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RESEARCH PRODUCT

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

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Abstract In this article our main concern is to prove the quantitative unique estimates for the p -Laplace equation, 1 p ∞ , with a locally Lipschitz drift in the plane. To be more precise, let u ∈ W l o c 1 , p ( R 2 ) be a nontrivial weak solution to div ( | ∇ u | p − 2 ∇ u ) + W ⋅ ( | ∇ u | p − 2 ∇ u ) = 0  in  R 2 , where W is a locally Lipschitz real vector satisfying ‖ W ‖ L q ( R 2 ) ≤ M 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 ‖ u ‖ L ∞ ( R 2 ) ≤ C 0 , then u satisfies the following asymptotic estimates at R ≫ 1 inf | z 0 | = R sup | z − z 0 | 1 | u ( z ) | ≥ e − C R 1 − 2 q log R , where C > 0 depends only on p , q , M and C 0 . When q = max { p , 2 } and p ∈ ( 1 , 2 ] , if | u ( z ) | ≤ | z | m for | z | > 1 with some m > 0 , then we have inf | z 0 | = R sup | z − z 0 | 1 | u ( z ) | ≥ C 1 e − C 2 ( log R ) 2 , where C 1 > 0 depends only on m , p and C 2 > 0 depends on m , p , M . 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.