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

• Published in: Electronic journal of differential equations Vol. 2025; no. 1-??; p. 83
• Main Author: Kim, Yong-Cheol
• Format: Journal Article
• Language: English
• Published: 08.08.2025
• ISSN: 1072-6691; 1072-6691
• DOI: 10.58997/ejde.2025.83

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 \((V_-,V_+)\in L^1_{loc}(\mathbb{R}^n)\times L^q_{loc}(\mathbb{R}^n)\) 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 \(V_+/b\in L^q_{ loc}(\mathbb{R}^n)\), \(g\in W^{s,p}(\mathbb{R}^n)\) 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 \(L^q_{loc}(\mathbb{R}^n)\) (\(q >\frac{n}{ps} >1\), \(0< s< 1\)). For more information see https://ejde.math.txstate.edu/Volumes/2025/83/abstr.html