scholarly journals Periodic solutions for one-dimensional nonlinear nonlocal problem with drift including singular nonlinearities

2021 ◽  
pp. 1-33
Author(s):  
Lisbeth Carrero ◽  
Alexander Quaas
Keyword(s):  
Fractional Laplacian ◽  
A Priori ◽  
Nonlocal Problem ◽  
Degree Theory ◽  
Gradient Term ◽  
A Priori Bounds ◽  
Multiplicity Results ◽  
One Dimensional ◽  
Priori Estimates ◽  
Singular Nonlinearities

In this paper, we prove existence results of a one-dimensional periodic solution to equations with the fractional Laplacian of order $s\in (1/2,1)$ , singular nonlinearity and gradient term under various situations, including nonlocal contra-part of classical Lienard vector equations, as well other nonlocal versions of classical results know only in the context of second-order ODE. Our proofs are based on degree theory and Perron's method, so before that we need to establish a variety of priori estimates under different assumptions on the nonlinearities appearing in the equations. Besides, we obtain also multiplicity results in a regime where a priori bounds are lost and bifurcation from infinity occurs.

A direct blowing-up and rescaling argument on nonlocal elliptic equations

2016 ◽  
Vol 27 (08) ◽  
pp. 1650064 ◽  
Author(s):  
Wenxiong Chen ◽  
Congming Li ◽  
Yan Li
Keyword(s):  
Positive Solutions ◽  
Fractional Laplacian ◽  
A Priori Estimates ◽  
A Priori ◽  
Degree Theory ◽  
Nonlocal Operator ◽  
Nonlocal Operators ◽  
Priori Estimates ◽  
Blowing Up ◽  
Existence Of Positive Solutions

In this paper, we develop a direct blowing-up and rescaling argument for nonlinear equations involving nonlocal elliptic operators including the fractional Laplacian. Instead of using the conventional extension method introduced by Caffarelli and Silvestre to localize the problem, we work directly on the nonlocal operator. Using the defining integral, by an elementary approach, we carry on a blowing-up and rescaling argument directly on the nonlocal equations and thus obtain a priori estimates on the positive solutions. Based on this estimate and the Leray–Schauder degree theory, we establish the existence of positive solutions. We believe that the ideas introduced here can be applied to problems involving more general nonlocal operators.

scholarly journals A priori bounds of the solution of a one point IBVP for a singular fractional evolution equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Said Mesloub ◽  
Hassan Eltayeb Gadain
Keyword(s):  
Differential Equations ◽  
Evolution Equation ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Initial Boundary ◽  
A Priori Bounds ◽  
Fractional Evolution Equation ◽  
Priori Estimates ◽  
Initial Boundary Value

Abstract A priori bounds constitute a crucial and powerful tool in the investigation of initial boundary value problems for linear and nonlinear fractional and integer order differential equations in bounded domains. We present herein a collection of a priori estimates of the solution for an initial boundary value problem for a singular fractional evolution equation (generalized time-fractional wave equation) with mass absorption. The Riemann–Liouville derivative is employed. Results of uniqueness and dependence of the solution upon the data were obtained in two cases, the damped and the undamped case. The uniqueness and continuous dependence (stability of solution) of the solution follows from the obtained a priori estimates in fractional Sobolev spaces. These spaces give what are called weak solutions to our partial differential equations (they are based on the notion of the weak derivatives). The method of energy inequalities is used to obtain different a priori estimates.

scholarly journals Existence and global stability of periodic solution for delayed discrete high-order Hopfield-type neural networks

2005 ◽  
Vol 2005 (3) ◽  
pp. 281-297 ◽  
Author(s):  
Hong Xiang ◽  
Ke-Ming Yan ◽  
Bai-Yan Wang
Keyword(s):  
Neural Networks ◽  
Periodic Solution ◽  
Global Stability ◽  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
High Order ◽  
Priori Estimates

By using coincidence degree theory as well as a priori estimates and Lyapunov functional, we study the existence and global stability of periodic solution for discrete delayed high-order Hopfield-type neural networks. We obtain some easily verifiable sufficient conditions to ensure that there exists a unique periodic solution, and all theirs solutions converge to such a periodic solution.

scholarly journals Existence results and a priori estimates for solutions of quasilinear problems with gradient terms

2019 ◽  
Vol 39 (2) ◽  
pp. 195-206
Author(s):  
Roberta Filippucci ◽  
Chiara Lini
Keyword(s):  
Existence Theorem ◽  
Liouville Theorem ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Limit Problem ◽  
Gradient Term ◽  
Priori Estimates ◽  
Bounded Smooth ◽  
Quasilinear Problems

In this paper we establish a priori estimates and then an existence theorem of positive solutions for a Dirichlet problem on a bounded smooth domain in \(\mathbb{R}^N\) with a nonlinearity involving gradient terms. The existence result is proved with no use of a Liouville theorem for the limit problem obtained via the usual blow up method, in particular we refer to the modified version by Ruiz. In particular our existence theorem extends a result by Lorca and Ubilla in two directions, namely by considering a nonlinearity which includes in the gradient term a power of \(u\) and by removing the growth condition for the nonlinearity \(f\) at \(u=0\).

scholarly journals ON THE SYMMETRY AND UNIQUENESS OF SOLUTIONS OF THE GINZBURG–LANDAU EQUATIONS FOR SMALL DOMAINS

2001 ◽  
Vol 03 (01) ◽  
pp. 1-14 ◽  
Author(s):  
A. AFTALION ◽  
E. N. DANCER
Keyword(s):  
Order Parameter ◽  
A Priori Estimates ◽  
A Priori ◽  
Small Radius ◽  
Uniqueness Of Solutions ◽  
Ginzburg Landau ◽  
One Dimensional ◽  
Priori Estimates ◽  
Ginzburg Landau Equations ◽  
Dimensional Domain

In this paper, we study the Ginzburg–Landau equations for a two dimensional domain which has small size. We prove that if the domain is small, then the solution has no zero, that is no vortex. More precisely, we show that the order parameter Ψ is almost constant. Additionnally, we obtain that if the domain is a disc of small radius, then any non normal solution is symmetric and unique. Then, in the case of a slab, that is a one dimensional domain, we use the same method to derive that solutions are symmetric. The proofs use a priori estimates and the Poincaré inequality.

A SOBOLEV SPACE APPROACH TO BOUNDARY VALUE PROBLEMS ON THE HALF-LINE

2005 ◽  
Vol 07 (01) ◽  
pp. 1-36 ◽  
Author(s):  
PATRICK J. RABIER ◽  
CHARLES A. STUART
Keyword(s):  
Boundary Condition ◽  
Functional Equation ◽  
Existence Of Solutions ◽  
A Priori ◽  
Degree Theory ◽  
A Priori Bounds ◽  
Initial Value ◽  
Second Order Equations ◽  
Ode Systems ◽  
Space Approach

This paper addresses the existence of solutions u ∈ H1(ℝ+;ℝN) of ODE systems [Formula: see text], with boundary condition u1(0) = ξ, where u1is a (vector) component of u. Under general conditions, the problem corresponds to a functional equation involving a Fredholm operator with calculable index, which is proper on the closed bounded subsets of H1(ℝ+;ℝN). When the index is 0 and the solutions are bounded a priori, the existence follows from an available degree theory for such operators. Specific conditions are given that guarantee the existence of a priori bounds and second order equations with Dirichlet, Neumann or initial value conditions are discussed as applications.

Multiple positive solutions for a nonlocal PDE with critical Sobolev-Hardy and singular nonlinearities via perturbation method

2020 ◽  
Vol 23 (3) ◽  
pp. 837-860 ◽  
Author(s):  
Adel Daoues ◽  
Amani Hammami ◽  
Kamel Saoudi
Keyword(s):  
Positive Solutions ◽  
Fractional Laplacian ◽  
Singular Term ◽  
Nonlocal Problem ◽  
Nonlocal Operator ◽  
Approximation Methods ◽  
Multiple Positive Solutions ◽  
Smooth Functions ◽  
Singular Nonlinearities ◽  
Sobolev Exponent

AbstractIn this paper we investigate the following nonlocal problem with singular term and critical Hardy-Sobolev exponent$$\begin{array}{} ({\rm P}) \left\{ \begin{array}{ll} (-\Delta)^s u = \displaystyle{\frac{\lambda}{u^\gamma}+\frac{|u|^{2_\alpha^*-2}u}{|x|^\alpha}} \ \ \text{ in } \ \ \Omega, \\ u >0 \ \ \text{ in } \ \ \Omega, \quad u = 0 \ \ \text{ in } \ \ \mathbb{R}^{N}\setminus \Omega, \end{array} \right. \end{array}$$where Ω ⊂ ℝN is an open bounded domain with Lipschitz boundary, 0 < s < 1, λ > 0 is a parameter, 0 < α < 2s < N, 0 < γ < 1 < 2 < $\begin{array}{} \displaystyle 2_s^* \end{array}$, where $\begin{array}{} \displaystyle 2_s^* = \frac{2N}{N-2s} ~\text{and}~ 2_\alpha^* = \frac{2(N-\alpha)}{N-2s} \end{array}$ are the fractional critical Sobolev and Hardy Sobolev exponents respectively. The fractional Laplacian (–Δ)s with s ∈ (0, 1) is the nonlinear nonlocal operator defined on smooth functions by$$\begin{array}{} \displaystyle (-\Delta)^s u(x)=-\frac{1}{2} \displaystyle\int_{\mathbb{R}^N} \frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2s}}{\rm d }y, \;\; \text{ for all }\, x \in \mathbb{R}^N. \end{array}$$By combining variational and approximation methods, we provide the existence of two positive solutions to the problem (P).

A priori bounds and existence of positive solutions to a p-kirchhoff equations

2021 ◽  
pp. 2150082
Author(s):  
Pengfei Li ◽  
Junhui Xie
Keyword(s):  
Positive Solutions ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Auxiliary Problem ◽  
Degree Theory ◽  
Kirchhoff Equations ◽  
Priori Estimates ◽  
Existence Of Positive Solutions ◽  
Dirichlet Boundary Problem

In this paper, we consider a [Formula: see text]-Kirchhoff problem with Dirichlet boundary problem in a bounded domain. Under suitable conditions, we get a priori estimates for positive solutions to an auxiliary problem by the well-known blow-up argument. As an application, a existence result for positive solutions is proved by the topological degree theory.

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