American Institute of Mathematical Sciences

Advanced
September  2018, 17(5): 1993-2010. doi: 10.3934/cpaa.2018095

The Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators with certain potentials

Woocheol Choi 1, and  Yong-Cheol Kim 2,,

1. 

Department of Mathematics Education, Incheon National University, Incheon 22012, Republic of Korea
2. 

Department of Mathematics Education, Korea University, Seoul 02841, Republic of Korea

* Corresponding author: Yong-Cheol Kim

Received  August 2017 Revised  January 2018 Published  April 2018

In this paper, we prove the Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators $L_K+V$ with nonnegative potentials $V∈ L^q_{\rm{loc}}(\mathbb{R}^n)$ for $q>\frac{n}{2s}$ with $0 < s < 1$ and $n>2s$; that is to say, we obtain the existence of a fundamental solution $\mathfrak{e}_V$ for $L_K+V$ satisfying
$\begin{equation*}\bigl(L_K+V\bigr)\mathfrak{e}_V = \delta _0\,\,\text{ in $\mathbb{R}^n$ }\end{equation*}$
in the distribution sense, where $\delta _0$ denotes the Dirac delta mass at the origin. In addition, we obtain a decay of the fundamental solution $\mathfrak{e}_V$.
Keywords: Fundamental solution, nonlocal Schrödinger operators, potential.
Mathematics Subject Classification: Primary: 47G20, 45K05, 35J60, 35B65, 35D30; Secondary: 60J75.
Citation: Woocheol Choi, Yong-Cheol Kim. The Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators with certain potentials. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1993-2010. doi: 10.3934/cpaa.2018095
References:
[1]

C. Bucur, Some observations on the Green function for the ball in the fractional Laplace framework, Comm. Pure and Appl. Anal., 15 (2016), 657-699. 

[2]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. P.D.E., 32 (2007), 1245-1260. 

[3]

W. Choi and Y.-C. Kim, Lp-mapping properties for nonlocal Schrödinger operators with certain potential, preprint, arXiv: math/0605406.

[4]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. 

[5]

L. Ehrenpreis, Solution of some problems of division. Ⅰ. Division by a polynomial of derivation, Amer. J. Math., 76 (1954), 883-903. 

[6]

L. Ehrenpreis, Solution of some problems of division. Ⅰ. Division by a polynomial of derivation, Amer. J. Math., 77 (1955), 286-292. 

[7]

M. Fall and T. Weth, Liouville theorems for a general class of nonlocal operators, Potential. Anal., 45 (2016), 187-200. 

[8]

Q. Han and F. Lin, Elliptic Partial Differential Equations Courant Lecture Notes in Mathematics, American Mathematical Society, 1997.

[9]

T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317-1368. 

[10] N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York, 1972. 
[11]

B. Malgrange, Existence et approximation des solutions des équations aux d'erivées partielles et des équations de convolution, Ann. Inst. Fourier, 6 (1955/56), 271-355. 

[12] M. Reed and B. Simon, Functional Analysis I,Methods of Modern Mathematical Physics, Academic Press, 1970. 
[13]

W. Rudin, Functional Analysis 2nd edition, International Series in Pure and Applied Mathematics. McGraw-Hill, 1991.

[14]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. 

[15] E. M. Stein, Singular Integrals and Differentiability, Princeton Univ. Press, 1970. 

show all references

References:
[1]

C. Bucur, Some observations on the Green function for the ball in the fractional Laplace framework, Comm. Pure and Appl. Anal., 15 (2016), 657-699. 

[2]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. P.D.E., 32 (2007), 1245-1260. 

[3]

W. Choi and Y.-C. Kim, Lp-mapping properties for nonlocal Schrödinger operators with certain potential, preprint, arXiv: math/0605406.

[4]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. 

[5]

L. Ehrenpreis, Solution of some problems of division. Ⅰ. Division by a polynomial of derivation, Amer. J. Math., 76 (1954), 883-903. 

[6]

L. Ehrenpreis, Solution of some problems of division. Ⅰ. Division by a polynomial of derivation, Amer. J. Math., 77 (1955), 286-292. 

[7]

M. Fall and T. Weth, Liouville theorems for a general class of nonlocal operators, Potential. Anal., 45 (2016), 187-200. 

[8]

Q. Han and F. Lin, Elliptic Partial Differential Equations Courant Lecture Notes in Mathematics, American Mathematical Society, 1997.

[9]

T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317-1368. 

[10] N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York, 1972. 
[11]

B. Malgrange, Existence et approximation des solutions des équations aux d'erivées partielles et des équations de convolution, Ann. Inst. Fourier, 6 (1955/56), 271-355. 

[12] M. Reed and B. Simon, Functional Analysis I,Methods of Modern Mathematical Physics, Academic Press, 1970. 
[13]

W. Rudin, Functional Analysis 2nd edition, International Series in Pure and Applied Mathematics. McGraw-Hill, 1991.

[14]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. 

[15] E. M. Stein, Singular Integrals and Differentiability, Princeton Univ. Press, 1970. 
[1]

Mouhamed Moustapha Fall, Veronica Felli. Unique continuation properties for relativistic Schrödinger operators with a singular potential. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5827-5867. doi: 10.3934/dcds.2015.35.5827
[2]

Younghun Hong. Strichartz estimates for $N$-body Schrödinger operators with small potential interactions. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5355-5365. doi: 10.3934/dcds.2017233
[3]

Michael Goldberg. Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 109-118. doi: 10.3934/dcds.2011.31.109
[4]

Yu Su. Ground state solution of critical Schrödinger equation with singular potential. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3347-3371. doi: 10.3934/cpaa.2021108
[5]

Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control and Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119
[6]

Jin-Cai Kang, Xiao-Qi Liu, Chun-Lei Tang. Ground state sign-changing solution for Schrödinger-Poisson system with steep potential well. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022112
[7]

Woocheol Choi, Yong-Cheol Kim. $L^p$ mapping properties for nonlocal Schrödinger operators with certain potentials. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5811-5834. doi: 10.3934/dcds.2018253
[8]

István Győri, László Horváth. On the fundamental solution and its application in a large class of differential systems determined by Volterra type operators with delay. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1665-1702. doi: 10.3934/dcds.2020089
[9]

Grégoire Allaire, M. Vanninathan. Homogenization of the Schrödinger equation with a time oscillating potential. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 1-16. doi: 10.3934/dcdsb.2006.6.1
[10]

Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003
[11]

Zhiyan Ding, Hichem Hajaiej. On a fractional Schrödinger equation in the presence of harmonic potential. Electronic Research Archive, 2021, 29 (5) : 3449-3469. doi: 10.3934/era.2021047
[12]

Hiroshi Isozaki, Hisashi Morioka. A Rellich type theorem for discrete Schrödinger operators. Inverse Problems and Imaging, 2014, 8 (2) : 475-489. doi: 10.3934/ipi.2014.8.475
[13]

Jaime Cruz-Sampedro. Schrödinger-like operators and the eikonal equation. Communications on Pure and Applied Analysis, 2014, 13 (2) : 495-510. doi: 10.3934/cpaa.2014.13.495
[14]

Fritz Gesztesy, Roger Nichols. On absence of threshold resonances for Schrödinger and Dirac operators. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3427-3460. doi: 10.3934/dcdss.2020243
[15]

Fengping Yao. Optimal regularity for parabolic Schrödinger operators. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1407-1414. doi: 10.3934/cpaa.2013.12.1407
[16]

Jean Bourgain. On quasi-periodic lattice Schrödinger operators. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 75-88. doi: 10.3934/dcds.2004.10.75
[17]

Jean Bourgain. On random Schrödinger operators on $\mathbb Z^2$. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 1-15. doi: 10.3934/dcds.2002.8.1
[18]

Mouhamed Moustapha Fall. Regularity estimates for nonlocal Schrödinger equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1405-1456. doi: 10.3934/dcds.2019061
[19]

Hongwei Wang, Amin Esfahani. On the Cauchy problem for a nonlocal nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022039
[20]

GUANGBING LI. Positive solution for quasilinear Schrödinger equations with a parameter. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1803-1816. doi: 10.3934/cpaa.2015.14.1803

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (247)
  • HTML views (216)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy