Preprint

Inserted: 19 oct 2026
Last Updated: 19 oct 2026

Year: 2026

Abstract:

In this paper, we study the higher Hölder regularity of a solution to the following $p-$Laplacian minimization problem with singular term \[ \min \bigg\{\int_\Omega w\,|\nabla u|^p \,+\, f\,u^\gamma\,:\,u \in W^{1,p}(\Omega),\,\,u \geq 0,\,\,u=\phi \,\,\,\mbox{on}\,\,\partial\Omega \bigg\} \] where $0<\gamma<1$, $1<p<\infty$, $f$ is a positive function in $L^q(\Omega)$ with $q > \max\{\frac{p}{p-\gamma},\frac{N}{p}\}$, $\phi$ is a nonnegative functions in $W^{1,p}(\Omega)$, and $w \in C^2(\Omega)$ with $w \geq c_w >0$. More precisely, we will show the following statements: \[ \begin{cases} f \in L^q(\Omega) \Rightarrow u \in C^{0,\frac{p-\frac{N}{q}}{p-\gamma}}_{loc}(\Omega),\qquad & \,\,\,\,\,q \leq \frac{N}{\gamma},\\ f \in L^q(\Omega) \Rightarrow u \in C^{1,\min\bigg\{\alpha_\star,\frac{\gamma-\frac{N}{q}}{p-\gamma}\bigg\}}_{loc}(\Omega),\qquad & q > \frac{N}{\gamma},\,\,\,\,\,\,\,\,\,p \geq 2,\\ f \in L^q(\Omega) \Rightarrow u \in C^{1,\min\bigg\{\alpha_\star,\frac{\gamma-\frac{N}{q}}{2-\gamma}\bigg\}}_{loc}(\Omega),\qquad & q > \frac{N}{\gamma},\,\,\,\,\,\,\,1<p < 2. \end{cases} \] Here, we denotes by $\alpha_\star$ a constant such that any solution to the weighted $p-$Laplacian equation $\nabla \cdot [w
\nabla u
^{p-2}\nabla u]=0$ is locally of class $C^{1,\alpha_\star}$. If $q >\frac{N}{\gamma}$, we will show that $u$ has more regularity on the free boundary $\partial\{u>0\}$, it is of class $C^{1,\alpha}$ with $\alpha=\min\bigg\{\frac{\gamma - \frac{N}{q}}{p-\gamma},1\bigg\}$.

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• p-Laplacian with singular term