We establish the existence and uniqueness of a positive continuous solution for some semilinear elliptic equations on an unbounded domain D in R2 with a nonempty compact boundary. The reaction term is allowed to change sign and satisfies a global Lipschitz condition with respect to the second variable, weighted by a function in a suitable Kato class K∞(D). Furthermore, we provide explicit two-sided global estimates that characterize the asymptotic behavior of the solution. Our approach is based on properties of the class K∞(D) and a contraction mapping principle in a weighted Banach space. Our results complete existing work in the literature by considering more general domains and not necessarily radial weight functions.
Bachar et al. (Wed,) studied this question.