scholarly journals Periodic solutions for a fractional asymptotically linear problem

2018 ◽  
Vol 149 (03) ◽  
pp. 593-615
Author(s):  
Vincenzo Ambrosio ◽  
Giovanni Molica Bisci
Keyword(s):  
Neumann Boundary ◽  
Index Theory ◽  
Careful Analysis ◽  
Asymptotically Linear ◽  
Existence And Multiplicity ◽  
Degenerate Elliptic ◽  
Subcritical Nonlinearity ◽  
Fractional Spaces ◽  
Non Local ◽  
Asymptotically Linear Problem

We study the existence and multiplicity of periodic weak solutions for a non-local equation involving an odd subcritical nonlinearity which is asymptotically linear at infinity. We investigate such problem by applying the pseudo-index theory developed by Bartolo, Benci and Fortunato [11] after transforming the problem to a degenerate elliptic problem in a half-cylinder with a Neumann boundary condition, via a Caffarelli-Silvestre type extension in periodic setting. The periodic nonlocal case, considered here, presents, respect to the cases studied in the literature, some new additional difficulties and a careful analysis of the fractional spaces involved is necessary.

scholarly journals Homoclinic Orbits for a Class of Nonperiodic Hamiltonian Systems with Some Twisted Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Qi Wang ◽  
Qingye Zhang
Keyword(s):  
Hamiltonian Systems ◽  
Index Theory ◽  
Maslov Index ◽  
Homoclinic Orbits ◽  
Asymptotically Linear ◽  
Existence And Multiplicity ◽  
Hamiltonian Functions

By the Maslov index theory, we will study the existence and multiplicity of homoclinic orbits for a class of asymptotically linear nonperiodic Hamiltonian systems with some twisted conditions on the Hamiltonian functions.

A note on the monotonicity of the Maslov-type index of linear Hamiltonian systems with applications

2005 ◽  
Vol 135 (6) ◽  
pp. 1263-1277 ◽  
Author(s):  
Chun-Gen Liu
Keyword(s):  
Hamiltonian Systems ◽  
Index Theory ◽  
Matrix Functions ◽  
Asymptotically Linear ◽  
Existence And Multiplicity ◽  
Linear Hamiltonian Systems

In this note, we first consider the monotonicity of the Maslov-type index theory. More precisely, for any two 1-periodic symmetric continuous matrix functions B0(t) and B1(t) with B0(t) < B1(t), we consider the relations between the Maslov-type indices (i(B0), ν (B0)) and (i(B1), ν (B1)). We then apply this theory to study the existence and multiplicity of some kinds of asymptotically linear Hamiltonian systems

Some remarks on the solvability of non-local elliptic problems with the Hardy potential

2014 ◽  
Vol 16 (04) ◽  
pp. 1350046 ◽  
Author(s):  
B. Barrios ◽  
M. Medina ◽  
I. Peral
Keyword(s):  
Real Number ◽  
Sobolev Inequality ◽  
Positive Real Number ◽  
Fractional Power ◽  
Existence Of Solution ◽  
Sharp Constant ◽  
Hardy Potential ◽  
Positive Real ◽  
Existence And Multiplicity ◽  
Non Local

The aim of this paper is to study the solvability of the following problem, [Formula: see text] where (-Δ)s, with s ∈ (0, 1), is a fractional power of the positive operator -Δ, Ω ⊂ ℝN, N > 2s, is a Lipschitz bounded domain such that 0 ∈ Ω, μ is a positive real number, λ < ΛN,s, the sharp constant of the Hardy–Sobolev inequality, 0 < q < 1 and [Formula: see text], with αλ a parameter depending on λ and satisfying [Formula: see text]. We will discuss the existence and multiplicity of solutions depending on the value of p, proving in particular that p(λ, s) is the threshold for the existence of solution to problem (Pμ).

scholarly journals Sign-Changing Solutions for a Class of Zero Mass Nonlocal Schrödinger Equations

2019 ◽  
Vol 19 (1) ◽  
pp. 113-132 ◽  
Author(s):  
Vincenzo Ambrosio ◽  
Giovany M. Figueiredo ◽  
Teresa Isernia ◽  
Giovanni Molica Bisci
Keyword(s):  
Careful Analysis ◽  
Continuous Functions ◽  
Nehari Manifold ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Deformation Lemma ◽  
Fractional Schrödinger Equations ◽  
Fractional Spaces ◽  
The Nehari Manifold ◽  
Sign Changing Solutions

Abstract We consider the following class of fractional Schrödinger equations: (-\Delta)^{\alpha}u+V(x)u=K(x)f(u)\quad\text{in }\mathbb{R}^{N}, where {\alpha\in(0,1)} , {N>2\alpha} , {(-\Delta)^{\alpha}} is the fractional Laplacian, V and K are positive continuous functions which vanish at infinity, and f is a continuous function. By using a minimization argument and a quantitative deformation lemma, we obtain the existence of a sign-changing solution. Furthermore, when f is odd, we prove that the above problem admits infinitely many nontrivial solutions. Our result extends to the fractional framework some well-known theorems proved for elliptic equations in the classical setting. With respect to these cases studied in the literature, the nonlocal one considered here presents some additional difficulties, such as the lack of decompositions involving positive and negative parts, and the non-differentiability of the Nehari Manifold, so that a careful analysis of the fractional spaces involved is necessary.

scholarly journals On Positive Radial Solutions for a Class of Elliptic Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Ying Wu ◽  
Guodong Han
Keyword(s):  
Fixed Point Index ◽  
Nonlinear Term ◽  
Elliptic Boundary ◽  
Index Theory ◽  
First Eigenvalue ◽  
Radial Solutions ◽  
Elliptic Boundary Value ◽  
Existence And Multiplicity ◽  
Fixed Point Index Theory ◽  
Positive Radial Solutions

A class of elliptic boundary value problem in an exterior domain is considered under some conditions concerning the first eigenvalue of the relevant linear operator, where the variables of nonlinear termfs,uneed not to be separated. Several new theorems on the existence and multiplicity of positive radial solutions are obtained by means of fixed point index theory. Our conclusions are essential improvements of the results in Lan and Webb (1998), Lee (1997), Mao and Xue (2002), Stańczy (2000), and Han and Wang (2006).

scholarly journals Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-17
Author(s):  
Chiara Zanini ◽  
Fabio Zanolin
Keyword(s):  
Complex Dynamics ◽  
Neumann Boundary ◽  
Step Function ◽  
One Dimensional ◽  
Nonlinear Schrödinger ◽  
Existence And Multiplicity ◽  
Twist Maps ◽  
Topological Horseshoes ◽  
Recent Developments ◽  
The One

We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation -ε2u′′+V(x)u=f(u), where the potential V(x) approximates a two-step function. The term f(u) generalizes the typical p-power nonlinearity considered by several authors in this context. Our approach is based on some recent developments of the theory of topological horseshoes, in connection with a linked twist maps geometry, which are applied to the discrete dynamics of the Poincaré map. We discuss the periodic and the Neumann boundary conditions. The value of the term ε>0, although small, can be explicitly estimated.

Multiplicity results for systems of asymptotically linear second order equations

2002 ◽  
Vol 2 (4) ◽  
Author(s):  
Anna Capietto ◽  
Francesca Dalbono
Keyword(s):  
Lower Bounds ◽  
Second Order ◽  
Upper And Lower Bounds ◽  
Asymptotically Linear ◽  
Multiplicity Results ◽  
Number Of Zeros ◽  
Existence And Multiplicity ◽  
Abstract Continuation Theorem ◽  
Nodal Properties ◽  
Second Order Equations

AbstractWe prove the existence and multiplicity of solutions, with prescribed nodal properties, for a BVP associated with a system of asymptotically linear second order equations. The applicability of an abstract continuation theorem is ensured by upper and lower bounds on the number of zeros of each component of a solution.

scholarly journals Periodic solutions for a class of second-order differential delay equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xuan Wu ◽  
Huafeng Xiao
Keyword(s):  
Periodic Solutions ◽  
Index Theory ◽  
Second Order ◽  
Delay Equations ◽  
Asymptotically Linear ◽  
Differential Delay ◽  
Differential Delay Equations ◽  
Existence Of Periodic Solutions ◽  
The Difference

<p style='text-indent:20px;'>In this paper, we study the existence of periodic solutions of the following differential delay equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} z^{\prime\prime}(t) = \sum\limits_{k = 1}^{M-1}(-1)^kf(z(t-k)), \notag \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ f\in C(\mathbf{R}^N, \mathbf{R}^N) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ M,N\in \mathbf{N} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ M $\end{document}</tex-math></inline-formula> is odd. By making use of <inline-formula><tex-math id="M4">\begin{document}$ S^1 $\end{document}</tex-math></inline-formula>-geometrical index theory, we obtain an estimation about the number of periodic solutions in term of the difference between eigenvalues of asymptotically linear matrices at the origin and at infinity.</p>

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