The Dirichlet-problem for harmonic maps from the disk into a lorentzian warped product
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.
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 and a stratified Morse function . 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 .