Remarks on the antimaximum principle

Abstract

We present several observations on the antimaximum principle (AMP) for the model problem $-\Delta_p u = \lambda |u|^{p-2} u + f$ in a bounded smooth domain $\Omega$, subject to the zero Dirichlet boundary conditions, and where the source function $f$ is nontrivial, nonnegative, and sufficiently regular. Denote by $\lambda_f$ the endpoint of validity of the AMP, so that every solution of the problem is negative in $\Omega$ for any $\lambda \in (\lambda_1,\lambda_f)$. Our discussion covers the following aspects: identification of a class of sources over which the AMP is uniform, lower semicontinuity of the map $f \mapsto \lambda_f$, bounds on $\lambda_f$, the nonexistence of negative solutions for sufficiently large $\lambda$ (extended AMP), the anticomparison principle, and the weakening of the source regularity from the Lebesgue to Morrey spaces. Some of the results are stated only in the linear case $p=2$. As a part of the discussion, we provide a few related open problems.