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{equation*} \left\{ \begin{array}{ll} Δu +ρ\left(V_1(x)e^{u}- V_2(x)e^{-τu}\right)=0 &\text{in } Ω_ε:=Ω\setminus \displaystyle \bigcup_{i=1}^m \overline{B(ξ_i,ε_i)}\\ u=0&\text{on }\partialΩ_ε, \end{array}\right. \end{equation*} where $\rho >0$, $V_1,V_2>0$ are smooth potentials in $\Omega$, $\tau >0$, $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^2$ and $B({\xi }_i,{\epsilon }_i)$ is a ball centered at ${\xi }_i\in \Omega$ with radius ${\epsilon }_i>0$, $i=1,\dots ,m$. When $\rho >0$ is small enough and $m_1\in \{1,\dots ,m-1\}$, there exist radii $\epsilon =({\epsilon }_1,\dots ,{\epsilon }_m)$ small enough such that the problem has a solution which blows-up positively at the points ${\xi }_1,\dots ,{\xi }_{m_1}$ and negatively at the points ${\xi }_{m_1+1},\dots ,{\xi }_m$ as $\rho \to 0$. The result remains true in cases $m_1=0$ with $V_1\equiv 0$ and $m_1=m$ with $V_2\equiv 0$, which are Liouville type equations.

14 pages. arXiv admin note: substantial text overlap with arXiv:1904.00127