Holder regularity of weak solutions to the quasi-linear elliptic equations involving a generalized variable exponent Laplacian
Applicable Nonlinear Analysis, Volume 2, Issue 3, December 2025, Pages: 200–206
MYKOLA IVANOVICH YAREMENKO
National Technical University of Ukraine, “Igor Sikorsky Kyiv Polytechnic Institute” Kyiv, Ukraine, 37, Prospect Beresteiskyi (former Peremohy), Kyiv, Ukraine
Abstract
We prove the Holder continuity of a weak solution \textbf{\textit{\(u\in W_{1}^{p\left(\cdot \right)} \left(\Omega \right)\)}} to quasi-linear elliptic equations in the diverged form \textbf{\textit{\(div\left(A\left(x,\; u,\; \nabla u\right)\right)+b\left(x,\; u,\; \nabla u\right)=0\)}}, in the regular bounded domain \textbf{\textit{\(\Omega \) }}in \textbf{\textit{\(R^{l} \)}} for \textbf{\textit{\(l\ge 3\)}}, where functions \textbf{\textit{\(A=a_{i} \left(x,\; u,\; k\right)\)}} and \textbf{\textit{\(b\left(x,\; u,\; k\right)\)}} are correctly defined for all \(x\in clos\left(\Omega \right)\) and every \textbf{\textit{\(u\)}}, \textbf{\textit{\(k\)}}, and \textbf{\textit{\(A\)}}, \textbf{\textit{\(b\) }}are measurable. We establish the conditions on coefficients \textbf{\textit{\(A\)}} and \textbf{\textit{\(b\)}} under which the weak solution \textbf{\textit{\(u\in W_{1}^{p\left(\cdot \right)} \left(\Omega \right)\) }} belongs to \(C_{0,\; \alpha } \left(\Omega \right)\) for certain \(\alpha \) depending only on \(M\), \(\nu \left(M\right)\), \(\mu \left(M\right)\), \(p\left(\cdot \right)\), and \(\left|\nabla p\right|\) , where \textbf{\textit{\(ess{\mathop{\max }\limits_{\Omega }} \left|u\right|\le M<\infty \)}}.
Cite this Article as
Mykola Ivanovich Yaremenko, Holder regularity of weak solutions to the quasi-linear elliptic equations involving a generalized variable exponent Laplacian, Applicable Nonlinear Analysis, 2(3), 200–206, 2025