A further remark on the density estimate for degenerate Allen-Cahn equations: ${\mathrm{\Delta }}_p$-type equations for $1<p<\frac{n}{n-1}$ with rough coefficients

In this short remark on a previous paper ^[1], we continue the study of Allen-Cahn equations associated with Ginzburg-Landau energies

$$J(v,\mathrm{\Omega })={\int }_{\mathrm{\Omega }}\{F(\mathrm{\nabla }v,v,x)+W(v,x)\}dx,$$

involving a Dirichlet energy $F(\overrightarrow{\xi },\tau ,x)\sim |\overrightarrow{\xi }{|}^p$ and a degenerate double-well potential $W(\tau ,x)\sim (1-{\tau }^2{)}^m$. In contrast to ^[1], we remove all regularity assumptions on the Ginzburg-Landau energy. Then, with further assumptions that $1<p<\frac{n}{n-1}$ and that $W(\tau ,x)$ is monotone in $\tau$ on both sides of $0$, we establish a density estimate for the level sets of nontrivial minimizers $|u|\le 1$.