Blow up Points and the Morse Indices of Solutions to the Liouville Equation in Two-Dimension
Futoshi Takahashi · Advanced Nonlinear Studies · 2012
Abstract We consider the Liouville equation −Δu = λeu in Ω, u =0 on∂Ω, on a smooth bounded domain Ω in ℝ 2 , where λ > 0 is a parameter. Let {u n } be an m-point blowing up solution sequence of the problem for λ = λn ↓ 0, which satisfies for m ∈ℕ. We prove that the number of blow up points m is less than or equal to the Morse index of u n for n sufficiently large. As a corollary, we show that if a solution u n of Morse index one has the property that , then the number of blow up points of the sequence is exactly one. Note that in the last result, we do not need any geometrical assumption such as the convexity of the domain.