Liouville theorems for nonnegative solutions to static weighted Schrödinger–Hartree–Maxwell type equations with combined nonlinearities

Abstract

In this paper, we are concerned with the physically interesting static weighted Schrödinger–Hartree–Maxwell type equations

$$\begin{aligned} (-\Delta )^{\frac{\alpha }{2}}u(x)=c_{1}|x|^{a}\left( \frac{1}{|x|^{\sigma }}*|u|^{q_{1}}\right) u^{p_{1}}(x)+c_{2}|x|^{b}u^{p_{2}}(x) \quad \text {in} \quad \mathbb {R}^{n} \end{aligned}$$

with combined nonlinearities, where $n\ge 2$, $0<\alpha \le 2$, $0<\sigma <n$, $c_{1}, \, c_{2}\ge 0$ with $c_{1}+c_{2}>0$, $0\le a,b<+\infty$, $0<q_{1}\le \frac{2n-\sigma }{n-\alpha }$, $0<p_{1}\le \frac{n+\alpha -\sigma +2a}{n-\alpha }$ and $0<p_{2}\le \frac{n+\alpha +2b}{n-\alpha }$. We derive Liouville theorems (i.e., non-existence of nontrivial nonnegative solutions) in the subcritical cases (see Theorem 1.1). The argument used in our proof is the method of scaling spheres developed in Dai and Qin (Liouville type theorems for fractional and higher order Hénon–Hardy equations via the method of scaling spheres, arXiv:1810.02752). As a consequence, we also derive Liouville theorem for weighted Schrödinger–Hartree–Maxwell type systems. Our results extend the Liouville theorems in Dai and Liu (Calc Var Partial Differ Equ 58(4): Paper No. 156, 24 pp, 2019) and Dai et al. (Classification of nonnegative solutions to static Schrödinger–Hartree–Maxwell type equations, arXiv:1909.00492) for $0<\alpha \le 2$.