Existence of Positive Solutions for a Semilinear Dirichlet Problem with Small Nonnegative Forcing Term

Source abstract

In this paper, we study the existence of positive solutions for the semilinear elliptic problem -Δu = f(u) + λh(x) in D, u = 0 on ∂D, where is a bounded domain with smooth boundary, is continuous and not identically zero, and is a continuous nonnegative function satisfying a suitable condition near . We prove that there exists such that for every , the problem admits a positive solution. The proof relies solely on the classical sub- and supersolution method. No growth condition on at infinity is required.