Positive solutions to logistic type equations with harvesting

TL;DR

This paper establishes the existence of positive solutions for logistic equations with harvesting in unbounded and bounded domains, relaxing growth conditions and employing comparison, variational, and truncation methods.

Contribution

It introduces a new solvable equation under relaxed growth assumptions to find positive solutions for complex logistic models with harvesting.

Findings

Positive solutions exist in both unbounded and bounded domains.

Relaxed growth conditions enable broader applicability.

New method simplifies solving complex logistic equations.

Abstract

We use comparison principles, variational arguments and a truncation method to obtain positive solutions to logistic type equations with harvesting both in $\mathbb{R}^N$ and in a bounded domain $\Omega\subset\mathbb{R}^N$, with $N\geq 3$, when the carrying capacity of the environment is not constant. By relaxing the growth assumption on the coefficients of the differential equation we derive a new equation which is easily solved. The solution of this new equation is then used to produce a positive solution of our original problem.