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The Conley index in Hilbert spaces and its applications

K. Gęba, M. Izydorek, A. Pruszko (1999)

Studia Mathematica

We present a generalization of the classical Conley index defined for flows on locally compact spaces to flows on an infinite-dimensional real Hilbert space H generated by vector fields of the form f: H → H, f(x) = Lx + K(x), where L: H → H is a bounded linear operator satisfying some technical assumptions and K is a completely continuous perturbation. Simple examples are presented to show how this new invariant can be applied in searching critical points of strongly indefinite functionals having...

The CR Yamabe conjecture the case n = 1

Najoua Gamara (2001)

Journal of the European Mathematical Society

Let ( M , θ ) be a compact CR manifold of dimension 2 n + 1 with a contact form θ , and L = ( 2 + 2 / n ) Δ b + R its associated CR conformal laplacien. The CR Yamabe conjecture states that there is a contact form θ ˜ on M conformal to θ which has a constant Webster curvature. This problem is equivalent to the existence of a function u such that L u = u 1 + 2 / n , u > 0 on M . D. Jerison and J. M. Lee solved the CR Yamabe problem in the case where n 2 and ( M , θ ) is not locally CR equivalent to the sphere S 2 n + 1 of 𝐂 n . In a join work with R. Yacoub, the CR Yamabe problem...

The existence of positive solution to some asymptotically linear elliptic equations in exterior domains.

Gongbao Li, Gao-Feng Zheng (2006)

Revista Matemática Iberoamericana

In this paper, we are concerned with the asymptotically linear elliptic problem -Δu + λ0u = f(u), u ∈ H01(Ω) in an exterior domain Ω = RnO (N ≥ 3) with O a smooth bounded and star-shaped open set, and limt→+∞ f(t)/t = l, 0 < l < +∞. Using a precise deformation lemma and algebraic topology argument, we prove under our assumptions that the problem possesses at least one positive solution.

The geometric complex for algebraic curves with cone-like singularities and admissible Morse functions

Ursula Ludwig (2010)

Annales de l’institut Fourier

In a previous note the author gave a generalisation of Witten’s proof of the Morse inequalities to the model of a complex singular curve X and a stratified Morse function f . In this note a geometric interpretation of the complex of eigenforms of the Witten Laplacian corresponding to small eigenvalues is provided in terms of an appropriate subcomplex of the complex of unstable cells of critical points of f .

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