Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity

We study the semilinear elliptic equation -Delta u = u(alpha)vertical bar log u vertical bar(beta) in B-1 \ {0}, where B-1 subset of R-n, with n >= 3, n/n-2 < alpha <n+2/n-2 and -infinity < beta < infinity. Our main result establishes that the nonnegative solution u is an element of C-2(B-1 \ {0}) of the above equation either has a removable singularity at the origin or it behaves like u(x) = A(1 + o(1))vertical bar x vertical bar(-2/alpha-1)(log 1/vertical bar x vertical bar)(-beta/alpha-1) as x -> 0, with A = [(2/alpha-1)(1-beta)(n - 2 - 2/alpha-1)](1/alpha-1).