Existence for non‑radial potentials when B > 2

Determine general existence conditions for solutions to the Liouville equation −Δu(x) = 4π B V(x) e^{u(x)} in R^2 when B > 2 and V ∈ L^1_loc(R^2) is not radially symmetric. Specify hypotheses on V under which a solution u with ∫_{R^2} V(x) e^{u(x)} dx = 1 exists, thereby extending existence results beyond the radially symmetric case.

Background

The paper establishes existence for radially symmetric V under explicit integrability and sign conditions and notes that their variational approach relies critically on radial symmetry. They also discuss counterexamples and geometric obstructions indicating that behavior of V at the origin and infinity alone may be insufficient to ensure existence for non‑radial V.

This question seeks a non‑radial counterpart to the existence results proved in Theorem 1.1, aiming for criteria on general locally integrable V that guarantee solvability when B > 2.