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We use the formalism of the R{\'e}nyi entropies to establish the symmetryrange of extremal functions in a family of subcriti-calCaffarelli-Kohn-Nirenberg inequalities. By extremal functions we mean functionswhich realize the equality case in the inequalities, written with optimalconstants. The method extends recent results on criticalCaffarelli-Kohn-Nirenberg inequalities. Using heuristics given by a nonlineardiffusion equation, we give a variational proof of a symmetry result, byestablishing a rigidity theorem: in the symmetry region, all positive criticalpoints have radial symmetry and are therefore equal to the unique positive,radial critical point, up to scalings and multiplications. This result issharp. The condition on the parameters is indeed complementary of the conditionwhich determines the region in which symmetry breaking holds as a consequenceof the linear instability of radial optimal functions. Compared to the criticalcase, the subcritical range requires new tools. The Fisher information has tobe replaced by R{\'e}nyi entropy powers, and since some invariances are lost,the estimates based on the Emden-Fowler transformation have to be modified.

Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities