Maximum and antimaximum principles for the p-Laplacian with weighted Steklov boundary conditions

Abstract

We study the maximum and antimaximum principles for the p-Laplacian operator under Steklov boundary conditions with an indefinite weight -Δpu + |u|p-2 u = 0 in Ω, |∇u|p-2 ∂u/∂v = λm(x)|u|p-2 u + h(x) on ∂Ω, where Ω is a smooth bounded domain of ℝN, N > 1. After reviewing some elementary properties of the principal eigenvalues of the p-Laplacian under Steklov boundary conditions with an indefinite weight, we investigate the maximum and antimaximum principles for this problem. Also we give a characterization for the interval of the validity of the uniform antimaximum principle.