Fully nonlinear logistic equations with sanctuary

For the fully nonlinear stationary logistic equation F(x,D²u)+μ u=k(x)uᵖ F ( x , D 2 u ) + μ u = k ( x ) u p with $$p>1$$ p > 1 and k(x)≥ 0 k ( x ) ≥ 0 , in a bounded domain with Dirichlet boundary condition, we determine, in terms of μ μ , the existence and uniqueness or the nonexistence of a positive solution. Furthermore, we study the asymptotic behavior of the solutions when μ μ approaches the boundary points of the existence range.