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.

19 pages