Concentrating solutions of some singularly perturbed elliptic equations

Andrea Malchiodi

Front. Math. China ›› 2008, Vol. 3 ›› Issue (2) : 239 -252.

PDF (184KB)
Front. Math. China ›› 2008, Vol. 3 ›› Issue (2) : 239 -252. DOI: 10.1007/s11464-008-0015-z
Research Article

Concentrating solutions of some singularly perturbed elliptic equations

Author information +
History +
PDF (184KB)

Abstract

We study singularly perturbed elliptic equations arising from models in physics or biology, and investigate the asymptotic behavior of some special solutions. We also discuss some connections with problems arising in differential geometry.

Keywords

Geometric partial differential equation / singularly perturbed / elliptic equation / concentration phenomenon

Cite this article

Download citation ▾
Andrea Malchiodi. Concentrating solutions of some singularly perturbed elliptic equations. Front. Math. China, 2008, 3(2): 239-252 DOI:10.1007/s11464-008-0015-z

登录浏览全文

4963

注册一个新账户 忘记密码

References

[1]

Ambrosetti A., Badiale M., Cingolani S. Semiclassical states of nonlinear Schrödinger equations. Arch Rational Mech Anal, 1997, 140: 285-300.

[2]

Ambrosetti A., Felli V., Malchiodi A. Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. J Eur Math Soc, 2005, 7: 117-144.

[3]

Ambrosetti A., Malchiodi A. Perturbation Methods and Semilinear Elliptic Problems on ℝN. Progr in Math, 2005, Boston: Birkhäuser.

[4]

Ambrosetti A., Malchiodi A., Ni W. M. Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, Part I. Comm Math Phys, 2003, 235: 427-466.

[5]

Ambrosetti A, Malchiodi A, Ni W M Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, Part II. Indiana Univ Math J, 2004, 53(2): 297-329.

[6]

Ambrosetti A., Malchiodi A., Ruiz D. Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity. J d’Analyse Math, 2006, 98: 317-348.

[7]

Ambrosetti A, Malchiodi A, Secchi S Multiplicity results for some nonlinear singularly perturbed elliptic problems on. Arch Rat Mech Anal, 2001, 159(3): 253-271.

[8]

Arioli G., Szulkin A. A semilinear Schrödinger equation in the presence of a magnetic field. Arch Rat Mech Anal, 2003, 170: 277-295.

[9]

Badiale M., D’Aprile T. Concentration around a sphere for a singularly perturbed Schrödinger equation. Nonlinear Anal, Ser A: Theory Methods, 2002, 49(7): 947-985.

[10]

Bartsch T., Peng S. Semiclassical symmetric Schrödinger equations: existence of solutions concentrating simultaneously on several spheres. Z Angew Math Phys, 2007, 58(5): 778-804.

[11]

Benci V., D’Aprile T. The semiclassical limit of the nonlinear Schrödinger equation in a radial potential. J Differential Equations, 2002, 184(1): 109-138.

[12]

Berestycki H., Lions P.-L. Nonlinear scalar field equations. I. Existence of a ground state. Arch Ration Mech Anal, 1983, 82(4): 313-345.

[13]

Berestycki H., Lions P.-L. Nonlinear scalar field equations. II. Existence of infinitely many solutions. Arch Ration Mech Anal, 1983, 82(4): 347-375.

[14]

Byeon J., Wang Z. Q. Standing waves with a critical frequency for nonlinear Schrödinger equations. Arch Ration Mech Anal, 2002, 165: 295-316.

[15]

Casten R. G., Holland C. J. Instability results for reaction diffusion equations with Neumann boundary conditions. J Diff Eq, 1978, 27(2): 266-273.

[16]

Dancer E. N. Stable and finite Morse index solutions on ℝN or on bounded domains with small diffusion. Trans Amer Math Soc, 2005, 357(3): 1225-1243.

[17]

Dancer E. N., Yan S. Multipeak solutions for a singularly perturbed Neumann problem. Pacific J Math, 1999, 189(2): 241-262.

[18]

Dancer E. N., Yan S. A new type of concentration solutions for a singularly perturbed elliptic problem. Trans Amer Math Soc, 2007, 359(4): 1765-1790.

[19]

D’Aprile T. On a class of solutions with non-vanishing angular momentum for nonlinear Schrödinger equations. Diff Int Equ, 2003, 16(3): 349-384.

[20]

Del Pino M., Felmer P. Local mountain passes for semilinear elliptic problems in unbounded domains. Calc Var, 1996, 4: 121-137.

[21]

Del Pino M., Felmer P. Semi-classical states for nonlinear schrödinger equations. J Funct Anal, 1997, 149: 245-265.

[22]

Del Pino M., Felmer P., Wei J. On the role of the mean curvature in some singularly perturbed Neumann problems. SIAM J Math Anal, 1999, 31: 63-79.

[23]

Del Pino M., Kowalczyk M., Wei J. Concentration at curves for nonlinear Schrödinger equations. Comm Pure Appl Math, 2007, 60(1): 113-146.

[24]

Floer A., Weinstein A. Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J Funct Anal, 1986, 69: 397-408.

[25]

Gierer A., Meinhardt H. A theory of biological pattern formation. Kybernetik (Berlin), 1972, 12: 30-39.

[26]

Grossi M. Some results on a class of nonlinear Schrödinger equations. Math Zeit, 2000, 235: 687-705.

[27]

Grossi M., Pistoia A., Wei J. Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory. Calc Var Partial Differential Equations, 2000, 11(2): 143-175.

[28]

Gui C. Multipeak solutions for a semilinear Neumann problem. Duke Math J, 1996, 84: 739-769.

[29]

Gui C., Wei J. On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems. Canad J Math, 2000, 52(3): 522-538.

[30]

Gui C., Wei J., Winter M. Multiple boundary peak solutions for some singularly perturbed Neumann problems. Ann Inst H Poincaré Anal Non Linéaire, 2000, 17(1): 47-82.

[31]

Jeanjean L., Tanaka K. A positive solution for a nonlinear Schrödinger equation on ℝN. Indiana Univ Math J, 2005, 54(2): 443-464.

[32]

Kato T. Perturbation Theory for Linear Operators. Grundlehren der Mathematischen Wissenschaften, 1976 2nd ed Berlin-New York: Springer-Verlag.

[33]

Kwong M. K. Uniqueness of positive solutions of −Δu + u + up = 0 in RN. Arch Ration Mech Anal, 1989, 105: 243-266.

[34]

Li Y. Y. On a singularly perturbed equation with Neumann boundary conditions. Comm Partial Differential Equations, 1998, 23: 487-545.

[35]

Li Y. Y., Nirenberg L. The Dirichlet problem for singularly perturbed elliptic equations. Comm Pure Appl Math, 1998, 51: 1445-1490.

[36]

Lin C. S., Ni W.-M., Takagi I. Large amplitude stationary solutions to a chemotaxis systems. J Differential Equations, 1988, 72: 1-27.

[37]

Mahmoudi F., Malchiodi A. Concentration on minimal submanifolds for a singularly perturbed Neumann problem. Adv in Math, 2007, 209-2: 460-525.

[38]

Mahmoudi F., Mazzeo R., Pacard F. Constant mean curvature hypersurfaces condensing along a submanifold. Geom Funct Anal, 2006, 16: 924-958.

[39]

Malchiodi A. Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains. Geometric and Functional Analysis, 2005, 15–6: 1162-1222.

[40]

Malchiodi A., Montenegro M. Boundary concentration phenomena for a singularly perturbed elliptic problem. Comm Pure Appl Math, 2002, 15: 1507-1568.

[41]

Malchiodi A., Montenegro M. Multidimensional boundary-layers for a singularly perturbed Neumann problem. Duke Math J, 2004, 124(1): 105-143.

[42]

Malchiodi A., Ni W. M., Wei J. Multiple clustered layer solutions for semilinear Neumann problems on a ball. Ann Inst H Poincaré Anal Non Linéaire, 2005, 22: 143-163.

[43]

Malchiodi A., Ni W. M., Wei J. Boundary clustered interfaces for the Allen-Cahn equation. Pacific J of Math, 2007, 229(2): 447-468.

[44]

Malchiodi A, Wei J. Boundary interface for the Allen-Cahn equation. J Fixed Point Theory and Appl (in press)

[45]

Matano H. Asymptotic behavior and stability of solutions of semilinear diffusion equations. Publ Res Inst Math Sci, 1979, 15(2): 401-454.

[46]

Mazzeo R., Pacard F. Foliations by constant mean curvature tubes. Comm Anal Geom, 2005, 13(4): 633-670.

[47]

Molle R., Passaseo D. Concentration phenomena for solutions of superlinear elliptic problems. Ann Inst H Poincaré Anal Non Linéaire, 2006, 23(1): 63-84.

[48]

Nardulli S. Le profil isopérimétrique des variétés riemanniennes compactes pour les petits volumes. Ph D thesis, 2006, Orsay: Universit Paris-Sud XI.

[49]

Ni W. M. Diffusion, cross-diffusion, and their spike-layer steady states. Notices Amer Math Soc, 1998, 45(1): 9-18.

[50]

Ni W. M., Takagi I. On the shape of least-energy solution to a semilinear Neumann problem. Comm Pure Appl Math, 1991, 41: 819-851.

[51]

Ni W. M., Takagi I. Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math J, 1993, 70: 247-281.

[52]

Ni W. M., Takagi I., Yanagida E. Stability of least energy patterns of the shadow system for an activator-inhibitor model. Recent topics in mathematics moving toward science and engineering. Japan J Indust Appl Math, 2001, 18(2): 259-272.

[53]

Oh Y.-G. On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potentials. Comm Math Phys, 1990, 131: 223-253.

[54]

Strauss W. A. Existence of solitary waves in higher dimensions. Comm Math Phys, 1977, 55(2): 149-162.

[55]

Turing A. M. The chemical basis of morphogenesis. Phil Trans Royal Soc London, Ser B, Biological Sciences, 1952, 237: 37-72.

[56]

Wang Z.-Q. On the existence of multiple, single-peaked solutions for a semilinear Neumann problem. Arch Ration Mech Anal, 1992, 120: 375-399.

[57]

Wei J. On the boundary spike layer solutions of a singularly perturbed semilinear Neumann problem. J Differential Equations, 1997, 134: 104-133.

[58]

Ye R. Foliation by constant mean curvature spheres. Pacific J Math, 1991, 147(2): 381-396.

[59]

Ye R. Constant mean curvature foliation: Singularity structure and curvature estimate. Pacific J Math, 1996, 174(2): 569-587.

AI Summary AI Mindmap
PDF (184KB)

875

Accesses

0

Citation

Detail

Sections
Recommended

AI思维导图

/