Boundary Behavior and Estimates for Solutions of Equations Containing the p-Laplacian
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We use ``Hardy-type'' inequalities to derive Lq estimates for solutions of equations containing the p-Laplacian with p>1. We begin by deriving some inequalities using elementary ideas from an early article [B3] which has been largely overlooked. Then we derive Lq estimates of the boundary behavior of test functions of finite energy, and consequently of principal (positive) eigenfunctions of functionals containing the p-Laplacian. The estimates contain exponents known to be sharp when p=2. These lead to estimates of the effect of boundary perturbation on the fundamental eigenvalue. Finally, we present global Lq estimates of solutions of the Cauchy problem for some initial-value problems containing the p-Laplacian.