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

TL;DR

This paper proves interior Hölder continuity and local boundedness of weak solutions to nonlocal p-Laplacian Schrödinger equations with potentials in the Muckenhoupt class, extending regularity results to broader potential classes.

Contribution

It establishes Hölder regularity and boundedness for solutions with potentials in the A_1-Muckenhoupt class, using a novel logarithmic estimate approach.

Findings

Weak solutions are Hölder continuous inside the domain.

Weak subsolutions are locally bounded.

Logarithmic estimates for supersolutions are derived.

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 $L^q_{loc}({\Bbb R}^n)$ ($q>\frac{n}{ps}>1, 0<s<1$).