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:

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.

Existence and concentration of ground state solutions for a class of fractional Schrödinger equations

2021 ◽  
pp. 1-25
Author(s):  
Zhuo Chen ◽  
Chao Ji
Keyword(s):  
Continuous Function ◽  
Periodic Function ◽  
Variational Methods ◽  
Ground State ◽  
Schrodinger Equation ◽  
Fractional Laplacian ◽  
Positive Parameter ◽  
Subcritical Growth ◽  
Ground State Solutions ◽  
Fractional Schrödinger Equations

In this paper, by using variational methods, we study the existence and concentration of ground state solutions for the following fractional Schrödinger equation ( − Δ ) α u + V ( x ) u = A ( ϵ x ) f ( u ) , x ∈ R N , where α ∈ ( 0 , 1 ), ϵ is a positive parameter, N > 2 α, ( − Δ ) α stands for the fractional Laplacian, f is a continuous function with subcritical growth, V ∈ C ( R N , R ) is a Z N -periodic function and A ∈ C ( R N , R ) satisfies some appropriate assumptions.

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 - .

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.

POSITIVE GROUND STATES FOR A CLASS OF SUPERLINEAR -LAPLACIAN COUPLED SYSTEMS INVOLVING SCHRÖDINGER EQUATIONS

2019 ◽  
Vol 109 (2) ◽  
pp. 193-216 ◽  
Author(s):  
J. C. DE ALBUQUERQUE ◽  
JOÃO MARCOS DO Ó ◽  
EDCARLOS D. SILVA
Keyword(s):  
Large Class ◽  
Ground State ◽  
Variational Approach ◽  
Ground States ◽  
Nehari Manifold ◽  
Coupled Systems ◽  
Coupling Term ◽  
Ground State Solutions ◽  
Image Position ◽  
The Nehari Manifold

We study the existence of positive ground state solutions for the following class of $(p,q)$-Laplacian coupled systems $$\begin{eqnarray}\left\{\begin{array}{@{}lr@{}}-\unicode[STIX]{x1D6E5}_{p}u+a(x)|u|^{p-2}u=f(u)+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}(x)|u|^{\unicode[STIX]{x1D6FC}-2}u|v|^{\unicode[STIX]{x1D6FD}}, & x\in \mathbb{R}^{N},\\ -\unicode[STIX]{x1D6E5}_{q}v+b(x)|v|^{q-2}v=g(v)+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}(x)|v|^{\unicode[STIX]{x1D6FD}-2}v|u|^{\unicode[STIX]{x1D6FC}}, & x\in \mathbb{R}^{N},\end{array}\right.\end{eqnarray}$$ where $1<p\leq q<N$. Here the coefficient $\unicode[STIX]{x1D706}(x)$ of the coupling term is related to the potentials by the condition $|\unicode[STIX]{x1D706}(x)|\leq \unicode[STIX]{x1D6FF}a(x)^{\unicode[STIX]{x1D6FC}/p}b(x)^{\unicode[STIX]{x1D6FD}/q}$, where $\unicode[STIX]{x1D6FF}\in (0,1)$ and $\unicode[STIX]{x1D6FC}/p+\unicode[STIX]{x1D6FD}/q=1$. Using a variational approach based on minimization over the Nehari manifold, we establish the existence of positive ground state solutions for a large class of nonlinear terms and potentials.

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 Computing the Ground and First Excited States of the Fractional Schrödinger Equation in an Infinite Potential Well

2015 ◽  
Vol 18 (2) ◽  
pp. 321-350 ◽  
Author(s):  
Siwei Duo ◽  
Yanzhi Zhang
Keyword(s):  
Schrödinger Equation ◽  
Excited States ◽  
Schrodinger Equation ◽  
Fractional Laplacian ◽  
Ground States ◽  
Euler Method ◽  
Potential Well ◽  
Fractional Schrödinger Equation ◽  
Fractional Schrodinger Equation ◽  
Infinite Potential Well

AbstractIn this paper, we numerically study the ground and first excited states of the fractional Schrödinger equation in an infinite potential well. Due to the nonlocality of the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions of the fractional Schrödinger equation analytically. We first introduce a normalized fractional gradient flow and then discretize it by a quadrature rule method in space and the semi-implicit Euler method in time. Our numerical results suggest that the eigenfunctions of the fractional Schrödinger equation in an infinite potential well differ from those of the standard (non-fractional) Schrödinger equation. We find that the strong nonlocal interactions represented by the fractional Laplacian can lead to a large scattering of particles inside of the potential well. Compared to the ground states, the scattering of particles in the first excited states is larger. Furthermore, boundary layers emerge in the ground states and additionally inner layers exist in the first excited states of the fractional nonlinear Schrödinger equation. Our simulated eigenvalues are consistent with the lower and upper bound estimates in the literature.

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