scholarly journals Concentration phenomena for fractional magnetic NLS equations

2021 ◽  
pp. 1-39
Author(s):  
Vincenzo Ambrosio
Keyword(s):  
Schrödinger Equation ◽  
Electric Potential ◽  
Schrodinger Equation ◽  
Nehari Manifold ◽  
Local Condition ◽  
Subcritical Growth ◽  
Concentration Phenomena ◽  
Penalization Technique ◽  
Complex Valued ◽  
Kato’S Inequality

We study the multiplicity and concentration of complex-valued solutions for a fractional magnetic Schrödinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik–Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems.

Least action nodal solutions for a quasilinear defocusing Schrödinger equation with supercritical nonlinearity

2019 ◽  
Vol 21 (05) ◽  
pp. 1850026 ◽  
Author(s):  
Minbo Yang ◽  
Carlos Alberto Santos ◽  
Jiazheng Zhou
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nehari Manifold ◽  
Nodal Solution ◽  
Nodal Solutions ◽  
Subcritical Growth ◽  
Least Action ◽  
Continuous Potential ◽  
Supercritical Nonlinearity

In this paper, we consider the existence of least action nodal solutions for the quasilinear defocusing Schrödinger equation in [Formula: see text]: [Formula: see text] where [Formula: see text] is a positive continuous potential, [Formula: see text] is of subcritical growth, [Formula: see text] and [Formula: see text] are two non-negative parameters. By considering a minimizing problem restricted on a partial Nehari manifold, we prove the existence of least action nodal solution via deformation flow arguments and [Formula: see text]-estimates.

Least Energy Nodal Solutions for a Defocusing Schrödinger Equation with Supercritical Exponent

2018 ◽  
Vol 62 (1) ◽  
pp. 1-23 ◽  
Author(s):  
Minbo Yang ◽  
Carlos Alberto Santos ◽  
Jiazheng Zhou
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Positive Function ◽  
Nehari Manifold ◽  
Nodal Solution ◽  
Nodal Solutions ◽  
Quasilinear Schrödinger Equation ◽  
Subcritical Growth ◽  
Image Position ◽  
Least Energy

AbstractIn this paper we consider the existence of least energy nodal solution for the defocusing quasilinear Schrödinger equation $$-\Delta u - u \Delta u^2 + V(x)u = a(x)[g(u) + \lambda \vert u \vert ^{p-2}u] \hbox{in} {\open R}^N,$$ where λ≥0 is a real parameter, V(x) is a non-vanishing function, a(x) can be a vanishing positive function at infinity, the nonlinearity g(u) is of subcritical growth, the exponent p≥22*, and N≥3. The proof is based on a dual argument on Nehari manifold by employing a deformation argument and an $L</italic>^{\infty}({\open R}^{N})$-estimative.

scholarly journals Phenomena of Blowup and Global Existence of the Solution to a Nonlinear Schrödinger Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Xiaowei An ◽  
Desheng Li ◽  
Xianfa Song
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Sufficient Conditions ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Complex Valued ◽  
Sharp Thresholds

We consider the following Cauchy problem:-iut=Δu-V(x)u+f(x,|u|2)u+(W(x)⋆|u|2)u,x∈ℝN,t>0,u(x,0)=u0(x),x∈ℝN,whereV(x)andW(x)are real-valued potentials andV(x)≥0andW(x)is even,f(x,|u|2)is measurable inxand continuous in|u|2, andu0(x)is a complex-valued function ofx. We obtain some sufficient conditions and establish two sharp thresholds for the blowup and global existence of the solution to the problem.

Ground state solution for the Schrödinger equation with Hardy potential and critical Sobolev exponent

2021 ◽  
pp. 1-19
Author(s):  
Jing Zhang ◽  
Lin Li
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Critical Sobolev Exponent ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
State Solution ◽  
Hardy Potential ◽  
Asymptotically Periodic ◽  
Sobolev Exponent

In this paper, we consider the following Schrödinger equation (0.1) − Δ u − μ u | x | 2 + V ( x ) u = K ( x ) | u | 2 ∗ − 2 u + f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where N ⩾ 4, 0 ⩽ μ < μ ‾, μ ‾ = ( N − 2 ) 2 4 , V is periodic in x, K and f are asymptotically periodic in x, we take advantage of the generalized Nehari manifold approach developed by Szulkin and Weth to look for the ground state solution of (0.1).

SCHRÖDINGER EQUATION FOR QUANTUM FRACTAL SPACE–TIME OF ORDER n VIA THE COMPLEX-VALUED FRACTIONAL BROWNIAN MOTION

2001 ◽  
Vol 16 (31) ◽  
pp. 5061-5084 ◽  
Author(s):  
GUY JUMARIE
Keyword(s):  
Brownian Motion ◽  
Schrödinger Equation ◽  
Fractional Order ◽  
Schrodinger Equation ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Dimensional System ◽  
One Dimensional ◽  
Klein Gordon ◽  
Complex Valued

First remark: Feynman's discovery in accordance of which quantum trajectories are of fractal nature (continuous everywhere but nowhere differentiable) suggests describing the dynamics of such systems by explicitly introducing the Brownian motion of fractional order in their equations. The second remark is that, apparently, it is only in the complex plane that the Brownian motion of fractional order with independent increments can be generated, by using random walks defined with the complex roots of the unity; in such a manner that, as a result, the use of complex variables would be compulsory to describe quantum systems. Here one proposes a very simple set of axioms in order to expand the consequences of these remarks. Loosely speaking, a one-dimensional system with real-valued coordinate is in fact the average observation of a one-dimensional system with complex-valued coordinate: It is a strip modeling. Assuming that the system is governed by a stochastic differential equation driven by a complex valued fractional Brownian of order n, one can then obtain the explicit expression of the corresponding covariant stochastic derivative with respect to time, whereby we switch to the extension of Lagrangian mechanics. One can then derive a Schrödinger equation of order n in quite a direct way. The extension to relativistic quantum mechanics is outlined, and a generalized Klein–Gordon equation of order n is obtained. As a by-product, one so obtains a new proof of the Schrödinger equation.

scholarly journals EXISTENCE AND CONCENTRATION OF SOLUTION FOR A NON-LOCAL REGIONAL SCHRÖDINGER EQUATION WITH COMPETING POTENTIALS

2018 ◽  
Vol 61 (2) ◽  
pp. 441-460 ◽  
Author(s):  
CLAUDIANOR O. ALVES ◽  
CÉSAR E. TORRES LEDESMA
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Fractional Laplacian ◽  
Positive Parameter ◽  
Concentration Phenomena ◽  
Formula Group ◽  
Non Local ◽  
Image Position

AbstractIn this paper, we study the existence and concentration phenomena of solutions for the following non-local regional Schrödinger equation $$\begin{equation*} \left\{ \begin{array}{l} \epsilon^{2\alpha}(-\Delta)_\rho^{\alpha} u + Q(x)u = K(x)|u|^{p-1}u,\;\;\mbox{in}\;\; \mathbb{R}^n,\\ u\in H^{\alpha}(\mathbb{R}^n) \end{array} \right. \end{equation*}$$ where ϵ is a positive parameter, 0 < α < 1, $1<p<\frac{n+2\alpha}{n-2\alpha}$, n > 2α; (−Δ)ρα is a variational version of the regional fractional Laplacian, whose range of scope is a ball with radius ρ(x) > 0, ρ, Q, K are competing functions.

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