On the classification of solutions of cosmic strings equation (Q2009084)

The authors investigate and classify the weak stable or finite Morse index solutions of the semi-linear partial differential equation \[ \Delta u+e^u+|x|^{\alpha}e^{\beta u}=0,\,\,\,\text{in}\,\,\,\mathbb{R}^N,\,\,\,N\ge 2,\tag{1} \] where \(\alpha>-2\) and \(\beta>0\). If \(N\in\left[2,2+\frac{4(\alpha+2)}{\max\{1,\beta\}}\right)\) they prove that equation (1) has no weak stable solution. If \(\alpha+2\ge 2\beta\) and \(2<N<2+8\min\{\beta,\frac{1}{\beta}\}\) they show that there is no weak solution of (1) which is stable outside a compact set, and so (1) has no weak finite Morse index solution.