  The paper makes use of recent results in the theory of Banach lattices and
positive operators to deal with abstract semilinear equations. The aim is to
work with minimal or no regularity conditions on the boundary of the domains,
where the usual arguments based on maximum principles do not apply. A key
result is an application of Kato's inequality to prove a comparison theorem for
eigenfunctions that only requires interior regularity and avoids the use of the
Hopf boundary maximum principle. We demonstrate the theory on an abstract
degenerate logistic equation by proving the existence, uniqueness and stability
of non-trivial positive solutions. Examples of operators include the Dirichlet
Laplacian on arbitrary bounded domains, a simplified construction of the Robin
Laplacian on arbitrary domains with boundary of finite measure and general
elliptic operators in divergence form.
