Department of Mathematics Education
Korea University
Seoul 02841, Republic of Korea
email: ychkim@korea.ac.kr
Yong-Cheol Kim
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 8, 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.
DOI: 10.58997/ejde.2025.83
Show me the PDF file (509 KB), TEX file for this article.
|
Yong-Cheol Kim Department of Mathematics Education Korea University Seoul 02841, Republic of Korea email: ychkim@korea.ac.kr |
|---|
Return to the EJDE web page