Classification of nonnegative solutions to static Schrödinger-Hartree-Maxwell type equations

Abstract

In this paper, we are mainly concerned with the physically interesting static Schrödinger-Hartree-Maxwell type equations \begin{equation*} (-Δ)^{s}u(x)=\left(\frac{1}{|x|^σ}\ast |u|^{p}\right)u^{q}(x) \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n} \end{equation*} involving higher-order or higher-order fractional Laplacians, where $n\ge 1$, $0<s:=m+\frac{\alpha }{2}<\frac{n}{2}$, $m\ge 0$ is an integer, $0<\alpha \le 2$, $0<\sigma <n$, $0<p\le \frac{2n-\sigma }{n-2s}$ and $0<q\le \frac{n+2s-\sigma }{n-2s}$. We first prove the super poly-harmonic properties of nonnegative classical solutions to the above PDEs, then show the equivalence between the PDEs and the following integral equations \begin{equation*} u(x)=\int_{\mathbb{R}^n}\frac{R_{2s,n}}{|x-y|^{n-2s}}\left(\int_{\mathbb{R}^{n}}\frac{1}{|y-z|^σ}u^p(z)dz\right)u^{q}(y)dy. \end{equation*} Finally, we classify all nonnegative solutions to the integral equations via the method of moving spheres in integral form. As a consequence, we obtain the classification results of nonnegative classical solutions for the PDEs. Our results completely improved the classification results in \cite{CD,DFQ,DL,DQ,Liu}. In critical and super-critical order cases (i.e., $\frac{n}{2}\le s:=m+\frac{\alpha }{2}<+\infty$), we also derive Liouville type theorem.

arXiv admin note: text overlap with arXiv:1905.04300