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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, Nemytskiń≠, Uryson, etc.)
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