Brouwer Degree Research Articles

Existence of least energy nodal solutions for a double-phase problem with nonlocal nonlinearity

In this paper, we study the following double-phase problem with nonlocal nonlinearity where is a bounded domain, , 1<p<q<N, , is the Riesz potential of order . By using the constrained variational method and Brouwer degree theory, we prove the existence of least energy nodal solutions.

Globally asymptotic stability and synchronization analysis of uncertain multi‐agent systems with multiple time‐varying delays and impulses

AbstractThis article interrogates the issue of global asymptotic stability and synchronization analysis of uncertain multi‐agent systems (MASs) with multiple time‐varying delays (discrete and distributed) and impulsive perturbations. In this problem, we prove the existence and uniqueness of the equilibrium point using Brouwer degree properties for the addressed MASs. By applying the approach of Lyapunov–Krasovskii function into the considered system, we achieve the global asymptotic stability and the synchronization criteria by designing the pining control strategy into the MAS. Finally, we provide two numerical calculations along with the computational simulations to check the validity of the theoretical findings reported in this article.

Sign-changing solutions for a fractional Schrödinger–Poisson system

In this paper, we deal with the existence and multiplicity of sign-changing solutions for fractional Schrödinger–Poisson system: (℘) where , and f is a continuous function. Based on perturbation approach and the method of invariant sets of descending flow, we obtain the existence and multiplicity of sign-changing solutions of system (). In addition, by applying the constrained variational method incorporated with Brouwer degree theory, we prove that system () possesses at least one ground state sign-changing solution. Furthermore, we show that the least energy of sign-changing solutions exceed twice than the least energy, and when f is odd, system () admits infinitely many nontrivial solutions.

Winding numbers, discriminants and topological phase transitions

The discovery that the band structures of certain insulators are marked by nontrivial topological properties has spawned a wave of publications dealing with such materials. The mathematical framework that underlies many of these publications is often very abstract, and it tends to override the essential physics, thus precluding an intuitive approach to this fascinating topic. Based on known and novel types of model systems, we point out several aspects of topological insulators, which may be approached by elementary and intuitive concepts like the Brouwer degree, winding numbers and the characterization of topological phase transitions using secular equations and discriminant theory.

A Degree Associated to Linear Eigenvalue Problems in Hilbert Spaces and Applications to Nonlinear Spectral Theory

We extend to the infinite dimensional context the link between two completely different topics recently highlighted by the authors: the classical eigenvalue problem for real square matrices and the Brouwer degree for maps between oriented finite dimensional real manifolds. Thanks to this extension, we solve a conjecture regarding global continuation in nonlinear spectral theory that we have formulated in a recent article. Our result (the ex conjecture) is applied to prove a Rabinowitz type global continuation property of the solutions to a perturbed motion equation containing an air resistance frictional force.

Degree, quaternions and periodic solutions.

The paper computes the Brouwer degree of some classes of homogeneous polynomials defined on quaternions and applies the results, together with a continuation theorem of coincidence degree theory, to the existence and multiplicity of periodic solutions of a class of systems of quaternionic valued ordinary differential equations. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.

Higher Order Analysis on the Existence of Periodic Solutions in Continuous Differential Equations via Degree Theory

Recently, the higher order averaging method for studying periodic solutions of both Lipschitz differential equations and discontinuous piecewise smooth differential equations was developed in terms of Brouwer degree theory. Between the Lipschitz and the discontinuous piecewise smooth differential equations, there is a huge class of differential equations lacking in a higher order analysis on the existence of periodic solutions, namely the class of continuous non-Lipschitz differential equations. In this paper, based on the coincidence degree theory for nonlinear operator equations, we perform a higher order analysis of continuous (non-Lipschitz) perturbed differential equations and derive sufficient conditions for the existence of periodic solutions for such systems. We apply our results to study continuous (non-Lipschitz) higher order perturbations of a harmonic oscillator.

Bifurcation of Periodic Solutions and Its Maximum Number in a Circular Mesh Antenna System with 1:2 Internal Resonance

In this paper, we study the bifurcation of periodic solutions for a four-dimensional deployable circular mesh antenna system. The tools for proving these results are the averaging theory and Brouwer degree theory. Based on constructing displacement maps, we study the bifurcation of the periodic solutions of linear center, and to discuss the maximum number of periodic solutions in certain parameter control conditions. The results in this paper are helpful to the study of nonlinear dynamic characteristics and vibration control of deployable circular mesh antenna model.

Spectral flow, Brouwer degree and Hill's determinant formula

In 2005 a new topological invariant defined in terms of the Brouwer degree of a determinant map, was introduced by Musso, Pejsachowicz and the first name author for counting the conjugate points along a semi-Riemannian geodesic. This invariant was defined in terms of a suspension of a complexified family of linear second order Dirichlet boundary value problems.In this paper, starting from this result, we generalize this invariant to a general self-adjoint Morse-Sturm system and we prove a new spectral flow formula. Finally we discuss the relation between this spectral flow formula and the Hill's determinant formula and we apply this invariant for detecting instability of periodic orbits of a Hamiltonian system.

Nodal solutions for the Schrödinger–Poisson equations with convolution terms

This paper deals with the following nonlinear Schrödinger–Poisson system with convolution terms: (SPC)−Δu+V(|x|)u+bϕu=Iα∗|u|p|u|p−2uinR3,−Δϕ=u2inR3,where b>0 is a parameter, V∈C([0,∞),R+),α∈(0,3), Iα:R3→R is the Riesz potential and p∈(3+α3,3+α). The presence of nonlocal terms ϕu and Iα∗|u|p|u|p−2u makes the variational functional of (SPC) totally different from the case of b=0 or the case with pure power nonlinearity. Taking advantage of the results from the matrix theory and the Brouwer degree theory, we introduce some new analytic techniques to prove that for any given integer k≥1, (SPC) admits a sign changing radial solution ukb for p>4, which changes sign exactly k times. Furthermore, for any sequence {bn} with bn→0+ as n→∞, there is a subsequence, still denoted by {bn}, such that ukbn converges to uk0 in H1(R3) as n→∞, where uk0 also changes sign exactly k times and is a sign-changing radial solution of the Choquard equation −Δu+V(|x|)u=Iα∗|u|p|u|p−2uinR3.Our result generalizes the existing ones for the Schrödinger–Poisson equations and Choquard equations, and seems to be the first result of such radial solutions for an equation with two competing convolution terms. Besides, we show that the degeneracy for this existence result happens for p<2.

On algebraic problems behind the Brouwer degree of equivariant maps

Given a finite group G and two unitary G-representations V and W, possible restrictions on topological degrees of equivariant maps between representation spheres S(V) and S(W) are usually expressed in a form of congruences modulo the greatest common divisor of lengths of orbits in S(V) (denoted by α(V)). Effective applications of these congruences is limited by answers to the following questions: (i) under which conditions, is α(V)>1? and (ii) does there exist an equivariant map with the degree easy to calculate? In the present paper, we address both questions. We show that α(V)>1 for each irreducible non-trivial C[G]-module if and only if G is solvable. This provides a new solvability criterion for finite groups. For non-solvable groups, we use 2-transitive actions to construct complex representations with non-trivial α-characteristic. Regarding the second question, we suggest a class of Norton algebras without 2-nilpotents giving rise to equivariant quadratic maps, which admit an explicit formula for the degree.

On Nodal Solutions of the Nonlinear Choquard Equation

Abstract This paper deals with the general Choquard equation - Δ ⁢ u + V ⁢ ( | x | ) ⁢ u = ( I α * | u | p ) ⁢ | u | p - 2 ⁢ u in ⁢ ℝ N , -\Delta u+V(|x|)u=(I_{\alpha}*|u|^{p})|u|^{p-2}u\quad\text{in }\mathbb{R}^{N}, where V ∈ C ⁢ ( [ 0 , ∞ ) , ℝ + ) {V\in C([0,\infty),\mathbb{R}^{+})} is bounded below by a positive constant, and I α {I_{\alpha}} denotes the Riesz potential of order α ∈ ( 0 , N ) {\alpha\in(0,N)} . In view of the convolution term, the nonlocal property makes the variational functional completely different from the one for local pure power-type nonlinearity. By combining the Brouwer degree and developing some new techniques, a family of radial solutions with a prescribed number of zeros is constructed for p ∈ [ 2 , N + α N - 2 ) {p\in[2,\frac{N+\alpha}{N-2})} , while the degeneracy happens for p ∈ ( N + α N , 2 ) {p\in(\frac{N+\alpha}{N},2)} . This result complements and improves the ones in the literature in the aspect of the range of p.

Existence and nonexistence of nodal solutions for Choquard type equations with perturbation

In this paper, we consider the following Choquard type equations with a local perturbation−Δu+u=(∫RN|u(y)|p|x−y|N−αdy)|u|p−2u+|u|q−2uinRN, where N≥3,α∈(0,N),q∈(2,2NN−2). By employing constrained variational method, gluing approach and Brouwer degree theory, we prove that for any given positive integer k, the above equation with p∈(2,N+αN−2) has a least energy radial solution changing sign exactly k times, while when p∈(N+αN,2), the above equation does not admits such a solution.

Least energy sign-changing solution to a fractional $p$-Laplacian problem involving singularities

In this paper we study the existence of a least energy sign-changing solution to a nonlocal elliptic PDE involving singularity by using the Nehari manifold method, the constraint variational method and Brouwer degree theory.

Functions of bounded fractional variation and fractal currents

Extending the notion of bounded variation, a function $$u \in L_c^1(\mathbb {R}^n)$$ is of bounded fractional variation with respect to some exponent $$\alpha $$ if there is a finite constant $$C \ge 0$$ such that the estimate $$\begin{aligned} \biggl |\int u(x) \det D(f,g_1,\ldots ,g_{n-1})_x \, dx\biggr | \le C\text{ Lip }^\alpha (f) \text{ Lip }(g_1) \cdots \text{ Lip }(g_{n-1}) \end{aligned}$$ holds for all Lipschitz functions $$f,g_1,\ldots ,g_{n-1}$$ on $$\mathbb {R}^n$$ . Among such functions are characteristic functions of domains with fractal boundaries and Holder continuous functions. We characterize functions of bounded fractional variation as a certain subspace of Whitney’s flat chains and as multilinear functionals in the setting of Ambrosio–Kirchheim currents. Consequently we discuss extensions to Holder differential forms, higher integrability, an isoperimetric inequality, a Lusin type property and change of variables. As an application we obtain sharp integrability results for Brouwer degree functions with respect to Holder maps defined on domains with fractal boundaries.

Sign-Changing Solutions of Fractional 𝑝-Laplacian Problems

Abstract In this paper, we obtain the existence and multiplicity of sign-changing solutions of the fractional p-Laplacian problems by applying the method of invariant sets of descending flow and minimax theory. In addition, we prove that the problem admits at least one least energy sign-changing solution by combining the Nehari manifold method with the constrained variational method and Brouwer degree theory. Furthermore, the least energy of sign-changing solutions is shown to exceed twice that of the least energy solutions.

Discrete Neumann boundary value problem for a nonlinear equation with singular \u03d5-Laplacian

Let Isubsetmathbb{R} be an open interval with 0in I, and let gin C^{1}(I, (0,+infty)). Let Ninmathbb{N} be an integer with Ngeq4, [2, N-1]_{mathbb{Z}}:={2, 3,ldots,N-1}. We are concerned with the existence of solutions for the discrete Neumann problem \t\t\t{∇(kn−1△vk1−(△vk)2)=nkn−1[−g′(ψ−1(vk))1−(△vk)2+g(ψ−1(vk))H(ψ−1(vk),k)],k∈[2,N−1]Z,Δv1=0=ΔvN−1\\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}$$\\textstyle\\begin{cases} \\nabla(k^{n-1}\\frac{\\triangle v_{k}}{\\sqrt{1-(\\triangle v_{k})^{2}}} )=nk^{n-1}[-\\frac{ g'(\\psi^{-1}(v_{k}))}{\\sqrt{1-(\\triangle v_{k})^{2}}}+g(\\psi^{-1}(v_{k}))H(\\psi^{-1}(v_{k}),k)],\\quad k\\in[2, N-1]_{\\mathbb{Z}},\\\\ \\Delta v_{1}=0=\\Delta v_{N-1} \\end{cases} $$\\end{document} which is a discrete analogue of the Neumann problem about the rotationally symmetric spacelike graphs with a prescribed mean curvature function in some Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes, where psi(s):=int_{0}^{s}frac{dt}{g(t)}, psi ^{-1} is the inverse function of ψ, and H:mathbb{R}times[2, N-1]_{mathbb{Z}}tomathbb{R} is continuous with respect to the first variable. The proofs of the main results are based upon the Brouwer degree theory.

On variational and topological methods in nonlinear difference equations

In this paper, first we survey the recent progress in usage of the critical point theory to study the existence of multiple periodic and subharmonic solutions in second order difference equations and discrete Hamiltonian systems with variational structure. Next, we propose a new topological method, based on the application of the equivariant version of the Brouwer degree to study difference equations without an extra assumption on variational structure. New result on the existence of multiple periodic solutions in difference systems (without assuming that they are their variational) satisfying a Nagumo-type condition is obtained. Finally, we also put forward a new direction for further investigations.

Light-Ring Stability for Ultracompact Objects.

We prove the following theorem: axisymmetric, stationary solutions of the Einstein field equations formed from classical gravitational collapse of matter obeying the null energy condition, that are everywhere smooth and ultracompact (i.e., they have a light ring) must have at least two light rings, and one of them is stable. It has been argued that stable light rings generally lead to nonlinear spacetime instabilities. Our result implies that smooth, physically and dynamically reasonable ultracompact objects are not viable as observational alternatives to black holes whenever these instabilities occur on astrophysically short time scales. The proof of the theorem has two parts: (i)We show that light rings always come in pairs, one being a saddle point and the other a local extremum of an effective potential. This result follows from a topological argument based on the Brouwer degree of a continuous map, with no assumptions on the spacetime dynamics, and, hence, it is applicable to any metric gravity theory where photons follow null geodesics. (ii)Assuming Einstein's equations, we show that the extremum is a local minimum of the potential (i.e.,astable light ring) if the energy-momentum tensor satisfies the null energy condition.

Existence and multiplicity of solutions for discrete Neumann-Steklov problems with singular \u03d5-Laplacian

This paper establishes the existence and multiplicity of solutions for the discrete Neumann-Steklov problem with singular ϕ-Laplacian by using the method of lower and upper solutions, a priori estimates and Brouwer degree theory.