Existence Results for Solutions to Nonlinear Dirac Systems on Compact Spin Manifolds

2018 ◽  
Vol 18 (1) ◽  
pp. 87-104
Author(s):  
Xu Yang
Keyword(s):  
Dirac Operator ◽  
Growth Rates ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Spin Manifold ◽  
Dirac System ◽  
Nontrivial Solutions ◽  
Subcritical Growth ◽  
Dirac Systems ◽  
Spin Manifolds

AbstractIn this article, we study the existence of solutions for the Dirac system\left\{\begin{aligned} \displaystyle Du&\displaystyle=\frac{\partial H}{% \partial v}(x,u,v)\quad\text{on }M,\\ \displaystyle Dv&\displaystyle=\frac{\partial H}{\partial u}(x,u,v)\quad\text{% on }M,\end{aligned}\right.whereMis anm-dimensional compact Riemannian spin manifold,{u,v\in C^{\infty}(M,\Sigma M)}are spinors,Dis the Dirac operator onM, and the fiber preserving map{H:\Sigma M\oplus\Sigma M\rightarrow\mathbb{R}}is a real-valued superquadratic function of class{C^{1}}with subcritical growth rates. Two existence results of nontrivial solutions are obtained via Galerkin-type approximations and linking arguments.

Existence theorems for fractional p-Laplacian problems

2016 ◽  
Vol 15 (05) ◽  
pp. 607-640 ◽  
Author(s):  
Paolo Piersanti ◽  
Patrizia Pucci
Keyword(s):  
Weak Solutions ◽  
Existence Of Solutions ◽  
Existence Theorems ◽  
Positive Function ◽  
Existence Results ◽  
Laplacian Operator ◽  
Bounded Domains ◽  
Lipschitz Boundary ◽  
Nontrivial Solutions ◽  
Existence Of Weak Solutions

The paper focuses on the existence of nontrivial solutions of a nonlinear eigenvalue perturbed problem depending on a real parameter [Formula: see text] under homogeneous boundary conditions in bounded domains with Lipschitz boundary. The problem involves a weighted fractional [Formula: see text]-Laplacian operator. Denoting by [Formula: see text] a sequence of eigenvalues obtained via mini–max methods and linking structures we prove the existence of (weak) solutions both when there exists [Formula: see text] such that [Formula: see text] and when [Formula: see text]. The paper is divided into two parts: in the first part existence results are determined when the perturbation is the derivative of a globally positive function whereas, in the second part, the case when the perturbation is the derivative of a function that could be either locally positive or locally negative at [Formula: see text] is taken into account. In the latter case, it is necessary to extend the main results reported in [A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional [Formula: see text]-Laplacian problems via Morse theory, Adv. Calc. Var. 9(2) (2016) 101–125]. In both cases, the existence of solutions is achieved via linking methods.

Existence of solutions to a class of semilinear elliptic equations involving general subcritical growth

2014 ◽  
Vol 144 (4) ◽  
pp. 809-818 ◽  
Author(s):  
Yong-Yi Lan ◽  
Chun-Lei Tang
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Semilinear Elliptic Equation ◽  
Existence Results ◽  
Subcritical Growth ◽  
Analysis Techniques ◽  
Semilinear Elliptic

In this paper, we consider the semilinear elliptic equation −Δu = λf(x,u) with the Dirichlet boundary value, and under suitable assumptions on the nonlinear term f with a more general growth condition. Some existence results of solutions are given for all λ > 0 via the variational method and some analysis techniques.

scholarly journals Variational Approach for the Variable-Order Fractional Magnetic Schrödinger Equation with Variable Growth and Steep Potential in ℝ N ∗

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Jianwen Zhou ◽  
Bianxiang Zhou ◽  
Liping Tian ◽  
Yanning Wang
Keyword(s):  
Schrödinger Equation ◽  
Variational Approach ◽  
Schrodinger Equation ◽  
Existence Of Solutions ◽  
Nehari Manifold ◽  
Existence Results ◽  
Variable Order ◽  
Variable Exponents ◽  
Nontrivial Solutions ◽  
The Nehari Manifold

In this paper, we show the existence of solutions for an indefinite fractional Schrödinger equation driven by the variable-order fractional magnetic Laplace operator involving variable exponents and steep potential. By using the decomposition of the Nehari manifold and variational method, we obtain the existence results of nontrivial solutions to the equation under suitable conditions.

scholarly journals Nontrivial solutions of inverse discrete problems with sign-changing nonlinearities

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Alberto Cabada ◽  
Nikolay D. Dimitrov
Keyword(s):  
Spectral Theory ◽  
Green's Function ◽  
Difference Equations ◽  
Existence Of Solutions ◽  
Degree Theory ◽  
Existence Results ◽  
Discrete Problem ◽  
Nontrivial Solutions ◽  
Order Difference ◽  
Discrete Problems

Abstract This paper is concerned with the existence of solutions of an inverse discrete problem with sign-changing nonlinearity. This kind of problems includes, as a particular case, nth order difference equations coupled with suitable conditions on the boundary of the interval of definition. It would be valid for the case in which the related Green’s function is positive on a subset of its rectangle of definition. The existence results follow from spectral theory, as an application of the Krein–Rutman theorem and by means of degree theory.

scholarly journals Existence of Solutions for Choquard Type Elliptic Problems with Doubly Critical Nonlinearities

2019 ◽  
Vol 21 (1) ◽  
pp. 77-93
Author(s):  
Yansheng Shen
Keyword(s):  
Variational Methods ◽  
Minimization Problem ◽  
Existence Of Solutions ◽  
Elliptic Problems ◽  
Type Equation ◽  
Nontrivial Solutions ◽  
Critical Nonlinearities

Abstract In this article, we first study the existence of nontrivial solutions to the nonlocal elliptic problems in ℝ N {\mathbb{R}^{N}} involving fractional Laplacians and the Hardy–Sobolev–Maz’ya potential. Using variational methods, we investigate the attainability of the corresponding minimization problem, and then obtain the existence of solutions. We also consider another Choquard type equation involving the p-Laplacian and critical nonlinearities in ℝ N {\mathbb{R}^{N}} .

scholarly journals Existence Results for the Conformal Dirac–Einstein System

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Chiara Guidi ◽  
Ali Maalaoui ◽  
Vittorio Martino
Keyword(s):  
Perturbation Methods ◽  
Existence Of Solutions ◽  
Coupled System ◽  
Existence Results ◽  
First Variation ◽  
The First Variation

AbstractWe consider the coupled system given by the first variation of the conformal Dirac–Einstein functional. We will show existence of solutions by means of perturbation methods.

scholarly journals GLOBAL EXISTENCE RESULTS FOR FUNCTIONAL DIFFERENTIAL INCLUSIONS WITH STATE-DEPENDENT DELAY

2014 ◽  
Vol 19 (4) ◽  
pp. 524-536 ◽  
Author(s):  
Mouffak Benchohra ◽  
Johnny Henderson ◽  
Imene Medjadj
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Global Existence ◽  
Differential Inclusions ◽  
Existence Of Solutions ◽  
Existence Results ◽  
State Dependent ◽  
State Dependent Delay ◽  
Functional Differential ◽  
Functional Differential Inclusions

Our aim in this work is to study the existence of solutions of a functional differential inclusion with state-dependent delay. We use the Bohnenblust–Karlin fixed point theorem for the existence of solutions.

scholarly journals Existence results for second order functional differential and integrodifferential inclusions in Banach spaces

2001 ◽  
Vol 32 (4) ◽  
pp. 315-325
Author(s):  
M. Benchohra ◽  
S. K. Ntouyas
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Initial Value Problems ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Second Order ◽  
Initial Value ◽  
Functional Differential ◽  
Condensing Maps

In this paper we investigate the existence of solutions on a compact interval to second order initial value problems for functional differential and integrodifferential inclusions in Banach spaces. We shall make use of a fixed point theorem for condensing maps due to Martelli.

scholarly journals On the fractional Dirac systems with non-singular operators

2019 ◽  
Vol 23 (Suppl. 6) ◽  
pp. 2159-2168
Author(s):  
Ahu Ercan
Keyword(s):  
Laplace Transforms ◽  
Dirac System ◽  
Singular Operators ◽  
Dirac Systems

In this manuscript, we consider the fractional Dirac system with exponential and Mittag-Leffler kernels in Riemann-Liouville and Caputo sense. We obtain the representations of the solutions for Dirac systems by means of Laplace transforms.

The Existence of Solutions for Nonlinear Operator Equations

2008 ◽  
Vol 15 (1) ◽  
pp. 45-52
Author(s):  
Marek Galewski
Keyword(s):  
Nonlinear Operator ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Nonlinear Operator Equation ◽  
Global Minima ◽  
Operator Equations ◽  
Nonlinear Operator Equations ◽  
Dual Variational Method ◽  
Dual Action ◽  
Primal And Dual

Abstract We provide the existence results for a nonlinear operator equation Λ*Φ′ (Λ𝑥) = 𝐹′(𝑥), in case 𝐹 – Φ is not necessarily convex. We introduce the dual variational method which is based on finding global minima of primal and dual action functionals on certain nonlinear subsets of their domains and on investigating relations between the minima obtained. The solution is a limit of a minimizng sequence whose existence and convergence are proved. The application for the non-convex Dirichlet problem with P.D.E. is given.

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