Higher order energy expansions for some singularly perturbed Neumann problems
Juncheng Wei, Matthias Winter · Comptes Rendus Mathématique · 2003
We consider the following singularly perturbed semilinear elliptic problem: ϵ 2 Δ u - u + u p = 0 in Ω , u > 0 in Ω and ∂ u ∂ ν = 0 on ∂ Ω , where Ω is a bounded smooth domain in ℝ N , ε >0 is a small constant and p is a subcritical exponent. Let J ϵ [ u ] : = ∫ Ω ( ϵ 2 2 | ∇ u | 2 + 1 2 u 2 - 1 p + 1 u p + 1 ) d x be its energy functional, where u ∈ H 1 ( Ω ) . Ni and Takagi proved that for a single boundary spike solution u ε , the following asymptotic expansion holds J ϵ [ u ϵ ] = ϵ N 1 2 I [ w ] - c 1 ϵ H ( P ϵ ) + o ( ϵ ) , where c