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 $$ \begin{cases} -\Delta\upsilon= e^{\upsilon}-s\phi_1-4\pi\alpha\delta_p-h(x)\,\,\,\, \,\textrm{in}\,\,\,\,\,\Omega,\\ \upsilon=0 \quad\quad\quad\quad\quad\quad \qquad\qquad\quad\quad\,\,\,\, \textrm{on}\,\ \,\partial\Omega, \end{cases} $$ 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 $\phi_1$ is a positive first eigenfunction of the problem $-\Delta\phi=\lambda\phi$ under Dirichlet boundary condition in $\Omega$. If $p$ is a strict local maximum point of $\phi_1$, we show that such a problem has a family of solutions $\upsilon_s$ with arbitrary $m$ bubbles accumulating to $p$, and the quantity $$ \int_{\Omega}e^{\upsilon_s} \rightarrow8\pi(m+1+\alpha)\phi_1(p) \ \text{ as $s\rightarrow+\infty$.} $$