Self-similar solutions for a semilinear heat equation with critical Sobolev exponent

The Cauchy problem for a semilinear heat equation with singular initial data {wt = Delta w + w(p) in R-N x (0,infinity) w(x,0) = lambda a(x/vertical bar x vertical bar)vertical bar x vertical bar(-2/(p-1)) in R-N \ {0} is studied, where N > 2, p = (N + 2) / (N - 2), lambda > 0 is a parameter, and a >= 0, a not equivalent to 0. We show that there exists a constant lambda* > 0 such that the problem has at least two positive self-similar solutions for lambda is an element of (0, lambda*) when N = 3,4,5, and that, when N >= 6 and a equivalent to 1, the problem has a unique positive radially symmetric self-similar solution for lambda is an element of (0, lambda*) with some lambda* C (0, lambda*). Our proofs are based on the variational methods and Pohozaev type arguments to the elliptic problem related to the profiles of self-similar solutions.