Quantitative uniqueness estimates for $p$-Laplace type equations in the plane

Abstract

In this article our main concern is to prove the quantitative unique estimates for the $p$-Laplace equation, $1<p<\infty$, with a locally Lipschitz drift in the plane. To be more precise, let $u\in W^{1,p}_{loc}(\mathbb{R}^2)$ be a nontrivial weak solution to \[ \text{div}(|\nabla u|^{p-2} \nabla u) + W\cdot(|\nabla u|^{p-2}\nabla u) = 0 \ \text{ in }\ \mathbb{R}^2, \] where $W$ is a locally Lipschitz real vector satisfying $\|W\|_{L^q(\mathbb{R}^2)}\leq \tilde{M}$ for $q\geq \max\{p,2\}$. Assume that $u$ satisfies certain a priori assumption at 0. For $q>\max\{p,2\}$ or $q=p>2$, if $\|u\|_{L^\infty(\mathbb{R}^2)}\leq C_0$, then $u$ satisfies the following asymptotic estimates at $R\gg 1$ \[ \inf_{|z_0|=R}\sup_{|z-z_0|<1} |u(z)| \geq e^{-CR^{1-\frac{2}{q}}\log R}, \] where $C$ depends only on $p$, $q$, $\tilde{M}$ and $C_0$. When $q=\max\{p,2\}$ and $p\in (1,2]$, under similar assumptions, we have \[ \inf_{|z_0|=R} \sup_{|z-z_0|<1} |u(z)| \geq R^{-C}, \] where $C$ depends only on $p$, $\tilde{M}$ and $C_0$. As an immediate consequence, we obtain the strong unique continuation principle (SUCP) for nontrivial solutions of this equation. We also prove the SUCP for the weighted $p$-Laplace equation with a locally positive locally Lipschitz weight.