Past and recent contributions to indefinite sublinear elliptic problems

Abstract

We review the indefinite sublinear elliptic equation $-\Delta u=a(x)u^q$ in a smooth bounded domain $\Omega \subset {\mathbb{R}}^N$, with Dirichlet or Neumann homogeneous boundary conditions. Here $0<q<1$ and $a$ is continuous and changes sign, in which case the strong maximum principle does not apply. As a consequence, the set of nonnegative solutions of these problems has a rich structure, featuring in particular both dead core and/or positive solutions. Overall, we are interested in sufficient and necessary conditions on $a$ and $q$ for the existence of positive solutions. We describe the main results from the past decades, and combine it with our recent contributions. The proofs are briefly sketched.

24 pages, 10 figures