A quantization property for blow up solutions of singular Liouville-type equations
The author considers a ``concentrating'' solution-sequence \(u_k\) satisfying \[ -\Delta u_k= |z|^{2\alpha_k} V_k(z) e^{u_k}\quad\text{in }\mathbb B_1, \] \[ |z|^{2\alpha_k} V_k(z) e^{u_k}\to \beta\delta_{z=0}\text{ weakly in the sense of measures in }\mathbb B_1, \] where \(\alpha_k\in \mathbb R^+\). Under some natural assumptions on \(\alpha_k\) and \(V)k(z)\), the author shows that necessarily \[ \beta\in 8\pi\,\mathbb N\cup\{8\pi(1+ \alpha)+ 8\pi \mathbb Z^+\}, \] where \(\alpha_k\to \alpha\). It is shown by explicit examples, that the obtained result is ``sharp''.
scientific article; zbMATH DE number 2141631
scientific article