Ground‐state solution of a nonlinear fractional Schrödinger–Poisson system
Abstract: In this paper, we study the nonlinear fractional Schrödinger-Poisson systemwhere (− Δ) s is the fractional Laplacian of order s, 3 4 < s < 1, and V ∶ R 3 → R is a potential function exhibiting a mixed behavior; that is, V is periodic in some directions while tends to a positive constant in other directions. Using the method of Nehari manifold and a concentration compactness argument of P.-L.Lions, we establish the existence of a signed ground-state solution for subcritical problem and for critical problem.
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Cited by 6 publications
References 45 publications
Option pricing in a stochastic delay volatility model
This work introduces a new stochastic volatility model with delay parameters in the volatility process, extending the Barndorff–Nielsen and Shephard model. It establishes an analytical expression for the log price characteristic function, which can be applied to price European options. Empirical analysis on S&P500 European call options shows that adding delay parameters reduces mean squared error. This is the first instance of providing an analytical formula for the log price characteristic function in a stochastic volatility model with multiple delay parameters. We also provide a Monte Carlo scheme that can be used to simulate the model.
Option pricing in a stochastic delay volatility model
This work introduces a new stochastic volatility model with delay parameters in the volatility process, extending the Barndorff–Nielsen and Shephard model. It establishes an analytical expression for the log price characteristic function, which can be applied to price European options. Empirical analysis on S&P500 European call options shows that adding delay parameters reduces mean squared error. This is the first instance of providing an analytical formula for the log price characteristic function in a stochastic volatility model with multiple delay parameters. We also provide a Monte Carlo scheme that can be used to simulate the model.
Existence and concentration of solutions for a fractional Schrödinger–Poisson system with discontinuous nonlinearity
In this paper, we study the following fractional Schrödinger–Poisson system with discontinuous nonlinearity: ε 2 s ( − Δ ) s u + V ( x ) u + ϕ u = H ( u − β ) f ( u ) , in R 3 , ε 2 s ( − Δ ) s ϕ = u 2 , in R 3 , u > 0 , in R 3 , $$\begin{cases}^{2s}{\left(-{\Delta}\right)}^{s}u+V\left(x\right)u+\phi u=H\left(u-\beta \right)f\left(u\right),\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \\ {\varepsilon }^{2s}{\left(-{\Delta}\right)}^{s}\phi ={u}^{2},\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \\ u{ >}0,\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \end{cases}$$ where ɛ > 0 is a small parameter, s ∈ ( 3 4 , 1 ) $s\in \left(\frac{3}{4},1\right)$ , β > 0, H is the Heaviside function, (−Δ) s u is the fractional Laplacian operator, V : R 3 → R $V :{\mathbb{R}}^{3}\to \mathbb{R}$ is a continuous potential and f : R → R $f :\mathbb{R}\to \mathbb{R}$ is superlinear continuous nonlinearity with subcritical growth at infinity. By using nonsmooth analysis, we investigate the existence and concentration of solutions for the above problem. Moreover, we obtain some properties of these solutions, such as convergence and decay estimate.
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