Hölder regularity of weak solutions to nonlocal $p$-Laplacian type Schrödinger equations with $A_1^p$-Muckenhoupt potentials

Abstract

In this paper, using the De Giorgi-Nash-Moser method, we obtain an interior Hölder continuity of weak solutions to nonlocal $p$-Laplacian type Schrödinger equations given by an integro-differential operator ${\mathrm{L}}_K^p$ ($p>1$) as follows; $$\{\begin{array}{llc}{\mathrm{L}}_K^{p}u+V|u{|}^{p-2}u=0& \text{ in Ω, }u=g& \text{ in Rn∖Ω }\end{array}$$ where $V=V_{+}-V_{-}$ with $(V_{-},V_{+})\in L_{loc}^1({\mathbb{R}}^n)\times L_{loc}^q({\mathbb{R}}^n)$ 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 $i\in \mathbb{N}$ ( here, $V_{+}^{b,i}:=V_{+}\max \{b,1/i\}/b$ for an almost everywhere positive bounded function $b$ on ${\mathbb{R}}^n$ with $V_{+}/b\in L_{loc}^q({\mathbb{R}}^n)$, $g\in W^{s,p}({\mathbb{R}}^n)$ and $\Omega \subset {\mathbb{R}}^n$ is a bounded domain with Lipschitz boundary.) In addition, we get the local boundedness of weak subsolutions of the nonlocal $p$-Laplacian type Schrödinger 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 Hölder regularity of the weak solutions. In particular, we note that all the above results are still working for any nonnegative potential in $L_{loc}^q({\mathbb{R}}^n)$ ($q>\frac{n}{ps}>1,0<s<1$).