Existence of Ground States of Fractional Schrödinger Equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Li Ma ◽  
Zhenxiong Li
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Nonlinear Term ◽  
Ground States ◽  
Schrödinger Equations ◽  
Fractional Schrödinger Equations ◽  
Whole Space ◽  
Nonlinear Fractional Schrödinger Equation ◽  
Fractional Schrodinger Equation

Abstract We consider ground states of the nonlinear fractional Schrödinger equation with potentials ( - Δ ) s ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( x , u ) , s ∈ ( 0 , 1 ) , (-\Delta)^{s}u+V(x)u=f(x,u),\quad s\in(0,1), on the whole space ℝ N {\mathbb{R}^{N}} , where V is a periodic non-negative nontrivial function on ℝ N {\mathbb{R}^{N}} and the nonlinear term f has some proper growth on u. Under uniform bounded assumptions about V, we can show the existence of a ground state. We extend the result of Li, Wang, and Zeng to the fractional case.

Ground States of some Fractional Schrödinger Equations on ℝN

2014 ◽  
Vol 58 (2) ◽  
pp. 305-321 ◽  
Author(s):  
Xiaojun Chang
Keyword(s):  
Variational Methods ◽  
Ground State ◽  
Schrodinger Equation ◽  
Fractional Laplacian ◽  
Ground States ◽  
Schrödinger Equations ◽  
Ground State Solutions ◽  
Fractional Schrödinger Equations ◽  
Image Position ◽  
Fractional Schrodinger Equation

AbstractIn this paper, we study a time-independent fractional Schrödinger equation of the form (−Δ)su + V(x)u = g(u) in ℝN, where N ≥, s ∈ (0,1) and (−Δ)s is the fractional Laplacian. By variational methods, we prove the existence of ground state solutions when V is unbounded and the nonlinearity g is subcritical and satisfies the following geometry condition:

scholarly journals Solutions of Unified Fractional Schrödinger Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
V. B. L. Chaurasia ◽  
Devendra Kumar
Keyword(s):  
Schrödinger Equation ◽  
Closed Form ◽  
Numerical Computation ◽  
Schrodinger Equation ◽  
Fourier Transforms ◽  
Schrödinger Equations ◽  
Laplace And Fourier Transforms ◽  
Fractional Schrödinger Equation ◽  
Fractional Schrödinger Equations ◽  
Fractional Schrodinger Equation

We obtain the solution of a unified fractional Schrödinger equation. The solution is derived by the application of the Laplace and Fourier transforms in closed form in terms of the Mittag-Leffler function. The result obtained here is quite general in nature and capable of yielding a very large number of results (new and known) hitherto scattered in the literature. Most of results obtained are in a form suitable for numerical computation.

scholarly journals Non-local to local transition for ground states of fractional Schrödinger equations on $$\mathbb {R}^N$$

2020 ◽  
Vol 22 (3) ◽  
Author(s):  
Bartosz Bieganowski ◽  
Simone Secchi
Keyword(s):  
Ground State ◽  
Ground States ◽  
Ground State Solution ◽  
State Solution ◽  
Local Problem ◽  
Schrödinger Equations ◽  
Ground State Solutions ◽  
Fractional Schrödinger Equations ◽  
Non Local ◽  
Local Transition

Abstract We consider the nonlinear fractional problem $$\begin{aligned} (-\Delta )^{s} u + V(x) u = f(x,u)&\quad \hbox {in } \mathbb {R}^N \end{aligned}$$ ( - Δ ) s u + V ( x ) u = f ( x , u ) in R N We show that ground state solutions converge (along a subsequence) in $$L^2_{\mathrm {loc}} (\mathbb {R}^N)$$ L loc 2 ( R N ) , under suitable conditions on f and V, to a ground state solution of the local problem as $$s \rightarrow 1^-$$ s → 1 - .

Orbital stability of standing waves for nonlinear fractional Schrödinger equation with unbounded potential

2019 ◽  
Vol 12 (03) ◽  
pp. 1950043
Author(s):  
Xiaohua Liu
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Standing Waves ◽  
Orbital Stability ◽  
Fractional Schrödinger Equation ◽  
Unbounded Potential ◽  
Nonlinear Fractional Schrödinger Equation ◽  
The Stability ◽  
First Time ◽  
Fractional Schrodinger Equation

In this paper, the orbital stability of standing waves for nonlinear fractional Schrödinger equation is considered. By constructing the constrained functional extreme-value problem, the existence of standing waves is studied. With the help of the orbital stability theories presented by Grillakis, Shatah and Strauss, the orbital stability of standing waves is determined by the sign of a discriminant. To our knowledge, it is the first time that the abstract orbital stability theories presented by Grillakis, Shatah and Strauss are applied to study the stability of solutions for fractional evolution equation.

scholarly journals Ground state solutions for fractional Schrödinger equation with variable potential and Berestycki–Lions type nonlinearity

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Jing Chen ◽  
Zu Gao
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Nonlocal Problem ◽  
Minimization Method ◽  
Fractional Schrödinger Equation ◽  
Ground State Solutions ◽  
The One ◽  
Least Energy Solutions ◽  
Fractional Schrodinger Equation

Abstract We consider the following nonlinear fractional Schrödinger equation: $$ (-\triangle )^{s} u+V(x)u=g(u) \quad \text{in } \mathbb{R} ^{N}, $$ ( − △ ) s u + V ( x ) u = g ( u ) in  R N , where $s\in (0, 1)$ s ∈ ( 0 , 1 ) , $N>2s$ N > 2 s , $V(x)$ V ( x ) is differentiable, and $g\in C ^{1}(\mathbb{R} , \mathbb{R} )$ g ∈ C 1 ( R , R ) . By exploiting the minimization method with a constraint over Pohoz̆aev manifold, we obtain the existence of ground state solutions. With the help of Pohoz̆aev identity we also process the existence of the least energy solutions for the above equation. Our results improve the existing study on this nonlocal problem with Berestycki–Lions type nonlinearity to the one that does not need the oddness assumption.

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