Radial symmetry and sharp asymptotic behaviors of nonnegative solutions to D^1, p-critical quasi-linear static Schr\"odinger-Hartree equation involving p-Laplacian -₏

In this paper, we mainly consider nonnegative weak solution to the D^1, p (^N) -critical quasi-linear static Schr\"odinger-Hartree equation with p-Laplacian -₏ and nonlocal nonlinearity: align* -ₚ u = (|x|^-2p |u|^p) |u|^p-2u &in \, \, RN, align* where 1<p<N2, N3 and u D^1, p (N). Being different to the D^1, p (^N) -critical local nonlinear term u^p^{-1} with p^: =NpN-p investigated in LDBS, BS16, VJ16 etc. , since the nonlocal convolution |x|^-2p*uᵖ appears in the Hartree type nonlinearity, it is impossible for us to use the scaling arguments and the Doubling Lemma as in 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 \|u \|₋^ₒ, (N) and hence derive the sharp estimates on upper bounds of asymptotic behaviors from the preliminary estimates as in 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 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 -ₚ u=V (x) u^p-1 with V L^N{p} (R^N). Then, by using the arguments from BS16, VJ16, we can deduce the sharp estimates on both upper and lower bounds for the decay rate of | u|. Finally, as a consequence, we can apply the method of moving planes to prove that all the nonnegative solutions are radially symmetric and strictly decreasing about some point x₀N.