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 $(-\Delta)^{s}u(x)=(\frac{1}{|x|^{\sigma}}\ast |u|^{p})u^{q}(x) \,\,\ {in} \,\,\, \mathbb{R}^{n}$ involving higher-order or higher-order fractional Laplacians, where $n\geq1$, $0<s:=m+\frac{\alpha}{2}<\frac{n}{2}$, $m\geq0$ is an integer, $0<\alpha\leq2$, $0<\sigma<n$, $0<p\leq\frac{2n-\sigma}{n-2s}$, and $0<q\leq\frac{n+2s-\sigma}{n-2s}$. 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 $u(x)=\int_{\mathbb{R}^n}\frac{R_{2s,n}}{|x-y|^{n-2s}}(\int_{\mathbb{R}^{n}}\frac{1}{|y-z|^{\sigma}}u^p(z)dz)u^{q}(y)dy.$ 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 and hence derive the sharp constants for related Hardy--Littlewood--Sobolev inequalities. Our results completely improved the classification results in [4, 23, 24, 25, 44] to the full range of $s$, $\sigma$, $p$, and $q$. In critical and supercritical-order cases (i.e., $\frac{n}{2}\leq s:=m+\frac{\alpha}{2}<+\infty$), we also derive Liouville type theorems.

Topics & Concepts