Brouwer Degree Research Articles

Multiple high energy solutions of a nonlinear Hardy–Sobolev critical elliptic equation arising in astrophysics

In this article, we study the existence and multiplicity of high energy solutions to the problem proposed as a model for the dynamics of galaxies: −Δu+V(x)u=|u|2∗−2u|y|,x=(y,z)∈Rm×Rn−m,where n>4, 2≤m<n, 2∗≔2(n−1)n−2 and potential function V(x):Rn→R. Benefiting from a global compactness result, we show that there exist at least two positive high energy solutions. Our proofs are based on barycenter function, quantitative deformation lemma and Brouwer degree theory.

Multiplicity and limit of solutions for logarithmic Schrödinger equations on graphs

Let Ω be a finite connected subset of a locally finite graph G = (V, E) with the vertex set V and the edge set E. We investigate the logarithmic Schrödinger equation on Ω with the nonlinear term |u|p−2u log u2. For p &amp;gt; 2, through two different approaches which are the Brouwer degree theory and mountain-pass theorem, we obtain the existence of ground state solutions. We also apply the Brouwer degree theory together with the constraint variational method to prove that the equation admits a sign-changing solution which implies the multiplicity of solutions to the equation. Finally, we illustrate that as p → 2, up to a subsequence, the solutions for p &amp;gt; 2 shall converge to a non-trivial solution of the equation with p = 2.

Multiple high‐energy solutions for an elliptic system with critical Hardy–Sobolev nonlinearity

This paper discusses the existence of multiple high‐energy solutions for a ‐Laplacian system involving critical Hardy‐Sobolev nonlinearity in . Considering that the “double” lack of compactness in the system is caused by the unboundedness of and the presence of the critical Hardy–Sobolev exponent, we demonstrate the version to of Struwe's classical global compactness result for double ‐Laplace operator. In virtue of the quantitative deformation lemma, a barycenter function, and the Brouwer degree theory, the existence of multiple high‐energy solutions is established. The results of this paper extend and complement the recent work.

An infinite dimensional version of the Kronecker index and its relation with the Leray–Schauder degree

Let \mathfrak{f} be a compact vector field of class C^{1} on a real Hilbert space \mathbb{H} . Denote by \mathbb{B} the open unit ball of \mathbb{H} and by \mathbb{S}= \partial\mathbb{B} the unit sphere. Given a point q \notin \mathfrak{f}(\mathbb{S}) , consider the self-map of \mathbb{S} defined by \mathfrak{f}_{q}^{\partial}(p)= \frac{\mathfrak{f}(p)-q}{\|\mathfrak{f}(p)-q\|}, \quad p \in \mathbb{S}. If \mathbb{H} is finite dimensional, then \mathbb{S} is an orientable, connected, compact differentiable manifold. Therefore, the Brouwer degree, \deg_{\mathrm{Br}}(\mathfrak{f}_{q}^{\partial}) is well defined, no matter what orientation of \mathbb{S} is chosen, assuming it is the same for \mathbb{S} as domain and codomain of \mathfrak{f}_{q}^{\partial} . This degree may be considered as a modern reformulation of the Kronecker index of the map \mathfrak{f}_{q}^{\partial} . Let \deg_{\mathrm{Br}}(\mathfrak{f},\mathbb{B},q) denote the Brouwer degree of \mathfrak{f} on \mathbb{B} with target q . It is known that one has the equality \deg_{\mathrm{Br}}(\mathfrak{f},\mathbb{B},q) = \deg_{\mathrm{Br}}(\mathfrak{f}_{q}^{\partial}). Our purpose is an extension of this formula to the infinite dimensional context. Namely, we will prove that \deg_{\mathrm{LS}}(\mathfrak{f},\mathbb{B},q) = \deg_{\mathrm{bf}}(\mathfrak{f}_{q}^{\partial}), where \deg_{\mathrm{LS}}(\cdot) denotes the Leray–Schauder degree and \deg_{\mathrm{bf}}(\cdot) is the degree earlier introduced by M. Furi and the first author, which extends, to the infinite dimensional case, the Brouwer degree and the Kronecker index. In other words, here, we extend to the Leray–Schauder degree the boundary dependence property which holds for the Brouwer degree in the finite dimensional context.

Dirichlet systems with discrete relativistic operator

AbstractWe are concerned with Dirichlet systems involving the relativistic discrete operator Here, for , we denote by the forward difference operator. Besides a “universal” existence result for a system with a general nonlinearity, we obtain multiplicity of solutions for systems with parameterized nonlinearities. Our approaches mainly rely on Brouwer degree arguments and critical point theory for convex, lower semicontinuous perturbations of ‐functionals.

Shape index, Brouwer degree and Poincaré–Hopf theorem

In this paper, we study the relationship of the Brouwer degree of a vector field with the dynamics of the induced flow. Analogous relations are studied for the index of a vector field. We obtain new forms of the Poincar é–Hopf theorem and of the Borsuk and Hirsch antipodal theorems. As an application, we calculate the Brouwer degree of the vector field of the Lorenz equations in isolating blocks of the Lorenz strange set.

Three-dimensional critical points and flow patterns in pulmonary alveoli with rhythmic wall motion

The dynamics of airflow in the pulmonary acini are of broad interest in understanding respiratory diseases and the fate of inhaled particles. This study investigates the three-dimensional (3d) alveolar flows with rhythmic cavity wall motion, using a finite element method based computational fluid dynamics. This study reports the new research findings on the critical points and associated flow patterns. The locations of critical points are found based on the Brouwer degree theory and Broyden’s method. The phase portrait is used to evaluate the flow patterns around the critical points and the stability (repelling/attracting property) of the critical points on the symmetry plane of the alveolus. Based on the Poincare–Bendixson theorem, the closed orbits on the symmetry plane are found which have the capability to alter the spiral direction of the spiral streamlines. In the 3d space, the alveolar flow is symmetric about the geometric symmetry plane of the alveolus. Different types of 3d critical points, including saddle, spiral, and spiral saddle, are revealed. There are only one saddle point and at least one spiral point or spiral saddle in the alveolar flow. Spiral points and spiral saddles are located on the vortex core line and their number is dependent on the Reynolds number and varies with time. The study of critical points and their evolution helps us to understand the mechanism of irreversible transport of particle tracers from a new perspective.

Perturbation of the spectra for asymptotically constant differential operators and applications

We study the spectra for a class of differential operators with asymptotically constant coefficients. These operators widely arise as the linearizations of nonlinear partial differential equations about patterns or nonlinear waves. We present a unified framework to prove the perturbation results on the related spectra. The proof is based on exponential dichotomies and the Brouwer degree theory. As applications, we employ the developed theory to study the stability of quasi-periodic solutions of the Ginzburg–Landau equation, fold-Hopf bifurcating periodic solutions of reaction–diffusion systems coupled with ordinary differential equations, and periodic annulus of the hyperbolic Burgers–Fisher model.

Topology of black hole thermodynamics in Lovelock gravity

In this work, we present a convenient method to perform the topological analysis of black hole thermodynamics. Utilizing the spinodal curve, thermodynamic critical points of a black hole are endowed with a topological quantity, Brouwer degree, which reflects intrinsic properties of the system under smooth deformations. Specially, in our setup, it can be easily calculated without exact solution of critical points. This enables us to conveniently investigate the topological transition between different thermodynamic systems, and give a topological classification for them. In this framework, topology of Lovelock AdS black holes with spherical horizon geometry is explored. Results show that charged black holes in arbitrary dimensions can be classified into the same topology class, whereas the $d=7$ and $d \geq 8$ uncharged black holes are in different topology classes. Moreover, we revisit the relation between different phase structures of these black holes from the viewpoint of topology. Some general topological properties of critical points are also discussed.

Multiple positive bound state solutions for fractional Schrödinger-Poisson system with critical growth

In this paper, we deal with the following critical fractional Schrödinger-Poisson system without subcritical perturbation $$ \begin{cases} (-\Delta)^{s} u + V(x)u + K(x)\phi u=|u|^{2_{s}^{*}-2}u, & x\in\mathbb{R}^{3},\\ (-\Delta)^{t}\phi=K(x)u^{2}, & x\in\mathbb{R}^{3}, \end{cases} $$ where $s\in(\frac{3}{4},1),$ $t\in(0,1)$, $2^{\ast}_{s}=\frac{6}{3-2s}$ is the critical Sobolev exponent. When $V(x)$ is positive bounded from below, combining with variation methods and Brouwer degree theory, we investigate the existence and multiplicity of positive bound solutions to this system. The results obtained in this paper extend and improve some recent works in which potential function $V(x)$ may vanish at the infinity.

Bound states of fractional Choquard equations with Hardy-Littlewood-Sobolev critical exponent

We deal with the following fractional Choquard equation(−Δ)su+V(x)u=(Iμ⁎|u|2μ,s⁎)|u|2μ,s⁎−2u,x∈RN, where Iμ(x) is the Riesz potential, s∈(0,1), 2s<N≠4s, 0<μ<min⁡{N,4s} and 2μ,s⁎=2N−μN−2s is the fractional critical Hardy-Littlewood-Sobolev exponent. By combining variational methods and the Brouwer degree theory, we investigate the existence and multiplicity of positive bound solutions to this equation when V(x) is a positive potential bounded from below. The results obtained in this paper extend and improve some recent works in the case where the coefficient V(x) vanishes at infinity.

Global stability of a tridiagonal competition model with seasonal succession.

In this paper, we investigate a tridiagonal three-species competition model with seasonal succession. The Floquet multipliers of all nonnegative periodic solutions of such a time-periodic system are estimated via the stability analysis of equilibria. Together with the Brouwer degree theory, sufficient conditions for existence and uniqueness of the positive periodic solution are given. We further obtain the global dynamics of coexistence and extinction for three competing species in this periodically forced environment. Finally, some numerical examples are presented to illustrate the effectiveness of our theoretical results.

The nodal solution for a problem involving the logarithmic and exponential nonlinearities

ABSTRACT In this work, we study the existence of the least energy sign-changing solution for the following degenerate Kirchhoff-type problem involving the fractional N/s-Laplacian with logarithmic and both subcritical and critical exponential nonlinearities: { M ( ∫ R 2 N | u ( x ) − u ( y ) | N s | x − y | 2 N d x d y ) ( − Δ ) N / s s u = | u | q − 2 uln ⁡ | u | 2 + μf ( u ) , in Ω , u = 0 , in R N ∖ Ω , where Ω ⊂ R N is a bounded domain. The proof is based on constrained variational method, fractional Trudinger–Moser inequality, quantitative deformation lemma and Brouwer's degree theory in Nehari sets. To be more precise, the least energy sign-changing solution is obtained by minimizing the energy functional on the sign-changing Nehari manifold.

Existence of least energy sign-changing solution for a class of fractional p&q-Laplacian problems with potentials vanishing at infinity

In this paper, we consider the fractional p &q -Laplacian equation: ( − Δ ) p s u + ( − Δ ) q s u + V ( x ) ( | u | p − 2 u + | u | q − 2 u ) = K ( x ) f ( u ) in R N , where s ∈ ( 0 , 1 ) , 1 < p < q < N s , and ( − Δ ) t s with t ∈ { p , q } is the fractional t-Laplacian operator, f is a C 1 real function and V, K are continuous, positive functions. By using constrained variational methods, a quantitative Deformation Lemma and Brouwer degree theory, we prove the existence of a least energy sign-changing solution.

The full power of the half-power

We use the complex square root to define a very simple homotopic invariant over the non-vanishing functions defined on the circle. As a consequence we provide easy proofs of the plane Brouwer fixed point theorem and the Fundamental Theorem of Algebra. The relation of this new invariant with the winding number and the Brouwer degree will be fully unveiled.

Some classical analysis results for continuous definable mappings

In this paper, we show that some fundamental results for smooth mappings (e.g., the Brouwer degree formula, the implicit function and inverse function theorems, the mean value theorem, Sard's theorem, Hadamard's global invertibility criteria, Pourciau's surjectivity and openness results) have natural extensions for continuous mappings that are definable in o-minimal structures. The arguments rely on nice properties of definable mappings and sets.

Brouwer degree for Kazdan-Warner equations on a connected finite graph

We study Kazdan-Warner equations on a connected finite graph via the method of the degree theory. Firstly, we prove that all solutions to the Kazdan-Warner equation with nonzero prescribed function are uniformly bounded and the Brouwer degree is well defined. Secondly, we compute the Brouwer degree case by case. As consequences, we give new proofs of some known existence results for the Kazdan-Warner equation on a connected finite graph.

Neumann problems of superlinear elliptic systems at resonance

We prove existence of weak solutions of Neumann problem of nonhomogeneous elliptic system with asymmetric nonlinearities that may resonant at − ∞ and superlinear at + ∞ . The proof is based on Mawhin's coincidence theory and the product formula of Brouwer degree.

Reassessment of contact restrictions and testing campaigns against COVID-19 via spatio-temporal modeling

Since the earliest outbreak of COVID-19, the disease continues to obstruct life normalcy in many parts of the world. The present work proposes a mathematical framework to improve non-pharmaceutical interventions during the new normal before vaccination settles herd immunity. The considered approach is built from the viewpoint of decision makers in developing countries where resources to tackle the disease from both a medical and an economic perspective are scarce. Spatial auto-correlation analysis via global Moran’s index and Moran’s scatter is presented to help modulate decisions on hierarchical-based priority for healthcare capacity and interventions (including possible vaccination), finding a route for the corresponding deployment as well as landmarks for appropriate border controls. These clustering tools are applied to sample data from Sri Lanka to classify the 26 Regional Director of Health Services (RDHS) divisions into four clusters by introducing convenient classification criteria. A metapopulation model is then used to evaluate the intra- and inter-cluster contact restrictions as well as testing campaigns under the absence of confounding factors. Furthermore, we investigate the role of the basic reproduction number to determine the long-term trend of the regressing solution around disease-free and endemic equilibria. This includes an analytical bifurcation study around the basic reproduction number using Brouwer Degree Theory and asymptotic expansions as well as related numerical investigations based on path-following techniques. We also introduce the notion of average policy effect to assess the effectivity of contact restrictions and testing campaigns based on the proposed model’s transient behavior within a fixed time window of interest.

The Brouwer degree associated to classical eigenvalue problems and applications to nonlinear spectral theory

The Brouwer degree associated to classical eigenvalue problems and applications to nonlinear spectral theory