English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, we consider nonlocal Schr\"odinger equations with certainpotentials $V$ given by an integro-differential operator $L_K$ as follows;\begin{equation*}L_K u+V u=f\,\,\text{ in $\BR^n$ }\end{equation*} where$V\in\rh^q$ for $q>\f{n}{2s}$ and $0<s<1$. We denote the solution of the aboveequation by $\cS_V f:=u$, which is called {\it the inverse of the nonlocalSchr\"odinger operator $L_K+V$ with potential $V$}; that is,$\cS_V=(L_K+V)^{-1}$. Then we obtain a weak Harnack inequality of weaksubsolutions of the nonlocal equation \begin{equation}\begin{cases}L_K u+Vu=0\,\,&\text{ in $\Om$,}  \quad u=g\,\,&\text{ in $\BR^n\s\Om$,} \end{cases}\end{equation} where $g\inH^s(\BR^n)$ and $\Om$ is a bounded open domain in $\BR^n$ with Lipschitzboundary, and also get an improved decay of a fundamental solution $\fe_V$ for$L_K+V$. Moreover, we obtain $L^p$ and $L^p-L^q$ mapping properties of theinverse $\cS_V$ of the nonlocal Schr\"odinger operator $L_K+V$.

$L^p$ mapping properties for nonlocal Schr\"odinger operators with certain potential