For p in (p_S, p_JL) there are countably many positive radial solutions w to the self-similar profile equation, and finitely many for p in (p_JL, p_L).
Quittner, Uniqueness of singular self-similar solutions of a semilin ear parabolic equation, Differential Integral Equations 31 (2018), 881–892
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On the multiplicity of self-similar solutions of the semilinear heat equation
For p in (p_S, p_JL) there are countably many positive radial solutions w to the self-similar profile equation, and finitely many for p in (p_JL, p_L).