Positive solutions to semipositone problems on Heisenberg group

In this article, we study the existence of positive weak solutions to a class of semipositone problems on the Heisenberg group HN. More precisely, we study the problem −ΔHu=g(ξ)fa(u) in HaN,(Pa) where a>0 is a real parameter, g is a positive function, and fa:R→R is a continuous and admits subcritical nonlinearity that takes negative values on part of its domain. The sign-changing nature of fa causes the strong maximum principle to fail, introducing substantial challenges in establishing positivity. Our first main result concerns the existence of weak solutions to (Pa), which we obtain via the mountain pass theorem. We then derive qualitative properties of these solutions and use them to prove that ua≥0 for sufficiently small values of a. Finally, by applying the Riesz potential formula, we conclude the strict positivity of solutions. This work extends the results of Alves et al. (Proc. Roy. Soc. Edinburgh Sect. A, 150(5):2349–2367, 2020) to the setting of the Heisenberg group.