Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 83, pp. 1-26.

Title: Holder regularity of weak solutions to nonlocal p-Laplacian type Schrodinger  equations with A_1^p-Muckenhoupt potentials

Author: Yong-Cheol Kim (Korea Institute for Advanced Study, Korea)

Abstract:
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 $L^p_K$
($p >1$),
$$\displaylines{
 L^p_K u+V|u|^{p-2} u=0 \quad\text{in } \Omega, \cr
u=g \quad \text{in } \mathbb{R}^n\backslash \Omega.
}$$
Where $V=V_+-V_-$ with $(V_-,V_+)\in L^1_{\rm loc}(\mathbb{R}^n)\times L^q_{\rm loc}(\mathbb{R}^n)$
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 $\Omega\subset\mathbb{R}^n$ 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$).

Submitted February 21, 2025. Published August 08, 2025.
Math Subject Classifications: 47G20, 45K05, 35J60, 35B65, 35D10, 60J75.
Key Words: Holder regularity; nonlocal p-Laplacian type Schrodinger equation;
 A_1^p-Muckenhoupt potential; De Giorgi-Nash-Moser method.