Past and recent contributions to indefinite sublinear elliptic problems

Abstract

We review the inde nite sublinear elliptic equation &Delta;u =a(x)u^q in a smooth bounded domain &Omega;CR^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 su_x000E_cient and necessary conditions on a and q for the existence of positive solu- tions. We describe the main results from the past decades, and combine it with our recent contributions. The proofs are briefly sketched.