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 < infinity, with a locally Lipschitz drift in the plane. To be more precise, let u is an element of W-loc(1,p)(R-2) be a nontrivial weak solution to div(vertical bar del u vertical bar(p-2)del u) + W . (vertical bar del u vertical bar(p-2)del u) = 0 in R-2 where W is a locally Lipschitz real vector satisfying parallel to W parallel to L-q(R-2) <= (M) over tilde for q >= max{p, 2}. Assume that u satisfies certain a priori assumption at 0. For q > max{p, 2} or q = p > 2, if parallel to u parallel to L-infinity(R-2) <= C0, then u satisfies the following asymptotic estimates at inf(vertical bar z0 vertical bar=R vertical bar z-z0 vertical bar<1) sup vertical bar u(z)vertical bar >= e(-CR1-2/q log R), where C > 0 depends only on p, q, (M) over tilde and C-0. When q = max{p, 2} and p is an element of(1, 2], if vertical bar u(z)vertical bar <= vertical bar z vertical bar(m) for vertical bar z vertical bar > 1 with some m > 0, then we have inf(vertical bar z0 vertical bar=R vertical bar z-z0 vertical bar<1) sup vertical bar u(z)vertical bar >= C(1)e(-C2(logR)2), where C-1 > 0 depends only on m, p and C-2 > 0 depends on m, p, (M) over tilde. 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. (C) 2016 Elsevier Ltd. All rights reserved.