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

Abstract

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{align*} -Δ_p u =\left(|x|^{-2p}\ast |u|^{p}\right)|u|^{p-2}u \qquad &\mbox{in} \,\, \mathbb{R}^N, \end{align*} where $1<p<\frac{N}{2}$, $N\ge 3$ and $u\in D^{1,p}({\mathbb{R}}^N)$. Being different to the $D^{1,p}({\mathbb{R}}^N)$-critical local nonlinear term $u^{p^{\star }-1}$ with $p^{\star }:=\frac{Np}{N-p}$ investigated in \cite{CFR,LDSMLMSB,GV,Ou,BS16,VJ16} etc., since the nonlocal convolution $|x{|}^{-2p}\ast u^p$ appears in the Hartree type nonlinearity, it is impossible for us to use the scaling arguments and the Doubling Lemma as in \cite{VJ16} to get preliminary estimates on upper bounds of asymptotic behaviors for any positive solutions $u$. Moreover, it is also quite difficult to obtain the boundedness of the quasi-norm $\Vert u{\Vert }_{L^{s,\infty }({\mathbb{R}}^N)}$ and hence derive the sharp estimates on upper bounds of asymptotic behaviors from the preliminary estimates as in \cite{VJ16}. Fortunately, by showing a better preliminary estimates on upper bounds of asymptotic behaviors through the De Giorgi-Moser-Nash iteration method and combining the result from \cite{XCL}, we are able to overcome these difficulties and establish regularity and the sharp estimates on both upper and lower bounds of asymptotic behaviors for any positive solution $u$ to more general equation $-{\Delta }_{p}u=V(x)u^{p-1}$ with $V\in L^{\frac{N}{p}}({\mathbb{R}}^N)$. Then, by using the arguments from \cite{BS16,VJ16}, we can deduce the sharp estimates on both upper and lower bounds for the decay rate of $|\nabla u|$. Finally, 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 $x_0\in {\mathbb{R}}^N$.

Extended results on radial symmetry and sharp asymptotic estimates for more general nonlocal quasi-linear equations were added in this version