Electronic Journal of Differential Equations (Mar 2020)
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 $$\displaylines{ -\Delta_p u + |u|^{p-2}u = 0 \quad \text{in }\Omega, \cr |\nabla u|^{p-2}\frac{\partial u}{\partial \nu} = \lambda m(x)|u|^{p-2}u + h(x) \quad\text{on }\partial\Omega, }$$ where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, 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.
