On the multiplicity of self-similar solutions of the semilinear heat equation

In studies of superlinear parabolic equations u_t=Δu+u^p,x∈R^N,t>0,where p>1, backward self-similar solutions play an important role. These are solutions of the form u(x,t)=(T−t)^{−1∕(p−1)}w(y), where y≔x∕T−t, T is a constant, and w is a solution of the equation Δw−y⋅∇w∕2−w∕(p−1)+w^p=0. We consider (classical) positive radial solutions w of this equation. Denoting by p_S, p_JL, p_L the Sobolev, Joseph-Lundgren, and Lepin exponents, respectively, we show that for p∈(p_S,p_JL) there are only countably many solutions, and for p∈(p_JL,p_L) there are only finitely many solutions. This result answers two basic open questions regarding the multiplicity of the solutions.