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.
Citation
Huyuan Chen. Feng Zhou. "Qualitative properties of singular solutions of Choquard equations." Adv. Differential Equations 28 (1/2) 35 - 72, January/February 2023. https://doi.org/10.57262/ade028-0102-35
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