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

Let $Ω$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following elliptic Dirichlet problem $\begin{cases} -Δ\upsilon= e^{\upsilon}-sϕ_1-4παδ_p-h(x)\,\,\,\, \,\textrm{in}\,\,\,\,\,Ω,\\[2mm] \upsilon=0 \quad\quad\quad\quad\quad\quad \qquad\qquad\quad\quad\,\,\,\, \textrm{on}\,\ \,\partialΩ, \end{cases}$ where $s>0$ is a large parameter, $h\in C^{0,γ}(\overlineΩ)$, $p\inΩ$, $α\in(-1,+\infty)\setminus\mathbb{N}$, $δ_p$ denotes the Dirac measure supported at point $p$ and $ϕ_1$ is a positive first eigenfunction of the problem $-Δϕ=λϕ$ under Dirichlet boundary condition in $Ω$. If $p$ is a strict local maximum point of $ϕ_1$, we show that such a problem has a family of solutions $\upsilon_s$ with arbitrary $m$ bubbles accumulating to $p$, and the quantity $\int_Ωe^{\upsilon_s}\rightarrow8π(m+1+α)ϕ_1(p)$ as $s\rightarrow+\infty$.

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