Quantitative uniqueness estimates for pp-Laplace type equations in the plane

In this article our main concern is to prove the quantitative unique estimates for the pp-Laplace equation, 1max{p,2}q>max{p,2} or q=p>2q=p>2, if ‖u‖L∞(R2)≤C0‖u‖L∞(R2)≤C0, then uu satisfies the following asymptotic estimates at R≫1R≫1inf|z0|=Rsup|z−z0|<1|u(z)|≥e−CR1−2qlogR, where C>0C>0 depends only on pp, qq, M̃ and C0C0. When q=max{p,2}q=max{p,2} and p∈(1,2]p∈(1,2], if |u(z)|≤|z|m|u(z)|≤|z|m for |z|>1|z|>1 with some m>0m>0, then we have inf|z0|=Rsup|z−z0|<1|u(z)|≥C1e−C2(logR)2, where C1>0C1>0 depends only on m,pm,p and C2>0C2>0 depends on m,p,M̃. 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 pp-Laplace equation with a locally positive locally Lipschitz weight.