Energy solutions and concentration problem of fractional Schrödinger equation

Existence and Uniqueness Theory for the Fractional Schrödinger Equation on the Torus

We study the Cauchy problem for the 1-d periodic fractional Schrödinger equation with cubic nonlinearity. In particular we prove local well-posedness in Sobolev spaces, for solutions evolving from rough initial data. In addition we show the existence of global-in-time infinite energy solutions. Our tools include a new Strichartz estimate on the torus along with ideas that Bourgain developed in ...

Existence of positive solutions for a boundary value problem of a nonlinear fractional differential equation

This paper presents conditions for the existence and multiplicity of positive solutions for a boundary value problem of a nonlinear fractional differential equation. We show that it has at least one or two positive solutions. The main tool is Krasnosel'skii fixed point theorem on cone and fixed point index theory.

Existence of Mild Solutions to a Cauchy Problem Presented by Fractional Evolution Equation with an Integral Initial Condition

In this article, we apply two new fixed point theorems to investigate the existence of mild solutions for a nonlocal fractional Cauchy problem with an integral initial condition in Banach spaces.

Triple Positive Solutions for Boundary Value Problem of a Nonlinear Fractional Differential Equation

Ground States for the Fractional Schrödinger Equation

In this article, we show the existence of ground state solutions for the nonlinear Schrödinger equation with fractional Laplacian (−∆)u+ V (x)u = λ|u|u in R for α ∈ (0, 1). We use the concentration compactness principle in fractional Sobolev spaces Hα for α ∈ (0, 1). Our results generalize the corresponding results in the case α = 1.