Asymptotic Analysis - Volume 73, issue 4

Authors: Le Peutrec, Dorian

Article Type: Research Article

Abstract: In this article, we are interested in the exponentially small eigenvalues of the self adjoint realization of the semiclassical Witten Laplacian associated with some Morse function, in the general framework of p-forms, on a connected compact Riemannian manifold without boundary. Our purpose is to notice that the knowledge of (the asymptotic formulae for) the smallest non-zero eigenvalues of the self adjoint realization of the semiclassical Witten Laplacian acting on functions, presented by Helffer, Klein and Nier in Matematica Contemporanea 26 (2004), 41–85, essentially contains all the necessary information to the treatment of the case of oriented surfaces, for p-forms.

Keywords: Witten complex, exponentially small eigenvalues, differential p-forms on surfaces

DOI: 10.3233/ASY-2011-1036

Citation: Asymptotic Analysis, vol. 73, no. 4, pp. 187-201, 2011

Authors: Wang, Liping

Article Type: Research Article

Abstract: We consider the Neumann problem for the Hénon equation  −Δu+u=|x|2α u(N+2)/(N−2) , u>0, in Ω,  ∂u/∂n=0   on ∂Ω,  (0.1) where Ω⊂RN ,N≥3 is a smooth and bounded domain, α>0 and n denotes the outward unit normal vector of ∂Ω. We show that problem (0.1) has infinitely many positive solutions, whose energy can be made arbitrarily large in some (partially symmetric) non-convex domains Ω. This seems to be a new phenomenon for the Hénon equation in bounded domains.

Keywords: Hénon equation, critical exponent, infinitely many positive solutions, energy arbitrarily large

DOI: 10.3233/ASY-2011-1037

Citation: Asymptotic Analysis, vol. 73, no. 4, pp. 203-223, 2011

Authors: Deng, Shengbing | Pistoia, Angela

Article Type: Research Article

Abstract: Let (M,g) be a smooth, compact Riemannian manifold of dimension n≥7. We consider the Paneitz–Branson type equation Δg 2 u−divg (A du)+au=|u|2♯ −2−ε u in M, where Δg =−divg ∇ is the Laplace–Beltrami operator, A is a smooth symmetrical (2,0)-tensor fields, a is a smooth function on M, 2♯ =2n/(n−4) is the critical exponent for the Sobolev embedding and ε is a small positive parameter. Under suitable conditions on the Trg A, we construct solutions uε which blow up at one point of the manifold as ε goes to zero.

Keywords: Paneitz–Branson type equations, critical exponent, blow up solutions, Liapunov–Schmidt reduction procedure

DOI: 10.3233/ASY-2011-1039

Citation: Asymptotic Analysis, vol. 73, no. 4, pp. 225-248, 2011

Article Type: Other

Citation: Asymptotic Analysis, vol. 73, no. 4, pp. 249-249, 2011