Blow up solutions for a Liouville equation with Hénon term (Q890777)

In this paper, the authors study the bubbling phenomena of the following boundary value problem \[ \begin{aligned}&\Delta u+\lambda |x|^{2\alpha }u^{p-1}e^{u^{p}}=0,\, u>0,\quad\text{in } \Omega; \\ &u=0\quad \text{on }\partial \Omega , \end{aligned}\tag{1} \] where \(\Omega \) is a bounded domain in \(\mathbb{R}^{2}\) with smooth boundary, \(\alpha \in (0,\infty )\backslash \mathbb{N}\), \(\lambda >0\) is a small parameter and \(0<p<2\). By applying the finite-dimensional reduction approach, they construct the bubbling solution \(u_{\lambda }\) of (1) which blows up at the origin and at some other finite points at \(\lambda \rightarrow 0\).