Existence of Positive Solutions for a Semilinear Dirichlet Problem with Small Nonnegative Forcing Term
DOI:
https://doi.org/10.65904/3083-1733.2026.2.10Keywords:
Semilinear elliptic problem, Dirichlet boundary condition, Positive solution, Sub- and supersolution method, Small forcing term, Maximum principleAbstract
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.
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