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Morse indices and the number of blow up points of blowing-up solutions for a Liouville equation with singular data

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  • Let $\Omega \subset \mathbb{R}^2$ be a smooth bounded domain and let $\Gamma = \left \{ p_1, \cdots, p_N \right \} \subset \Omega$ be the set of prescribed points. Consider the Liouville type equation \[ -\delta u = \lambda \Pi_{j = 1}^{N} |x - p_j|^{2\alpha_j} V(x) e^u \quad \mbox{in} \; \Omega, \quad u = 0 \quad \mbox{on} \; \partial \Omega, \] where $\alpha_j \; (j=1,\cdots, N)$ are positive numbers, $V(x) > 0$ is a given smooth function on $\bar{\Omega}$, and $\lambda > 0$ is a parameter. Let $\{ u_n \}$ be a blowing up solution sequence for $\lambda = \lambda_n \downarrow 0$ having the $m$-points blow up set $S = \{ q_1, \cdots, q_m \} \subset \Omega$, i.e., \[ \lambda_n \prod_{j = 1}^N |x - p_j|^{2 \alpha_j} V(x) e^{u_n} dx \rightharpoonup \sum_{i=1}^m b_i \delta_{q_i} \] in the sense of measures, where $b_i = 8\pi$ if $q_i \notin \Gamma$, $b_i = 8\pi(1 + \alpha_j)$ if $q_i = p_j$ for some $p_j \in \Gamma$. We show that the number of blow up points $m$ is less than or equal to the Morse index of $u_n$ for $n$ sufficiently large, provided $\alpha_j \in (0,+\infty) \setminus \mathbb{N}$ for all $j = 1, \cdots, N$. This is a generalization of the result [13] in which nonsingular case ($\alpha_j = 0$ for all $j$) was studied.
    Mathematics Subject Classification: Primary: 35B40, 35J20, 35J25.


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