Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

Hopf fibration and singularly perturbed elliptic equations
Pages: 823 - 838, Issue 4, August 2014

doi:10.3934/dcdss.2014.7.823      Abstract        References        Full text (513.4K)           Related Articles

Bernhard Ruf - Dip. di Matematica, Universita degli Studi, Via Saldini 50, 20133 Milano, Italy (email)
P. N. Srikanth - T.I.F.R. CAM, Bangalore, 560065, India (email)

1 A. Ambrosetti, A. Malchiodi and W.-M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres. II, Indiana Univ. Math. J., 53 (2004), 297-329.       
2 M. Badiale and T. d'Aprile, Concentration around a sphere for a singularly perturbed Schrödinger equation, Nonlinear Anal. Ser. A: Theory Methods, 49 (2002), 947-985.       
3 V. Benci and T. d'Aprile, The semiclassical limit of the nonlinear Schrödinger equation in a radial potential, J. Differential Equations, 184 (2002), 109-138.       
4 A. Besse, Einstein Manifolds, Springer, 1987.       
5 J. Byeon and J. Park, Singularly perturbed nonlinaer elliptic problems on manifolds, Calculus of Variations, 24 (2005), 459-477.       
6 D. Cao and E. S. Noussair, Existance of symmetri multi-peaked solutions to singularly perturbed semilinear elliptic problems, Comm. PDE, 25 (2000), 2185-2232.       
7 M. Clapp, M. Ghimenti and A. M. Micheletti, Solutions to a singularly perturbed supercritical elliptic equation on a Riemannian manifold concentrating at a submanifold, preprint
8 M. del Pino and P. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J., 48 (1999), 883-898.       
9 M. del Pino, M. Kowalczyk, Michal and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146.       
10 A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.       
11 B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.       
12 S. Ishihara, Quaternion Kählerian manifolds, J. Diff. Geometry, 9 (1974), 483-900.       
13 N. Johnson, http://www.nilesjohnson.net.
14 S. Karigiannis, http://www.math.uwaterloo.ca/~ karigian/talks/CUMC-2010.pdf.
15 C. S. Lin, W.-M. Ni and I. Takagi, Large amplituted stationary soutions to a chemotaxis system, J. Diff. Equ., 72 (1988), 1-27.       
16 A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains, Geometric and Functional Analysis GAFA, 15 (2005), 1162-1222.       
17 A. Malchiodi and M. Montenegro, Boundary layers of arbitrary dimension for a singularly perturbed Neumann problem, Mat. Contemp., 27 (2004), 117-146.       
18 A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143.       
19 B. B. Manna and P. N. Srikanth, On the solutions of a singular elliptic equation concentrating on two orthogonal spheres, preprint.
20 W.-M. Ni and I. Takagi, On the shape of least energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.       
21 W.-M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 68 (1995), 731-768.       
22 Y-G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$ , Comm. Partial Differential Equations, 13 (1988), 1499-1519.       
23 B. O'Neill, Semi-Riemannian Geometry, Academic Press, 1983.       
24 F. Pacella and P. N. Srikanth, A reduction method for semilinear elliptic equations and solutions concentrating on spheres, to appear.
25 B. Ruf and P. N. Srikanth, Singularly pertubed elliptic equations with solutions concentrating on a $1$-dimensional orbit, JEMS, 12 (2010), 413-427.       
26 B. Ruf and P. N. Srikanth, Concentration on Hopf-Fibres for singularly perturbed elliptic equations, preprint.
27 J. C. Wood, Harmonic morphisms between Riemannian manifolds, in Modern Trends in Geometry and Topology, Cluj Univ. Press, Cluj-Napoca, 2006, 397-414.       

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