhibited.

Unusual bifurcation of a Neumann boundary value problem

Boundary layer behaviour of a family of second order nonlinear differential equations with Neumann boundary condition arising from an order-reduction of a pseudo-differential equation in fluid dynamics is analysed. The problem is considered with two perturbation parameters μ,δ. Using asymptotic and numerical methods it is shown that by perturbing δ the problem in the limit μ→0 changes from a homogeneous problem with symmetric solutions (including outer solutions) of a simplified equation, to a non-homogeneous problem which, in general, does not have a non-zero outer limit solution (with boundary layers). By studying the bifurcation diagrams as μ and δ vary it is shown that the main branch of solutions will ‘tear’ for μ=n−2 where n is an even integer. Blow-up regions, isolated islands of solution, and for μ≪1 “exponentially small” regions, all occur: a structure that is difficult for conventional path following algorithms to find.

Nonzero Periodic Solutions of a Special System of Nonlinear Differential Equations

We prove a theorem on the existence of a nonzero periodic solution to a system of differential equations using the fixed-point method for a nonlinear operator defined on the product of two compact sets.

On an abstract bifurcation result concerning homogeneous potential operators with applications to PDEs

We study an abstract equation in a reflexive Banach space, depending on a real parameter λ. The equation is composed by homogeneous potential operators. By analyzing the Nehari sets, we prove a bifurcation result. In some particular cases we describe the full bifurcation diagram, and in general, we estimate the parameter λb for which the problem does not have non-zero solution where λ>λb. We give many applications to partial differential equations: Kirchhoff type equations, Schrödinger equations coupled with the electromagnetic field, Chern-Simons-Schrödinger systems and a nonlinear eigenvalue problem.

On a class of elliptic boundary-value problems with parameter and discontinuous non-linearity

We study an elliptic boundary-value problem in a bounded domain with inhomogeneous Dirichlet condition, discontinuous non-linearity and a positive parameter occurring as a factor in the non-linearity. The non-linearity is in the right-hand side of the equation. It is non-positive (resp. equal to zero) for negative (resp, non-negative) values of the phase variable. Let be a solution of the boundary-value problem with zero right-hand side (the boundary function is assumed to be positive). Putting , we reduce the original problem to a problem with homogeneous boundary condition. The spectrum of the transformed problem consists of the values of the parameter for which this problem has a non-zero solution (the function is a solution for all values of the parameter). Under certain additional restrictions we construct an iterative process converging to a minimal semiregular solution of the transformed problem for an appropriately chosen starting point. We prove that any non-empty spectrum of the boundary-value problem is a ray , where . As an application, we consider the Gol’dshtik mathematical model for separated flows of an incompressible fluid. We show that it satisfies the hypotheses of our theorem and has a non-empty spectrum.

Quasilinear equations involving indefinite nonlinearities and exponential critical growth in $${\mathbb {R}}^N$$

In this work, we establish the existence of nonzero solutions for a class of quasilinear elliptic equations involving indefinite nonlinearities with exponential critical growth of Trudinger–Moser type. Our proofs rely on variational arguments in a Orlicz–Sobolev space with a version of the Trudinger–Moser inequality.

Exterior Energy Bounds for the Critical Wave Equation Close to the Ground State

By definition, the exterior asymptotic energy of a solution to a wave equation on $\mathbb{R}^{1+N}$ is the sum of the limits as $t\to \pm\infty$ of the energy in the the exterior $\{|x|>|t|\}$ of the wave cone. In our previous work (JEMS 2012, arXiv:1003.0625), we have proved that the exterior asymptotic energy of a solution of the linear wave equation in odd space dimension $N$ is bounded from below by the conserved energy of the solution. In this article, we study the analogous problem for the linear wave equation with a potential \begin{equation} \label{abstractLW} \tag{*} \partial_t^2u+L_Wu=0,\quad L_W:=-\Delta -\frac{N+2}{N-2}W^{\frac{4}{N-2}} \end{equation} obtained by linearizing the energy critical wave equation at the ground-state solution $W$, still in odd space dimension. This equation admits nonzero solutions of the form $A+tB$, where $L_WA=L_WB=0$ with vanishing asymptotic exterior energy. We prove that the exterior energy of a solution of \eqref{abstractLW} is bounded from below by the energy of the projection of the initial data on the orthogonal complement of the space of initial data corresponding to these solutions. This will be used in a subsequent paper to prove soliton resolution for the energy-critical wave equation with radial data in all odd space dimensions. We also prove analogous results for the linearization of the energy-critical wave equation around a Lorentz transform of $W$, and give applications to the dynamics of the nonlinear equation close to the ground state in space dimensions $3$ and $5$.

Canonical Lorentzian spin structure and twistor spinors on the Fefferman space of a contact Riemannian manifold

The Fefferman space of a contact Riemannian manifold carries a Lorentzian spin structure canonically. On the Lorentzian spin manifold, we investigate the Dirac operator and the twistor operator closely. In particular, we show that, if the contact Riemannian manifold is integrable, then there exist non-zero global solutions of the twistor equation.

Isometry-invariant solutions to a critical problem on non-compact Riemannian manifolds

We analyse an elliptic equation with critical growth set on a d-dimensional (d≥3) Hadamard manifold (M,g). By adopting a variational perspective, we prove the existence of non-zero non-negative solutions invariant under the action of a specific family of isometries. Our result remains valid when the original nonlinearity is singularly perturbed. Preserving the same variational approach, but considering other groups of isometries, we finally show that when M=Rd, d>3, and the nonlinearity is odd, there exist at least (−1)d+[d−32] pairs of sign-changing solutions.

Free vibration analysis of the macro-micro-nano plates and shells made of a material with functionally graded porosity: A closed-form solution

This article deals with introducing a closed-form solution for considering the vibration of macro-micro-nano plates and shells made of unit material with functionally graded porosity (FGP). Classic continuum theory beside modified nonlocal strain gradient theory are used at macro and micro/nano scales, respectively. The governing equations are obtained in the frame work of third order plate and shell theories. In the solution procedure, a differential operator is defined and by using it, five equations of motion are converted to the five algebraic equations. Then, by satisfying the equations of boundary conditions and achieving the nonzero solution, the frequencies are extracted.

A non-homogeneous elliptic problem in low dimensions with three symmetric solutions

We investigate the existence of multiple solutions for a low-dimensional Neumann problem with non-standard growth set on a strip-like domain of Rn. Variational techniques, depending in particular on a minimax theorem and a version of the principle of symmetric criticality, are employed to establish the existence of three non-zero solutions endowed with axial symmetry. Some consequences are also derived.

Bounded solutions to an energy subcritical non-linear wave equation on [formula omitted

In this work we consider an energy subcritical semi-linear wave equation (3<p<5){∂t2u−Δu=ϕ(x)|u|p−1u,(x,t)∈R3×R;u|t=0=u0∈H˙sp(R3);∂tu|t=0=u1∈H˙sp−1(R3); where sp=3/2−2/(p−1) and the function ϕ:R3→[−1,1] is a radial continuous function with a limit at infinity. We prove that unless the elliptic equation −ΔW=ϕ(x)|W|p−1W has a nonzero radial solution W∈C2(R3)∩H˙sp(R3), any radial solution u with a finite uniform upper bound on the critical Sobolev norm ‖(u(⋅,t),∂tu(⋅,t))‖H˙sp×H˙sp(R3) for all t in the maximal lifespan must be a global solution in time and scatter.

MULTIPERIODIC SOLUTIONS OF SECOND-ORDER SYSTEMS WITH DIFFERENTIATION OPERATOR ON THE LYAPUNOV'S VECTOR FIELD

The problem of the existence and integral representation of a unique multiperiodic solution of a second-order linear inhomogeneous system with constant coefficients and a differentiation operator on the direction of the main diagonal of the space of time variables and of the vector fields in the form of Lyapunov systems with respect to space variables were considered. The multiperiodicity of zeros of this operator and the condition for the absence of a nonzero multiperiodic and real-analytic solution of the homogeneous system corresponding to the given system are established. An integral representation of solutions of an inhomogeneous linear autonomous system that multiperiodic in time variables and realanalytic in space variables is obtained. The existence theorem of a unique multiperiodic in time variables and real-analytic in space variables solutions of the original linear system in terms of the Green's function under sufficiently general conditions is substantiated.

A locally and cubically convergent algorithm for computing 𝒵‐eigenpairs of symmetric tensors

SummaryThis paper is concerned with computing ‐eigenpairs of symmetric tensors. We first show that computing ‐eigenpairs of a symmetric tensor is equivalent to finding the nonzero solutions of a nonlinear system of equations, and then propose a modified normalized Newton method (MNNM) for it. Our proposed MNNM method is proved to be locally and cubically convergent under some suitable conditions, which greatly improves the Newton correction method and the orthogonal Newton correction method recently provided by Jaffe, Weiss and Nadler since these two methods only enjoy a quadratic rate of convergence. As an application, the unitary symmetric eigenpairs of a complex‐valued symmetric tensor arising from the computation of quantum entanglement in quantum physics are calculated by the MNNM method. Some numerical results are presented to illustrate the efficiency and effectiveness of our method.

Properties of Structured Tensors and Complementarity Problems

In this paper, we present some new results on a class of tensors, which are defined by the solvability of the corresponding tensor complementarity problem. For such structured tensors, we give a sufficient condition to guarantee the nonzero solution of the corresponding tensor complementarity problem with a vector containing at least two nonzero components and discuss their relationships with some other structured tensors. Furthermore, with respect to the tensor complementarity problem with a nonnegative such structured tensor, we obtain the upper and lower bounds of its solution set, and by the way, we show that the eigenvalues of such a tensor are closely related to this solution set.

High order symmetric direct discontinuous Galerkin method for elliptic interface problems with fitted mesh

In this article, we aim to develop a high order symmetric direct discontinuous Galerkin method solving elliptic interface problems with zero or non-zero solution jump and flux jump interface conditions. We focus on the case, in which the mesh is partitioned along the curved interface. The two interface jump conditions are naturally and simultaneously built into the numerical flux definition on the curved triangular elements' edges that overlap with the interface. We obtain a stable and high order method, regardless of the combination of the two interface jump conditions. The two interface jump conditions are both weakly enforced in the scheme formulation. Optimal (k+1)th order L2 norm error estimate is proved for polygonal interfaces. A sequence of numerical examples is carried out to verify the optimal convergence of the symmetric direct discontinuous Galerkin method with high order P2, P3 and P4 approximations. Uniform optimal convergence order, which is independent of the diffusion coefficient ratio inside and outside of the interface, is obtained. The symmetric direct discontinuous Galerkin method is shown to be capable of handling interface problems with complicated geometries.