Fully nonlinear logistic equations with sanctuary

Abstract

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