Bubbling solutions for a planar exponential nonlinear elliptic equation with a singular source

Abstract

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^2$ with smooth boundary, we study the following elliptic Dirichlet problem $$\{\begin{array}{l}-\Delta \upsilon =e^{\upsilon }-s{\varphi }_1-4\pi \alpha {\delta }_p-h(x)\text{in}\Omega ,\\ \upsilon =0\text{on}\partial \Omega ,\end{array}$$ where $s>0$ is a large parameter, $h\in C^{0,\gamma }(\overline{\Omega })$, $p\in \Omega$, $\alpha \in (-1,+\infty )\setminus \mathbb{N}$, ${\delta }_p$ denotes the Dirac measure supported at point $p$ and ${\varphi }_1$ is a positive first eigenfunction of the problem $-\Delta \varphi =\lambda \varphi$ under Dirichlet boundary condition in $\Omega$. If $p$ is a strict local maximum point of ${\varphi }_1$, we show that such a problem has a family of solutions ${\upsilon }_s$ with arbitrary $m$ bubbles accumulating to $p$, and the quantity ${\int }_{\Omega }{e}^{{\upsilon }_s}\to 8\pi (m+1+\alpha ){\varphi }_1(p)$ as $s\to +\infty$.

This manuscript has been accepted for publication in Advances in Differential Equations