COMPACT FINITE DIFFERENCES METHOD AND CAPUTO FRACTIONAL DERIVATIVE DEFINITION FOR LINEAR FRACTIONAL SCHRÖDINGER EQUATIONS

2018 ◽  
Vol 17 (1) ◽  
pp. 19-45
Author(s):  
Neslihan Er ◽  
Hikmet Çağlar ◽  
Nazan Çağlar
Keyword(s):  
Fractional Derivative ◽  
Finite Differences ◽  
Caputo Fractional Derivative ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Fractional Schrödinger Equations ◽  
Finite Differences Method ◽  
Compact Finite Differences

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.

Sign in / Sign up

Export Citation Format

Share Document