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 $0<s<1$ is a potential such that $(V_-,V_+^{b,i})$ belongs to the $(A_1,A_1)$-Muckenhoupt class and $V_+^{b,i}$ is in the $A_1$-Muckenhoupt class for all $i\in {\Bbb N}$ ( here, $V_+^{b,i}:=V_+\max\{b,1/i\}/b$ for an almost everywhere positive bounded function $b$ on ${\Bbb R}^n$ with $V_+/b\in L^q_{loc}({\Bbb R}^n)$, $g\in W^{s,p}({\Bbb R}^n)$ and $\Omega\subset{\Bbb R}^n$ is a bounded domain with Lipschitz boundary.) In addition, we get the local boundedness of weak subsolutions of the nonlocal $p$-Laplacian type Schr\"odinger equations. In a different way from \cite{DKP1}, we obtain the logarithmic estimate of the weak supersolutions which play a crucial role in proving the H\"older regularity of the weak solutions. In particular, we note that all the above results are still working for any nonnegative potential in $L^q_{loc}({\Bbb R}^n)$ ($q>\frac{n}{ps}>1, 0<s<1$).