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 −Δu=λVeu−Δu=λVeu in a two-dimensional bounded smooth domain with a Dirichlet boundary condition. Here λ>0λ>0 is a parameter and VV is a positive C1C1 function on Ω̄. It is known that the solution concentrates on a critical point of a Hamiltonian as λ↓0λ↓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≡1V≡1. Technical modifications are used in the proof to control residual terms.