## The equator map and the negative exponential functional.(English)Zbl 0768.58012

The study of stability properties of the equator map $$u^*: B^ n \to S^ n$$, $$n \geq 3$$, provided insight into the regularity theory of harmonic maps between Riemannian manifolds (here $$B^ n$$ denotes the Euclidean unit $$n$$-ball, $$S^ n$$ the Euclidean unit $$n$$-sphere and $$u^*(x)=({x\over | x|},0))$$. In particular, $$u^*$$ is a weakly harmonic map; i.e., a weak solution of the Euler-Lagrange equation associated with the energy functional $$E(u) = {1\over 2}\int| Du|^ 2$$.
The main aim of this paper is to replace $$E(u)$$ by the negative- exponential energy functional $$NE(u)=\int e^{{1\over 2}| Du|^ 2}$$; the equator map $$u^*$$ is then shown to be an unstable critical point of $$NE(u)$$ for all $$n \geq 2$$.
Reviewer: A.Ratto (Brest)

### MSC:

 58E20 Harmonic maps, etc. 35B35 Stability in context of PDEs
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### References:

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