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On an initial value problem modeling evolution and selection in living systems
Hopf fibration and singularly perturbed elliptic equations
1. | Dip. di Matematica, Universita degli Studi, Via Saldini 50, 20133 Milano |
2. | T.I.F.R. CAM, Bangalore, 560065, India |
References:
[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.
doi: 10.1512/iumj.2004.53.2400. |
[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.
doi: 10.1016/S0362-546X(01)00717-9. |
[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.
doi: 10.1006/jdeq.2001.4138. |
[4] | |
[5] |
J. Byeon and J. Park, Singularly perturbed nonlinaer elliptic problems on manifolds, Calculus of Variations, 24 (2005), 459-477.
doi: 10.1007/s00526-005-0339-4. |
[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.
doi: 10.1080/03605300008821582. |
[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.
doi: 10.1512/iumj.1999.48.1596. |
[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.
doi: 10.1002/cpa.20135. |
[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.
doi: 10.1016/0022-1236(86)90096-0. |
[11] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
doi: 10.1007/BF01221125. |
[12] |
S. Ishihara, Quaternion Kählerian manifolds, J. Diff. Geometry, 9 (1974), 483-900. |
[13] |
, N. Johnson,, , ().
|
[14] |
, S. Karigiannis,, , ().
|
[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.
doi: 10.1016/0022-0396(88)90147-7. |
[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.
doi: 10.1007/s00039-005-0542-7. |
[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.
doi: 10.1215/S0012-7094-04-12414-5. |
[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.
doi: 10.1002/cpa.3160440705. |
[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.
doi: 10.1002/cpa.3160480704. |
[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.
doi: 10.1080/03605308808820585. |
[23] | |
[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.
doi: 10.4171/JEMS/203. |
[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. |
show all references
References:
[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.
doi: 10.1512/iumj.2004.53.2400. |
[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.
doi: 10.1016/S0362-546X(01)00717-9. |
[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.
doi: 10.1006/jdeq.2001.4138. |
[4] | |
[5] |
J. Byeon and J. Park, Singularly perturbed nonlinaer elliptic problems on manifolds, Calculus of Variations, 24 (2005), 459-477.
doi: 10.1007/s00526-005-0339-4. |
[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.
doi: 10.1080/03605300008821582. |
[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.
doi: 10.1512/iumj.1999.48.1596. |
[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.
doi: 10.1002/cpa.20135. |
[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.
doi: 10.1016/0022-1236(86)90096-0. |
[11] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
doi: 10.1007/BF01221125. |
[12] |
S. Ishihara, Quaternion Kählerian manifolds, J. Diff. Geometry, 9 (1974), 483-900. |
[13] |
, N. Johnson,, , ().
|
[14] |
, S. Karigiannis,, , ().
|
[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.
doi: 10.1016/0022-0396(88)90147-7. |
[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.
doi: 10.1007/s00039-005-0542-7. |
[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.
doi: 10.1215/S0012-7094-04-12414-5. |
[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.
doi: 10.1002/cpa.3160440705. |
[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.
doi: 10.1002/cpa.3160480704. |
[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.
doi: 10.1080/03605308808820585. |
[23] | |
[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.
doi: 10.4171/JEMS/203. |
[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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