Threshold condensation to singular support for a Riesz equilibrium problem

We compute the equilibrium measure in dimension $$d=s+4$$ associated to a Riesz s-kernel interaction with an external field given by power of Euclidean norm. Our study reveals that can be mixture continuous part and singular part. Depending on value power, threshold phenomenon occurs consists reduction or condensation In particular, logarithmic case $$s=0$$ ( $$d=4$$ ), there is sphere special radius when becomes quadratic. This contrasts $$d=s+3$$ studied previously, which showed fully dimensional supported ball. approach makes use, among other tools, Frostman Euler–Lagrange variational characterization, Funk–Hecke formula, Gegenbauer orthogonal polynomials, hypergeometric functions.