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On the Morse indices of sign changing solutions of nonlinear elliptic problems. (English) Zbl 0946.35023
Let \(f\colon {\mathbb{R}}\to {\mathbb{R}}\) be of class \({\mathcal C}^1\) with \(f(0)=0\) and consider the problem (D) \(-\Delta u=f(u)\) in a Lipschitz bounded domain of \({\mathbb{R}}^N\) with Dirichlet boundary data. The Morse index and the sign of the solutions of (D) are investigated by some new abstract critical point theorems for functionals on partially ordered Hilbert spaces. This additional information is used to establish the existence of multiple solutions of (D).
Under quite similar conditions on the partial order used and on the functional appearing in the abstract results established there, a “Mountain Pass Theorem in order intervals” was recently established by Shujie Li and the third author that was applied to study the existence of multiple solutions and sign-changing solutions of some elliptic problems.

MSC:
35J20 Variational methods for second-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35B50 Maximum principles in context of PDEs
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