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A general multiplicity theorem for certain nonlinear equations in Hilbert spaces

Authors:	Biagio Ricceri

Affiliation:	Department of Mathematics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy

Abstract:	In this paper, we prove the following general result. Let be a real Hilbert space and a continuously Gâteaux differentiable, nonconstant functional, with compact derivative, such that $begin{displaymath}limsup_{Vert xVertto +infty}{{J(x)}over {Vert xVert^2}}leq 0 .end{displaymath}$ Then, for each $rin ]inf_{X}J,sup_{X}J[$ for which the set $J^{-1}([r,+infty[)$ is not convex and for each convex set dense in , there exist $x_0in Scap J^{-1}(]-infty,r[)$ and such that the equation $begin{displaymath}x=lambda J'(x)+x_0end{displaymath}$ has at least three solutions.

Keywords:	Nonlinear equations Hilbert spaces local and global minima critical points level sets minimax theory Chebyshev sets

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