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

TL;DR

This paper establishes a singular Adams inequality for the biharmonic operator on the Heisenberg group and applies it to prove the existence of solutions for a nonlinear PDE with exponential nonlinearity and singular potential.

Contribution

It introduces a new singular Adams inequality on the Heisenberg group and demonstrates its application in solving a nonlinear PDE with singular potential and exponential growth.

Findings

Proved singular Adams inequality for biharmonic operator on Heisenberg group

Established existence of solutions for a PDE with exponential nonlinearity and singular potential

Addressed the case with critical exponential growth and singularity in the potential

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*} \Delta_{\mathbb{H}^n}^2 u=\frac{f(\xi,u)}{\rho(\xi)^a}\,\,\text{ in }\Omega,\,\, u|_{\partial\Omega}=0=\left.\frac{\partial u}{\partial \nu}\right|_{\partial\Omega}, \end{equation*} where $0\in \Omega \subseteq \mathbb{H}^4$ is a smooth bounded domain, $0\leq a<Q,\,(Q=10).$ The special feature of this problem is that it contains an exponential nonlinearity and singular potential.