2019/09/03 by Figueroa, P.
#35B44 #35J25 #35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1909.00905
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)=0 · amp;in Ωε:=Ω∖ \bigcupi=1m B(ξi,εi)
u=0 · amp;on ∂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.