Asymptotic non-degeneracy of multiple blowup solutions to the Liouville–Gel’fand problem with an inhomogeneous coefficient

We study asymptotic non-degeneracy of multi-point blowup solutions to the Liouville–Gel’fand problem − Δ u = λ V e u in a two-dimensional bounded smooth domain with a Dirichlet boundary condition. Here λ > 0 is a parameter and V is a positive C 1 function on Ω ̄ . It is known that the solution concentrates on a critical point of a Hamiltonian as λ ↓ 0 . We show that if this critical point is non-degenerate, then the associated solution is linearly non-degenerate, which is a natural extension of the case V ≡ 1 . Technical modifications are used in the proof to control residual terms.