For fully nonlinear Loewner-Nirenberg equations with boundary data w=0, the hyperbolic solution is unique when μ_Γ^+>1, while for μ_Γ^+≤1 all solutions form a one-parameter family depending only on x_n.
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The fully nonlinear Loewner-Nirenberg problem: Liouville theorems and counterexamples to local boundary estimates
For fully nonlinear Loewner-Nirenberg equations with boundary data w=0, the hyperbolic solution is unique when μ_Γ^+>1, while for μ_Γ^+≤1 all solutions form a one-parameter family depending only on x_n.