A quantitative symmetry result for $p$-Laplace equations with discontinuous nonlinearities

In this paper, we study positive solutions $u$ of the homogeneous Dirichlet problem for the $p$-Laplace equation $-{\mathrm{\Delta }}_{p}u=f(u)$ in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$, where $N\ge 2$, $1<p<+\mathrm{\infty }$ and $f$ is a discontinuous function. We address the quantitative stability of a Gidas-Ni-Nirenberg type symmetry result for $u$, which was established by Lions and Serra when $\mathrm{\Omega }$ is a ball. By exploiting a quantitative version of the Pólya-Szegö principle, we prove that the deviation of $u$ from its Schwarz symmetrization can be estimated in terms of the isoperimetric deficit of $\mathrm{\Omega }$.