Fully nonlinear logistic equations with sanctuary

Abstract

For the fully nonlinear stationary logistic equation $\mathcal{F}(x,D^{2}u)+\hat{I}¼u=k(x)u^p$ with $p>1$ and $k(x)\ge 0$, in a bounded domain with Dirichlet boundary condition, we determine, in terms of $\hat{I}¼$, the existence and uniqueness or the nonexistence of a positive solution. Furthermore, we study the asymptotic behavior of the solutions when $\hat{I}¼$ approaches the boundary points of the existence range.

19 pages