Hausdorff measure of the free boundary for the $p$-obstacle problem with subcritical exponents

Abstract

This paper investigates a class of $p$-obstacle problems with subcritical exponents having the form \begin{align} \mathrm{div}\left( a(x)|\nabla u|^{p-2}\nabla u\right) =m_1\chi_{\{u>0\}}-m_2u^{\lambda-1}\chi_{\{u>0\}} \ \text{in}\ \Omega,\notag \end{align} where $\Omega $ is a smooth bounded domain in $ \mathbb{R}^N (N \geq 2)$, $m_1,m_2$ are positive constants, the coefficient function $a \in C^2(\Omega)$ has a positive lower bound, and $2 \leq p < \lambda <p^*:= \frac{Np}{N-p}$ when $p<N$ and $N \geq 3$, or $2\leq p < \lambda <+\infty$ when $ N = 2$. By using the mountain-pass lemma, combined with the penalty method, we first establish the existence of non-negative weak solutions. Then, using the De Giorgi-Nash iteration, we prove the $L^\infty$ bound and local $C^{1,\alpha}$ continuity for the solutions. In addition, we prove local porosity of the free boundary based on the optimal growth and non-degeneracy of solutions near the free boundary. Furthermore, by means of Lebesgue measure estimates for gradient level sets, we show that at least one solution corresponds the free boundary having locally finite $(N-1)$-dimensional Hausdorff measure.