Holder regularity of weak solutions to nonlocal p-Laplacian type Schrodinger equations with A_1^p-Muckenhoupt potentials - Yong-Cheol Kim (Electronic Journal of Differential Equations)
In this article, using the De Giorgi-Nash-Moser method, we obtain an interior Holder continuity of weak solutions to nonlocal $p$-Laplacian type Schrodinger equations given by an integro-differential operator $L^p_K$ ($p >1$), $$\displaylines{ L^p_K u+V|u|^{p-2} u=0 \quad\text{in } Ω, \cr u=g \quad \text{in } \mathbb{R}^n\backslash Ω. }$$ Where $V=V_+-V_-$ with $(V_-,V_+)\in L^1_{\rm loc}(\mathbb{R}^n)× L^q_{\rm loc}(\mathbb{R}^n)$ 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 $V_+/b\in L^q_{\rm loc}(\mathbb{R}^n)$, $g\in W^{s,p}(\mathbb{R}^n)$ and $Ω\subset\mathbb{R}^n$ is a bounded domain with Lipschitz boundary. In addition, we prove local boundedness of weak subsolutions of the nonlocal $p$-Laplacian type Schrodinger equations. Also we obtain the logarithmic estimate of the weak supersolutions which play a crucial role in proving Holder regularity of the weak solutions. We note that all the above results also work for a nonnegative potential in $L^q_{\rm loc}(\mathbb{R}^n)$ ($q>\frac{n}{ps}>1$, $ 0< s< 1$).