Structure of solutions to a singular Liouville system arising from modeling dissipative stationary plasmas
\quad In this paper, the authors study a system of elliptic equations which arises from modeling dissipative stationary plasmas. More precisely, denote by \(B_R\) the ball of radius \(R>0\) in \(\mathbf{R}^2\), then the system takes the form \[ \;\;\;\;\left\{\begin{align*}{ &\Delta u+a e^u+ b e^v=4\pi k_1 \delta_0,\;\;x\in B_R, \cr &\Delta v+c e^u+ d e^v=4\pi k_2 \delta_0,\;\;x\in B_R,\cr & u<0,\;v<0\;\;\text{ in }\; B_R,\;\;\cr & u=v=0\;\;\text{ on }\; \partial B_R, }\end{align*}\right. \] where \(a,d\leq 0\), \(b,c>0\), \(k_1,k_2>0\) are constants and where \(\delta_0\) is the Dirac measure at the origin. \vskip 3 mm \quad With some assumptions on the parameters \(a,b,c,d\), the authors obtain existence and uniqueness results for radial solutions, and study blow up phenomena.
scientific article; zbMATH DE number 6157305