Fully nonlinear logistic equations with sanctuary

Abstract

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

Theorems & Definitions (33)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 4
• Proposition 5
• Corollary 6
• proof
• Proposition 7
• Proposition 8
• proof