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

Woocheol Choi; Yong-Cheol Kim

doi:10.3934/cpaa.2018095

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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, L^p-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 2^nd 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, L^p-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 2^nd 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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