Asymptotic symmetry of singular solutions of semilinear elliptic equations

For dimensions 3 n 6, we derive lower bound for positive solution of in the neighbourhood of singularity. Here μ>0. We prove that there exists a constant c1>0 depending on n, μ, K, K such that For dimension 3, we assume that K is Hölder continuous with exponent θ with while for dimensions n=4,5,6, assume that K C1 is bounded between two positive constantscxθ−1 K(x) Cxθ−1 for c, C>0 and . As an application, we derive the asymptotic symmetry and a priori estimates for the solutions.