What question did this study set out to answer?

This research aims to establish the existence and uniqueness of positive solutions for semilinear elliptic equations with changing sign nonlinearities on unbounded domains.

May 15, 2026Open Access

Positive Solutions of Semilinear Elliptic Equations with Sign-Changing Nonlinearities on Unbounded Domains in R2

Key Points

This research aims to establish the existence and uniqueness of positive solutions for semilinear elliptic equations with changing sign nonlinearities on unbounded domains.
Analyzed semilinear elliptic equations on unbounded domains D in R2.
Utilized a contraction mapping principle in a weighted Banach space.
Examined a reaction term satisfying a global Lipschitz condition and a Kato class K∞(D).
Demonstrated existence and uniqueness of a positive continuous solution.
Provided two-sided global estimates for the solution's asymptotic behavior.
Extended results to general domains without requiring radial weight functions.

Abstract

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.

Read Full Paperexternally

Perguntar à IA

Bookmark

View Full Paper