Characterizing the existence of large solutions for a class of sublinear problems with nonlinear diffusion

Abstract

In this paper we characterize the existence of large solutions for a general class of sublinear elliptic problems of logistic type related to the porous media equation. Our main result shows that large solutions do exist if, and only if, the nonlinear diffusion is not too large. As a byproduct of the general theory developed by the authors in [3], those large solutions must be unstable with respect to the positive solutions of the parabolic counterpart of the elliptic model. This seems to be the first result of this nature available in the literature. Most precisely, as the diffusion becomes non-linear the metasolutions become unstable, so arising a classical steady-state gaining the stability lost by the metasolution. In particular, a dynamical bifurcation occurs from the linear diffusion case.