Multiplicity of self-similar solutions for a critical equation

We consider the equation−Δu−12(x⋅∇u)=f(u)+β|u|2⁎−2u,x∈RN, with β>0, f superlinear and 2⁎:=2N/(N−2) for N⩾3. We prove that, for each k∈N, there exists β⁎=β⁎(k)>0 such that the equation has at least k pairs of solutions provided β∈(0,β⁎). In the proof we use variational methods for the (even) functional associated to the equation.