The Singular Perturbation Problem for a Class of Generalized Logistic Equations Under Non-classical Mixed Boundary Conditions

Abstract

This paper studies a singular perturbation result for a class of generalized diffusive logistic equations, d⁢ℒ⁢u=u⁢h⁢(u,x){d\mathcal{L}u=uh(u,x)}, under non-classical mixed boundary conditions, ℬ⁢u=0{\mathcal{B}u=0} on ∂⁡Ω{\partial\Omega}. Most of the precursors of this result dealt with Dirichlet boundary conditions and self-adjoint second order elliptic operators. To overcome the new technical difficulties originated by the generality of the new setting, we have characterized the regularity of ∂⁡Ω{\partial\Omega} through the regularity of the associated conormal projections and conormal distances. This seems to be a new result of a huge relevance on its own. It actually complements some classical findings of Serrin, [39], Gilbarg and Trudinger, [21], Krantz and Parks, [27], Foote, [18] and Li and Nirenberg [28] concerning the regularity of the inner distance function to the boundary.

Keywords

• singular perturbation
• positive solution
• generalized logistic equation
• non-classical mixed boundary conditions
• conormal vector field
• conormal projection
• conormal distance
• regularity
• 35b25
• 35b09
• 35j25
• 35q92