H\"older regularity of weak solutions to nonlocal $p$-Laplacian type Schr\"odinger equations with $A_1^p$-Muckenhoupt potentials

Abstract

In this paper, using the De Giorgi-Nash-Moser method, we obtain an interior H\"older continuity of weak solutions to nonlocal $p$-Laplacian type Schr\"odinger equations given by an integro-differential operator ${\rm L}^p_K$ ($p>1$) as follows; $$\begin{cases} {\rm L}^p_K u+V|u|^{p-2} u=0 &\text{ in $\Omega$, } u=g &\text{ in ${\Bbb R}^n\setminus\Omega$ } \end{cases}$$ where $V=V_+-V_-$ with $(V_-,V_+)\in L^1_{loc}({\Bbb R}^n)\times L^q_{loc}({\Bbb R}^n)$ for $q>\frac{n}{ps}>1$ and $0 \frac{n}{ps}>1, 0<s<1$).