Singular Adams inequality for biharmonic operator on Heisenberg Group and its applications

Abstract

The goal of this paper is to establish singular Adams type inequality for biharmonic operator on Heisenberg group. As an application, we establish the existence of a solution to \begin{equation*} Δ_{\mathbb{H}^n}^2 u=\frac{f(ξ,u)}{ρ(ξ)^a}\,\,\text{ in }Ω,\,\, u|_{\partialΩ}=0=\left.\frac{\partial u}{\partial ν}\right|_{\partialΩ}, \end{equation*} where $0\in \Omega \subseteq {\mathbb{H}}^4$ is a smooth bounded domain, $0\le a<Q,(Q=10).$ The special feature of this problem is that it contains an exponential nonlinearity and singular potential.

arXiv commentThere is small correction in this version. For our results to be meaningful, we need to work with the case Q=4 instead of Q=10. So throughout the article, We have now taken Q=4, that is we are working with one dimensional Heisenberg group. All the proofs remain unaffected due to this change