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

In this article our main concern is to prove the quantitative unique estimates for the $p$-Laplace equation, $1<p<\mathrm{\infty }$, with a locally Lipschitz drift in the plane. To be more precise, let $u\in W_{loc}^{1,p}({\mathbb{R}}^2)$ be a nontrivial weak solution to $$\text{div}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u)+W\cdot (\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u)=0\text{ in }{\mathbb{R}}^2,$$ where $W$ is a locally Lipschitz real vector satisfying $|W|_{L^q($\mathbb{R}$^2)}\leq \tilde{M}$for$q\geq \max{p,2}$\mathrm{.}Assumethat$u$ satisfies certain a priori assumption at 0. For $q>\max \{p,2\}$ or $q=p>2$, if $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathbb{R}}^2)}\le C_0$, then $u$ satisfies the following asymptotic estimates at $R\gg 1$ $$\underset{\mathrm{\mid }{z}_0\mathrm{\mid }=R}{\inf }\underset{\mathrm{\mid }z-z_0\mathrm{\mid }<1}{\sup }\mathrm{\mid }u(z)\mathrm{\mid }\ge e^{-C{R}^{1-\frac{2}{q}}\log R},$$ 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 $$\underset{\mathrm{\mid }{z}_0\mathrm{\mid }=R}{\inf }\underset{\mathrm{\mid }z-z_0\mathrm{\mid }<1}{\sup }\mathrm{\mid }u(z)\mathrm{\mid }\ge R^{-C},$$ 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.