Non-linear Schrödinger equation with non-local regional diffusion
type: Item
instance of: scholarly article
title: Non-linear Schrödinger equation with non-local regional diffusion
zbMATH Open document ID: 1326.35338
published in: Calculus of Variations and Partial Differential Equations
publication date: 2015-09-11T00:00:00Z
full work available at URL: http://hdl.handle.net/10533/147946
review text: The paper focuses on an existence result for the nonlinear Schrödinger equation written as \(\epsilon ^{2\alpha }(-\Delta )_{\rho }^{\alpha }u+u=f(u) \) in \(\mathbb{R}^{n}\), \(n\geq 2\), under appropriate hypotheses on \(\rho \) and on \(f\) and on some concentration properties of these weak solutions. Here, \(\epsilon >0\), \(\alpha \in (0,1)\) and \((-\Delta )_{\rho }^{\alpha }\) is a variational version of the non-local regional Laplace operator, with range of scope associated to \(\rho \in C(\mathbb{R}^{n},\mathbb{R}^{+})\). The authors introduce the energy functional associated to this nonlinear Schrödinger equation \[ I_{\rho }(u)=\frac{1}{2}\int_{\mathbb{R} ^{n}}\int_{B(0,\rho (x))}\frac{|u(x)-u(z)|^{2}}{|x-z|^{n+2\alpha }} dzdx+\int_{\mathbb{R}^{n}}u(x)^{2}dx-\int_{\mathbb{R}^{n}}F(u(x))dx, \] where \( F(t)=\int_{0}^{t}f(\xi )d\xi \). A weak solution in \(H^{\alpha }(\mathbb{R} ^{n})\) to the nonlinear Schrödinger equation is defined as a critical point of the energy functional \(I_{\rho }\). The first main result of the paper proves the existence of a non-trivial weak solution \(u\geq 0\) to the above-indicated nonlinear Schrödinger equation. For the proof, the authors apply the mountain pass theorem to the energy functional \(I_{\rho }\) and they use a comparison argument already introduced by the author, \textit{A. Quaas} and \textit{J. Tan} in [Proc. R. Soc. Edinb., Sect. A, Math. 142, No. 6, 1237-1262 (2012; Zbl 1290.35308)]. They also use some properties of the operator \((-\Delta )_{\rho }^{\alpha }\). The second main result proves a concentration behavior of ground states to the above-indicated Schrödinger equation. The authors indeed prove that when \(\epsilon\) goes to 0, there exists a subsequence of non-negative solutions which concentrate around a global minimum point of some limit concentration function \(\mathcal{H}\). The authors precise the speed of concentration. For the proof, they first define the notion of weak solution in \(H_{\rho }^…
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 35B38
Mathematics Subject Classification ID: 35A01
zbMATH DE Number: 6481160
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: existence result
zbMATH Keywords: weak solution
zbMATH Keywords: critical point
zbMATH Keywords: mountain pass theorem
zbMATH Keywords: ground state
zbMATH Keywords: concentration
author: P. Felmer
author: César Torres
MaRDI profile type: Publication
OpenAlex ID: W1979994218
Wikidata QID: Q115387374
cites work: Dual variational methods in critical point theory and applications
cites work: Bound state for the fractional Schrödinger equation with unbounded potential
cites work: Homoclinic type solutions for a semilinear elliptic PDE on ℝn
cites work: Hitchhiker's guide to the fractional Sobolev spaces
cites work: Some solutions to the space fractional Schrödinger equation using momentum representation method
cites work: Uniqueness of non-linear ground states for fractional Laplacians in \(\mathbb{R}\)
cites work: Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian
cites work: Integration by parts formula for regional fractional Laplacian
cites work: Reflected symmetric \(\alpha\)-stable processes and regional fractional Laplacian
cites work: Some physical applications of fractional Schrödinger equation
cites work: A class of integral equations and approximation of \(p\)-Laplace equations
cites work: Fractional quantum mechanics and Lévy path integrals
cites work: Critical point theory and Hamiltonian systems
cites work: The fractional Schrödinger equation for delta potentials
cites work: Q3749777
cites work: On a class of nonlinear Schrödinger equations
cites work: On concentration of positive bound states of nonlinear Schrödinger equations
cites work: Q3760934
DOI: 10.1007/S00526-014-0778-X
description: scientific article; zbMATH DE number 6481160