Affiliations: Instituto de Ciencias Físicas y Matemáticas, Facultad Ciencias, Universidad Austral de Chile, Campus Isla Teja s/n, Valdivia, Chile. E-mail: [email protected]
Abstract: We consider a sinh-Poisson type equation with variable intensities and Dirichlet boundary condition on a pierced domain
Δu+ρ(V1(x)eu−V2(x)e−τu)=0in Ωϵ:=Ω∖⋃i=1mB(ξi,ϵi)‾u=0on ∂Ωϵ,
where ρ>0, V1,V2>0 are smooth potentials in Ω, τ>0, Ω is a smooth bounded domain in R2 and B(ξi,ϵi) is a ball centered at ξi∈Ω with radius ϵi>0, i=1,…,m. When ρ>0 is small enough and m1∈{1,…,m−1}, there exist radii ϵ=(ϵ1,…,ϵm) small enough such that the problem has a solution which blows-up positively at the points ξ1,…,ξm1 and negatively at the points ξm1+1,…,ξm as ρ→0. The result remains true in cases m1=0 with V1≡0 and m1=m with V2≡0, which are Liouville type equations.
