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 \(D^{1,p}(\mathbb {R}^{N})\) $D^{1,p}({\mathbb{R}}^N)$ -critical quasi-linear static Schrödinger–Hartree equation with p-Laplacian \(-\Delta _{p}\) $-{\mathrm{\Delta }}_p$ and nonlocal nonlinearity: \(\begin{aligned} -\Delta _p u =\left( |x|^{-2p}*|u|^{p}\right) |u|^{p-2}u \qquad&\text{ in } \,\, \mathbb {R}^N, \end{aligned}\) $$\begin{array}{cc}\hfill -{\mathrm{\Delta }}_{p}u=\left({|x|}^{-2p},,\ast ,{|u|}^p\right){|u|}^{p-2}u& \text{in}{\mathbb{R}}^N,\hfill \end{array}$$ where \(1<p<\frac{N}{2}\) $1<p<\frac{N}{2}$ , \(N\ge 3\) $N\ge 3$ and \(u\in D^{1,p}(\mathbb {R}^N)\) $u\in D^{1,p}({\mathbb{R}}^N)$ . First, we establish regularity and the sharp estimates on asymptotic behaviors for any positive solution u (and \(|\nabla u|\) $|\mathrm{\nabla }u|$ ) to more general equation \(-\Delta _p u=V(x)u^{p-1}\) $-{\mathrm{\Delta }}_{p}u=V(x)u^{p-1}$ with \(V\in L^{\frac{N}{p}}(\mathbb {R}^{N})\) $V\in L^{\frac{N}{p}}({\mathbb{R}}^N)$ . 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 \(x_0\in \mathbb {R}^N\) $x_0\in {\mathbb{R}}^N$ . The radial symmetry and sharp asymptotic estimates for more general nonlocal quasi-linear equations were also included.