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 $$\begin{array}{c}\left\{\begin{array}{cc}\mathrm{\Delta }\mathit{u}+\mathit{\rho }\mathrm{(}{\mathit{V}}_1\mathrm{(}\mathit{x}\mathrm{)}{\mathit{e}}^{\mathit{u}}-{\mathit{V}}_2\mathrm{(}\mathit{x}\mathrm{)}{\mathit{e}}^{-\mathit{\tau }\mathit{u}}\mathrm{)}=0\hfill & \text{in }{\mathrm{\Omega }}_{\mathit{ϵ}}:=\mathrm{\Omega }\setminus {\bigcup }_{\mathit{i}=1}^{\mathit{m}}\overline{\mathit{B}\mathrm{(}{\mathit{\xi }}_{\mathit{i}}\mathrm{,}{\mathit{ϵ}}_{\mathit{i}}\mathrm{)}}\hfill \\ \mathit{u}=0\hfill & \text{on }\partial {\mathrm{\Omega }}_{\mathit{ϵ}}\mathrm{,}\hfill \end{array}\right.\end{array}$$ where $\mathit{\rho }\mathrm{>}0$ , ${\mathit{V}}_1\mathrm{,}{\mathit{V}}_2\mathrm{>}0$ are smooth potentials in Ω, $\mathit{\tau }\mathrm{>}0$ , Ω is a smooth bounded domain in ${\mathbb{R}}^2$ and $\mathit{B}\mathrm{(}{\mathit{\xi }}_{\mathit{i}}\mathrm{,}{\mathit{ϵ}}_{\mathit{i}}\mathrm{)}$ is a ball centered at ${\mathit{\xi }}_{\mathit{i}}\in \mathrm{\Omega }$ with radius ${\mathit{ϵ}}_{\mathit{i}}\mathrm{>}0$ , $\mathit{i}=1\mathrm{,}\dots \mathrm{,}\mathit{m}$ . When $\mathit{\rho }\mathrm{>}0$ is small enough and ${\mathit{m}}_1\in \{1\mathrm{,}\dots \mathrm{,}\mathit{m}-1\}$ , there exist radii $\mathit{ϵ}=\mathrm{(}{\mathit{ϵ}}_1\mathrm{,}\dots \mathrm{,}{\mathit{ϵ}}_{\mathit{m}}\mathrm{)}$ small enough such that the problem has a solution which blows-up positively at the points ${\mathit{\xi }}_1\mathrm{,}\dots \mathrm{,}{\mathit{\xi }}_{{\mathit{m}}_1}$ and negatively at the points ${\mathit{\xi }}_{{\mathit{m}}_1+1}\mathrm{,}\dots \mathrm{,}{\mathit{\xi }}_{\mathit{m}}$ as $\mathit{\rho }\to 0$ . The result remains true in cases ${\mathit{m}}_1=0$ with ${\mathit{V}}_1\equiv 0$ and ${\mathit{m}}_1=\mathit{m}$ with ${\mathit{V}}_2\equiv 0$ , which are Liouville type equations.