Asymptotic symmetry of singular solutions of semilinear elliptic equations

For dimensions 3⩽n⩽63⩽n⩽6, we derive lower bound for positive solution ofΔu−μu+K(x)un+2n−2=0in B2∖{0},u∈C2(B2∖{0}) in the neighbourhood of singularity. Here μ>0μ>0. We prove that there exists a constant c1>0c1>0 depending on n, μ  , |K||K|, |∇K||∇K| such thatu(x)>c1|x|−n−22in B1∖{0}. For dimension 3, we assume that K is Hölder continuous with exponent θ   with 12<θ⩽1 while for dimensions n=4,5,6n=4,5,6, assume that K∈C1K∈C1 is bounded between two positive constantsc|x|θ−1⩽|∇K(x)|⩽C|x|θ−1c|x|θ−1⩽|∇K(x)|⩽C|x|θ−1 for c  , C>0C>0 and n−22⩽θ⩽n−2.As an application, we derive the asymptotic symmetry and a priori estimates for the solutions.