Qualitative properties of singular solutions of Choquard equations

Abstract

Our aim of this paper is to study qualitative properties of isolated singular solutions to Choquard equation \begin{equation}\label{eq 0.1} -\Delta u+ u =I_\alpha[u^p] u^q+k\delta_0\quad {\rm in}\ \, \mathcal{D}'(\mathbb{R}^N), \tag{0.1} \end{equation} where $p, \, q\ge 1$, $N\ge2$, $\alpha\in(0,N)$, $k > 0$, $\delta_0$ is the Dirac mass concentrated at the origin and $I_\alpha[u^p](x)=\int_{\mathbb{R}^N} \frac{u(y)^p}{|x-y|^{N-\alpha}}\, dy.$ Multiple properties of very weak solutions of (0.1) are considered: (i) to obtain the existence of minimal solutions and extremal solutions for $N=2$, which are derived in [8] when $N\geq3$; (ii) to analyze the stability of minimal solutions and the semi stability of extremal solutions; (iii) to derive a second solution by the Mountain Pass theorem when $q=p-1$ and $N=2,3$; (iv) to obtain the radial symmetry of the positive singular solutions by the method of moving planes.