Remarks on positive solutions to a semilinear Neumann problem

Anna Maria Candela; Monica Lazzo

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1994)

  • Volume: 5, Issue: 3, page 237-246
  • ISSN: 1120-6330

Abstract

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In this paper we study the influence of the domain topology on the multiplicity of solutions to a semilinear Neumann problem. In particular, we show that the number of positive solutions is stable under small perturbations of the domain.

How to cite

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Candela, Anna Maria, and Lazzo, Monica. "Remarks on positive solutions to a semilinear Neumann problem." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 5.3 (1994): 237-246. <http://eudml.org/doc/244143>.

@article{Candela1994,
abstract = {In this paper we study the influence of the domain topology on the multiplicity of solutions to a semilinear Neumann problem. In particular, we show that the number of positive solutions is stable under small perturbations of the domain.},
author = {Candela, Anna Maria, Lazzo, Monica},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Neumann problem; Variational methods; Multiple solutions; peak solutions; multiple solutions; domain variation},
language = {eng},
month = {9},
number = {3},
pages = {237-246},
publisher = {Accademia Nazionale dei Lincei},
title = {Remarks on positive solutions to a semilinear Neumann problem},
url = {http://eudml.org/doc/244143},
volume = {5},
year = {1994},
}

TY - JOUR
AU - Candela, Anna Maria
AU - Lazzo, Monica
TI - Remarks on positive solutions to a semilinear Neumann problem
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1994/9//
PB - Accademia Nazionale dei Lincei
VL - 5
IS - 3
SP - 237
EP - 246
AB - In this paper we study the influence of the domain topology on the multiplicity of solutions to a semilinear Neumann problem. In particular, we show that the number of positive solutions is stable under small perturbations of the domain.
LA - eng
KW - Neumann problem; Variational methods; Multiple solutions; peak solutions; multiple solutions; domain variation
UR - http://eudml.org/doc/244143
ER -

References

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  1. BENCI, V. - CERAMI, G., The effect of the domain topology on the number of solutions of nonlinear elliptic problems. Arch. Rat. Mech. Anal., 114, 1991, 79-93. Zbl0727.35055MR1088278DOI10.1007/BF00375686
  2. BENCI, V. - CERAMI, G., Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology. To appear. Zbl0822.35046MR1384393DOI10.1007/BF01234314
  3. BENCI, V. - CERAMI, G. - PASSASEO, D., On the number of the positive solutions of some nonlinear elliptic problems.A tribute in honour of G. Prodi, Scuola Norm. Sup. Pisa, 1991, 93-107. Zbl0838.35040MR1205376
  4. BERESTYCKI, H. - GALLOUET, T. - KAVIAN, O., Equations de champs scalaires euclidiens nonlinéaires dans le plan. C. R. Acad. Sc. Paris, Série I Math., 297, 1983, 307-310. Zbl0544.35042MR734575
  5. BERESTYCKI, H. - LIONS, P. L., Nonlinear scalar field equations, I-Existence of a ground state. Arch. Rat. Mech. Anal., 82, 1983, 313-375. Zbl0533.35029MR695535DOI10.1007/BF00250555
  6. COLEMAN, S. - GLASER, V. - MARTIN, A., Action minima among solutions to a class of Euclidean scalar field equations. Comm. Math. Phys., 58, 1978, 211-221. MR468913
  7. GIERER, A. - MEINHARDT, H., A theory of biological pattern formation. Kybernetik (Berline), 12, 1972. 
  8. KWONG, M. K., Uniqueness of positive solutions of Δ ⁢ u - u + u p = 0 in R n . Arch. Rat. Mech. Anal., 105, 1989, 243-266. Zbl0676.35032MR969899DOI10.1007/BF00251502
  9. LAZZO, M., Morse theory and multiple positive solutions to a Neumann problem. Ann. Mat. Pura e Appl., to appear. Zbl0849.35034MR1378245DOI10.1007/BF01759261
  10. LIN, C. S. - NI, W. M. - TAKAGI, I., Large amplitude stationary solutions to a chemotaxis system. Jour. Diff. Eq., 72, 1988, 1-27. Zbl0676.35030MR929196DOI10.1016/0022-0396(88)90147-7
  11. MANCINI, G. - MUSINA, R., The role of the boundary in some semilinear Neumann problems. Rend. Sem. Mat. Padova, 88, 1992, 127-138. Zbl0814.35037MR1209119
  12. NI, W. M. - TAKAGI, I., On the shape of least-energy solutions to a semilinear Neumann problem. Comm. Pure Appl. Math., 45, 1991, 819-851. Zbl0754.35042MR1115095DOI10.1002/cpa.3160440705
  13. WANG, Z. Q., On the existence of multiple, single-peaked solutions for a semilinear Neumann problem. Arch. Rat. Mech. Anal., 120, 1992, 375-399. Zbl0784.35035MR1185568DOI10.1007/BF00380322

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