## Real analyticity and non-degeneracy.(English)Zbl 1040.35033

The following theorem is proved: Let $$U$$ be a real Banach space and $$F: U\times \mathbb{R}\to U$$ an analytic Fredholm map of index one with $$F(u_0,0)= 0$$, let $$\partial_uF(u_0,0)$$ be invertible, and let the set $$\{(u,t)\in U\times [0,1]: F(u,t)= 0\}$$ be compact. Then there exists a subset $$T$$ of $$[0,1)$$ consisting of isolated points such that for all $$t\in [0,1)\setminus T$$ the equation $$F(u,t)= 0$$ has an odd number of non-degenerated solutions. Note that this theorem is false for $$C^\infty$$ maps.
Applications are given to positive solutions of elliptic PDEs of the type $$-\Delta u= u^p$$ on bounded domains $$\Omega\subset \mathbb{R}^N$$, on strips $$S\subset \mathbb{R}^N$$ and on the whole space $$\mathbb{R}^N$$. For that the following surprising result is shown: The superposition operator $$u\mapsto u^p$$ is analytic on certain open sets in certain subspaces of the space of continuous functions $$u:\Omega\to \mathbb{R}$$ with $$u= 0$$ on $$\partial\Omega$$, even if $$p$$ is non-integer.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 58C10 Holomorphic maps on manifolds 46T25 Holomorphic maps in nonlinear functional analysis 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
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