Radial symmetry and sharp asymptotic behaviors of nonnegative solutions to $$D^{1,p}$$ -critical quasi-linear static Schrödinger–Hartree equation involving p-Laplacian $$-\Delta _{p}$$

In this paper, we mainly consider nonnegative weak solution to the \(D^{1,p}(\mathbb {R}^{N})\)-critical quasi-linear static Schrödinger–Hartree equation with p-Laplacian \(-\Delta _{p}\) and nonlocal nonlinearity:

$$\begin{aligned} -\Delta _p u =\left( |x|^{-2p}*|u|^{p}\right) |u|^{p-2}u \qquad&\text{ in } \,\, \mathbb {R}^N, \end{aligned}$$

where \(1<p<\frac{N}{2}\), \(N\ge 3\) and \(u\in D^{1,p}(\mathbb {R}^N)\). First, we establish regularity and the sharp estimates on asymptotic behaviors for any positive solution u (and \(|\nabla u|\)) to more general equation \(-\Delta _p u=V(x)u^{p-1}\) with \(V\in L^{\frac{N}{p}}(\mathbb {R}^{N})\). Then, as a consequence, we can apply the method of moving planes to prove that all the nontrivial nonnegative solutions are radially symmetric and strictly decreasing about some point \(x_0\in \mathbb {R}^N\). The radial symmetry and sharp asymptotic estimates for more general nonlocal quasi-linear equations were also included.