Radial symmetry and sharp asymptotic behaviors of nonnegative solutions to D1,p-critical quasi-linear static Schr\"{o}dinger-Hartree equation involving p-Laplacian -Δₚ

In this paper, we mainly consider nonnegative weak solution to the D1,p(N)-critical quasi-linear static Schr\"{o}dinger-Hartree equation with p-Laplacian -Δₚ and nonlocal nonlinearity: {align*} -Δ_p u =(|x|⁻²ᵖ |u|ᵖ)|u|ᵖ⁻²u &in \,\, R^N, {align*} where 1<p<N/2, N≥3 and u∈ D1,p(N). Being different to the D1,p(N)-critical local nonlinear term u^p^-1 with p^:=Np/N-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 \|_Ls,∞(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 -Δₚ u=V(x)uᵖ⁻¹ with V∈ LN/p(RN). 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 x₀∈N.