Classification of solutions of elliptic equations arising from a gravitational \(O(3)\) gauge field model
The authors consider the equation \[ \Delta v -\rho_\tau(x)f_\tau(v,a,\varepsilon)=4\pi \sum_{j=1}^{d_1}n_{j,1}\delta_{p_{j,1}}-4\pi \sum_{j=1}^{d_2}n_{j,2}\delta_{p_{j,2}},\quad x\in\mathbb{R}^2, \] where \[ \rho_\tau(x)=\left( \prod_{j=1}^{d_1}|x-p_{j,1}|^{(1-\tau)n_{j,1}}- \prod_{j=1}^{d_2}|x-p_{j,2}|^{(1-\tau)n_{j,2}} \right)^{-a} \] and \[ f_\tau(v,a,\varepsilon)=\dfrac{2e^{\frac{(1-\tau)}{2}av}\big((1+\tau)e^v-(1-\tau)\big)}{\varepsilon^2(1+e^v)^{1+a}}. \] Here \(p_{j,1}\) and \(p_{j,2}\) are different points, the unknown is \[ v: \mathbb{R}^2\setminus \{p_{j,k}: j=1,\ldots,d_k,\;k=1,2\} \to \mathbb{R}, \] the scaled gravitational constant \(a\in\mathbb{R}\) is nonnegative, \(\tau\in[-1,1],\) \(\delta_p\) is the Dirac delta and \(n_{j,k}\) are positive integers representing the multiplicity of the strings and the antistrings \(p_{j,k}.\) By using the decay rates of the solutions at infinity, the authors provide a complete classification of the solutions for all possible values of \(aN_1\) and \(aN_2,\) where the total string and anti-string numbers are defined by \[ N_1=n_{1,1}+\cdots+n_{d_{1},1},\quad N_2=n_{1,2}+\cdots+n_{d_{2},2}. \]
scientific article; zbMATH DE number 6836595
scientific article