# American Institute of Mathematical Sciences

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 & Applied Analysis, 2018, 17 (5) : 1993-2010. doi: 10.3934/cpaa.2018095
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##### References:
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