A note on a sinh-Poisson type equation with variable intensities on pierced domains

Abstract

We consider a sinh-Poisson type equation with variable intensities and Dirichlet boundary condition on a pierced domain Δ u + ρ ( V 1 ( x ) e u − V 2 ( x ) e − τ u ) = 0 in  Ω ϵ : = Ω ∖ ⋃ i = 1 m B ( ξ i , ϵ i ) ‾ u = 0 on  ∂ Ω ϵ , where ρ > 0, V 1 , V 2 > 0 are smooth potentials in Ω, τ > 0, Ω is a smooth bounded domain in R 2 and B ( ξ i , ϵ i ) is a ball centered at ξ i ∈ Ω with radius ϵ i > 0, i = 1 , … , m. When ρ > 0 is small enough and m 1 ∈ { 1 , … , m − 1 }, there exist radii ϵ = ( ϵ 1 , … , ϵ m ) small enough such that the problem has a solution which blows-up positively at the points ξ 1 , … , ξ m 1 and negatively at the points ξ m 1 + 1 , … , ξ m as ρ → 0. The result remains true in cases m 1 = 0 with V 1 ≡ 0 and m 1 = m with V 2 ≡ 0, which are Liouville type equations.