Search engine for discovering works of Art, research articles, and books related to Art and Culture
Javascript must be enabled to continue!
One-dimensional periodic fractional Schrödinger equations with exponential critical growth
View through CrossRef
In the present paper, we study the existence of nontrivial solutions of
the following one-dimensional fractional
Schr\“{o}dinger equation $$
(-\Delta)^{1/2}u+V(x)u=f(x,u), \
\ x\in \R, $$ where
$(-\Delta)^{1/2}$ stands for the
$1/2$-Laplacian, $V(x)\in
\mathcal{C}(\R,
(0,+\infty))$, and
$f(x,u):\R\times\R\to
\R$ is a continuous function with an exponential
critical growth. Comparing with the existing works in the field of
exponential-critical-growth fractional Schr\”{o}dinger
equations, we encounter some new challenges due to the weaker
assumptions on the reaction term $f$. By using some sharp energy
estimates, we present a detailed analysis of the energy level, which
allows us to establish the existence of nontrivial solutions for a wider
class of nonlinear terms. Furthermore, we use the non-Nehari manifold
method to establish the existence of Nehari-type ground state solutions
of the one-dimensional fractional Schr\”{o}dinger
equations.
Title: One-dimensional periodic fractional Schrödinger equations with exponential critical growth
Description:
In the present paper, we study the existence of nontrivial solutions of
the following one-dimensional fractional
Schr\“{o}dinger equation $$
(-\Delta)^{1/2}u+V(x)u=f(x,u), \
\ x\in \R, $$ where
$(-\Delta)^{1/2}$ stands for the
$1/2$-Laplacian, $V(x)\in
\mathcal{C}(\R,
(0,+\infty))$, and
$f(x,u):\R\times\R\to
\R$ is a continuous function with an exponential
critical growth.
Comparing with the existing works in the field of
exponential-critical-growth fractional Schr\”{o}dinger
equations, we encounter some new challenges due to the weaker
assumptions on the reaction term $f$.
By using some sharp energy
estimates, we present a detailed analysis of the energy level, which
allows us to establish the existence of nontrivial solutions for a wider
class of nonlinear terms.
Furthermore, we use the non-Nehari manifold
method to establish the existence of Nehari-type ground state solutions
of the one-dimensional fractional Schr\”{o}dinger
equations.
Related Results
Solving Undamped and Damped Fractional Oscillators via Integral Rohit Transform
Solving Undamped and Damped Fractional Oscillators via Integral Rohit Transform
Background: The dynamics of fractional oscillators are generally described by fractional differential equations, which include the fractional derivative of the Caputo or Riemann-Li...
Macroeconomic and Social Precursors of Suicide Rates in the Philippines: A Quantitative Analysis (Preprint)
Macroeconomic and Social Precursors of Suicide Rates in the Philippines: A Quantitative Analysis (Preprint)
BACKGROUND
Suicide is a complex, serious and multifaceted public health issue that poses significant challenges to societies worldwide. In fact, it represen...
Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications
Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications
It is known that, from a modeling point of view, fractional dynamic equations are more suitable compared to integer derivative models. In fact, a fractional dynamic equation is ref...
Soham Transform in Fractional Differential Equations
Soham Transform in Fractional Differential Equations
Objectives: Soham transforms is one of the appropriate tools for solving fractional differential equations that are flexible enough to adapt to different purposes. Methods: Integra...
Synchronization transition with coexistence of attractors in coupled discontinuous system
Synchronization transition with coexistence of attractors in coupled discontinuous system
The studies of extended dynamics systems are relevant to the understanding of spatiotemporal patterns observed in diverse fields. One of the well-established models for such comple...
A fractional-order 4D chaotic electronic circuit based on the Caputo–Fabrizio derivative: modeling, theoretical analysis and numerical simulation
A fractional-order 4D chaotic electronic circuit based on the Caputo–Fabrizio derivative: modeling, theoretical analysis and numerical simulation
Purpose
This paper introduces a novel four-dimensional chaotic electronic circuit modeled using the Caputo–Fabrizio (CF) Fractional derivative (FD), which featu...
On Λ-Fractional fluid mechanics
On Λ-Fractional fluid mechanics
Λ-fractional analysis has already been presented as the only fractional analysis conforming with the Differential Topology prerequisites. That is, the Leibniz rule and chain rule d...
Modulation Instability, Analytical, and Numerical Studies for Integrable Time Fractional Nonlinear Model through Two Explicit Methods
Modulation Instability, Analytical, and Numerical Studies for Integrable Time Fractional Nonlinear Model through Two Explicit Methods
The present work deals with the investigation of the time‐fractional Klein–Gordon (K‐G) model, which has great importance in theoretical physics with applications in various fields...