関数係数を持つ Liouville-Gel'fand 問題に対する多点爆発解の漸近的非退化性

抄録

We study asymptotic non-degeneracy of multi-point blowup solutions to the Liouville-Gel'fand problem $-\Delta u = \lambda V e^u$ (which is a nonlinear exponential type equation) in two-dimensional bounded smooth domain $\Omega$ with Dirichlet boundary condition. Here $\lambda>0$ is a parameter and $V$ is a positive $C^1$-class function on $\bar{\Omega}$. It is known that the solution concentrates on a critical point of a Hamiltonian as $\lambda \downarrow 0$. 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 of the constant coefficient. Technical modifications are used in the proof to control residual terms.