scholarly journals Momentum transforms and Laplacians in fractional spaces

2012 ◽  
Vol 16 (4) ◽  
pp. 1315-1348 ◽  
Author(s):  
Gianluca Calcagni ◽  
Giuseppe Nardelli
Keyword(s):  
Fractional Spaces

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 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 stability of difference schemes in fractional spaces

2013 ◽  
Vol 57 (3-4) ◽  
pp. 900-908
Author(s):  
Allaberen Ashyralyev ◽  
Ahmet Batyrov
Keyword(s):  
Difference Schemes ◽  
Stability Of Difference Schemes ◽  
Fractional Spaces

A note on fractional powers of strongly positive operators and their applications

2019 ◽  
Vol 22 (2) ◽  
pp. 302-325 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Ayman Hamad
Keyword(s):  
Banach Space ◽  
Elliptic Operators ◽  
Positive Operators ◽  
Fractional Powers ◽  
Fractional Spaces

Abstract The present paper deals with fractional powers of positive operators in a Banach space. The main theorem concerns the structure of fractional powers of positive operators in fractional spaces. As applications, the structure of fractional powers of elliptic operators is studied.

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