Multiple solutions for a singular nonhomogenous biharmonic equation in Heisenberg group

Abstract

In this paper, we study a singular nonhomogenous biharmonic problem with Dirichlet boundary condition in Heisenberg group$ \begin{equation*} \left \{\begin{array}{ll} \Delta^2_{{\mathbb{H}^1}}u = \frac{f(\xi, u)}{\rho(\xi)^\beta}+\epsilon h(\xi)\ &\mbox{in} \ \Omega, \\ u = \frac{\partial u}{\partial \nu} = 0, \ \ &\mbox{on}\ \partial\Omega, \end{array} \right. \end{equation*} $where $ \Omega\subset {\mathbb{H}^1} $ is a bounded smooth domain, $ \Delta^2_{{\mathbb{H}^1}}u = \Delta_{\mathbb{H}^1}(\Delta_{\mathbb{H}^1}u) $ denotes the biharmonic operator in Heisenberg group $ \mathbb{H}^1 = \mathbb{C} \times \mathbb{R} $, $ 0\leq \beta < 4 $ with $ 4 $ is the homogeneous dimension of $ {\mathbb{H}^1} $ and $ f:\Omega \times \mathbb{R}\rightarrow \mathbb{R} $ is a continuous function which satisfies subcritical and critical exponential growth condition, $ h(\xi)\in (D_0^{2, 2}(\Omega))^* $, $ h(\xi)\geq0 $ and $ h(\xi) \not\equiv0 $, $ \rho(\xi) = (|z|^4+t^2)^{\frac{1}{4}} $, $ \xi = (z, t)\in \mathbb{H}^1 $ with $ z = (x, y)\in\mathbb{R}^{2} $, $ \epsilon $ is a small positive parameter. We obtain the existence and multiplicity of solutions by the Ekeland variational principle, mountain pass theorem and singular Adams inequality in Heisenberg group.