Linearized pseudo-Einstein equations on the Heisenberg group

We study the pseudo-Einstein equation $R_{1 \bar{1}} = 0$ on the Heisenberg group $\mathbb{H}_1 = \mathbb{C} \times \mathbb{R}$. We consider first order perturbations $\theta_\epsilon = \theta_0 + \epsilon \theta$ and linearize the pseudo-Einstein equation about $\theta_0$ (the canonical Tanaka–Webster flat contact form on $\mathbb{H}_1$ thought of as a strictly pseudoconvex CR manifold). If $\theta = e^{2u} \theta_0$ the linearized pseudo-Einstein equation is $\Delta_b u − 4 |Lu|^2 = 0$ where $\Delta_b$ is the sublaplacian of $(\mathbb{H}_1, \theta_0)$ and L is the Lewy operator. We solve the linearized pseudo-Einstein equation on a bounded domain $\Omega \subset \mathbb{H}_1$ by applying subelliptic theory i.e. existence and regularity results for weak sub elliptic harmonic maps. We determine a solution $u$ to the linearized pseudo-Einstein equation, possessing Heisenberg spherical symmetry, and such that $u(x) \to − \infty$ as $|x| \to +\infty$.