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arXiv · 2410.09482

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

Giulio Ciraolo·Xiaoliang Li
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Abstract

In this paper, we study positive solutions $u$ of the homogeneous Dirichlet problem for the $p$-Laplace equation $-Δ_p \,u=f(u)$ in a bounded domain $Ω\subset\mathbb{R}^N$, where $N\ge 2$, $1<p<+\infty$ 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 $Ω$ 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 $Ω$.

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Giulio Ciraolo, Xiaoliang Li. 2024-10-12. A quantitative symmetry result for $p$-Laplace equations with discontinuous nonlinearities. https://arxiv.org/abs/2410.09482

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