Positive solutions to semipositone problems on Heisenberg group

Abstract

This article focuses on establishing a positive weak solution to a class of semipositone problems over the Heisenberg group $\mathbb{H}^N$. In particular, we are interested in the positive weak solution to the following problem: \begin{equation}\label{p1} -Δ_{\mathbb{H}}u= g(ξ)f_a(u) \text{ in } \mathbb{H}^N \tag{$P_a$}, \end{equation} where $a>0$ is a real parameter and $g$ is a positive function. The function $f_a: \mathbb{R} \rightarrow \mathbb{R}$ is continuous and of semipositone type which means it becomes negative on some parts of the domain. Due to this sign-changing nonlinearity, we can not directly apply the maximum principle to obtain the positivity of the solution to \eqref{p1}. For that purpose, we need some regularity results for our solutions. In this direction, we first prove the existence of weak solutions to \eqref{p1} via the mountain pass technique. Further, we establish some regularity properties of our solutions and using that we prove the $L^\infty$-norm convergence of the sequence of solutions $\{u_a\}$ to a positive function $u$ as $a \rightarrow 0$, which yields $u_a \geq 0$ for $a$ sufficiently small. Finally, we use the Riesz-representation formula to obtain the positivity of solutions under some extra hypothesis on $f_0$ and $g$. To the best of our knowledge, there is no article dealing with semipositone problems in Heisenberg group set up.