Stability under small perturbations of V

Investigate whether small perturbations of the potential V affect existence or uniqueness of solutions to the Liouville equation −Δu(x) = 4π B V(x) e^{u(x)} in R^2. In particular, determine whether radially symmetric solutions for V(x) = |f_0(x)|^2 e^{−B|x|^2} with f_0(z) = z^n can be perturbed to yield solutions for V(x) = |f_0(x) + δ g(x)|^2 e^{−B|x|^2}, where g is a lower‑degree complex polynomial and |δ| is sufficiently small.

Background

The authors provide examples indicating that small changes in V can dramatically alter existence or uniqueness (e.g., differing behaviors for V(r) = r^α e^{-r} versus V(r) = e^{-r^2}). They highlight a physically motivated family in Section 5.4 (nonlinear Landau levels) where V has the form |f|^2 e^{−B|x|^2}, raising the question of perturbation stability around radial configurations.

This problem calls for a perturbative theory for the Liouville equation with Gaussian weights and polynomial amplitudes, relevant to the Chern‑Simons–Schrödinger model.