Existence of Positive Solutions to Semilinear Equations with Sublinear Nonlinearities and Compact Positivity-Improving Resolvent

TL;DR

This paper establishes a general existence theorem for positive solutions to semilinear equations with sublinear nonlinearities, using an abstract operator framework with a compact positivity-improving resolvent.

Contribution

It extends classical elliptic theory results to an abstract setting without requiring additional regularizing properties of the resolvent.

Findings

Proves a Brézis--Oswald type existence theorem for positive solutions.

Develops a method combining sub- and supersolutions with spectral ideas.

Bridges order-theoretic methods and energy-based theory for sublinear problems.

Abstract

We prove a Br\'ezis--Oswald type existence theorem for positive solutions of semilinear equations in an abstract setting in which the underlying linear operator has a compact positivity-improving resolvent. The assumptions imposed on the sublinear nonlinearity are comparable to those used in the classical elliptic theory, but no additional regularizing properties of the resolvent, such as ultracontractivity or smoothing, are required. The proof combines the method of sub- and supersolutions with spectral ideas of Br\'ezis--Oswald type and several new arguments adapted to the abstract operator-theoretic framework. In this way, the paper provides a bridge between order-theoretic methods and the classical energy-based theory of sublinear problems.