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

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

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 $\mathfrak{e}_V$.
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 NezzaG. 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. KuusiG. 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 NezzaG. 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. KuusiG. 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. 
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