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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 & 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.   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

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

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

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

[14]

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

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

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.   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

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

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

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

[14]

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

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