Linearized pseudo-Einstein equations on the Heisenberg group

We study the pseudo-Einstein equation R11¯=0 on the Heisenberg group H1=C×R. We consider first order perturbations θϵ=θ0+ϵθ and linearize the pseudo-Einstein equation about θ0 (the canonical Tanaka-Webster flat contact form on H1 thought of as a strictly pseudoconvex CR manifold). If θ=e2uθ0 the linearized pseudo-Einstein equation is Δbu−4|Lu|2=0 where Δb is the sublaplacian of (H1,θ0) and L¯ is the Lewy operator. We solve the linearized pseudo-Einstein equation on a bounded domain Ω⊂H1 by applying subelliptic theory i.e. existence and regularity results for weak subelliptic harmonic maps. We determine a solution u to the linearized pseudo-Einstein equation, possessing Heisenberg spherical symmetry, and such that u(x)→−∞ as |x|→+∞.