Semilinear elliptic equations on rough domains

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