Symmetry and monotonicity of positive solutions for a Choquard equation involving the logarithmic Laplacian operator

Abstract

In this paper, we study a Schrödinger–Choquard equation involving the logarithmic Laplacian operator in \(\mathbb {R}^{n}\):

$$\begin{aligned} \mathcal {L}_\triangle u(x)+\omega u(x)=C_{n,s}(|x|^{2s-n}*u^{p})u^{r}, x\in \mathbb {R}^{n}, \end{aligned}$$

where \(0<s<1,\ p>1,\ r>0,\ n\ge 2,\ \omega >0\). Using the direct method of moving planes, we prove that if u satisfies some suitable asymptotic properties, then u must be radially symmetric and monotone decreasing about some point in the whole space. The key ingredients of the proofs are the narrow region principle and decay at infinity theorem; the ideas can be applied to problems involving more general nonlocal operators.