We consider maps between spheres \(S^n\) to \(S^\ell\) that minimize the Sobolev-space energy \(W^{s,n/s}\) for some \(s \in (0,1)\) in a given homotopy class. The basic question is: in which homotopy class does a minimizer exist? This is a nontrivial question since the energy under consideration is conformally invariant and bubbles can form. Sacks-Uhlenbeck theory tells us that minimizers exist in a set of homotopy classes that generates the whole homotopy group \(\pi_{n}(\S^\ell)\). In some situations explicit examples are known if n/s = 2 or s=1. In our talk we are interested in the stability of the above question in dependence of s. We can show that as s varies locally, the set of homotopy classes in which minimizers exist can be chosen stable. We also discuss that the minimum \(W^{s,n/s}\)-energy in homotopy classes is continuously depending on s. Joint work with K. Mazowiecka (U Warsaw) This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.