Positive solutions to semipositone problems on Heisenberg group
Vikram Naik, Rohit Kumar
TL;DR
This work investigates semipositone problems for the sub-Laplacian on the Heisenberg group $\mathbb{H}^N$, focusing on the existence of positive weak solutions when the nonlinear source $g(\xi)f_a(u)$ changes sign. The authors formulate an energy framework in the Beppo Levi space $\mathcal{D}^{1,2}(\mathbb{H}^N)$ and prove the Mountain Pass geometry along with the Palais–Smale condition, yielding a nontrivial weak solution for small $a$. They establish uniform $L^\infty$ bounds and regularity via a Moser-type iteration, and show that as $a\to0$ the solutions converge to a nonnegative limit; under an additional hypothesis and via the Riesz representation formula, positivity of the limit (and hence of the approximating solutions for small $a$) is obtained. The results extend the semipositone problem literature to the subelliptic setting of the Heisenberg group and open the door to further questions on critical growth, multi-parameter variants, and nonlinear operators such as the $p$-Laplacian on $\mathbb{H}^N$.
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.