Proceedings of the American Mathematical Society

On the existence and multiplicity of positive solutions for some indefinite nonlinear eigenvalue problem

Author(s): Manuel Delgado; Antonio Suárez
Journal: Proc. Amer. Math. Soc. 132 (2004), 1721-1728.
MSC (2000): Primary 35J65; Secondary 35B32, 35P30
Posted: January 20, 2004
Retrieve article in: PDF

Abstract: This paper is concerned with the existence, uniqueness and/or multiplicity, and stability of positive solutions of an indefinite weight elliptic problem with concave or convex nonlinearity. We use mainly bifurcation methods to obtain our results.

1.: S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations, 1 (1993), 439-475. MR 97a:35057
2.: H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709. MR 54:3519
3.: H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374. MR 99e:35057
4.: H. Berestycki, I. Capuzzo-Dolcetta, and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal., 4 (1994), 59-78. MR 96d:35041
5.: H. Berestycki, I. Capuzzo-Dolcetta, and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, NoDEA Nonlinear Differential Equations Appl., 2 (1995), 553-572. MR 96i:35033
6.: K. J. Brown and P. Hess, Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem, Differential Integral Equations, 3 (1990), 201-207. MR 90j:35009
7.: W. Chen and C. Li, Indefinite elliptic problems in a domain, Discrete Contin. Dynam. Systems, 3 (1997), 333-340. MR 98a:35035
8.: S. Cingolani and J. L. Gámez, Positive solutions of a semilinear elliptic equation on $\mathbb{R}^N$ with indefinite nonlinearity, Adv. Differential Equations, 1 (1996), 773-791. MR 97e:35046
9.: D. S. Cohen and T. W. Laestch, Nonlinear boundary value problems suggested by chemical reactor theory, J. Differential Equations, 7 (1970), 217-226. MR 41:3994
10.: M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. MR 44:5836
11.: J. L. Gámez, Sub- and super-solutions in bifurcation problems, Nonlinear Anal., 28 (1997), 625-632. MR 97j:35047
12.: R. Gómez-Reñasco and J. López-Gómez, The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction-diffusion equations, J. Differential Equations, 167 (2000), 36-72. MR 2001k:35168
13.: P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999-1030. MR 81m:35102
14.: B. Ko and K. J. Brown, The existence of positive solutions for a class of indefinite weight semilinear elliptic boundary value problems, Nonlinear Anal., 39 (2000), 587-597. MR 2000k:35098
15.: J. López-Gómez, The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Differential Equations, 127 (1996), 263-294. MR 97b:35037
16.: J. López-Gómez, On the existence of positive solutions for some indefinite superlinear elliptic problems, Comm. Partial Differential Equations, 22 (1997), 1787-1804. MR 99m:35075
17.: P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. MR 46:745

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35J65, 35B32, 35P30

Manuel Delgado
Affiliation: Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Fac. Matemáticas, C/. Tarfia s/n, C.P. 41012, Universidad de Sevilla, Spain
Email: madelgado@us.es

Antonio Suárez
Affiliation: Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Fac. Matemáticas, C/. Tarfia s/n, C.P. 41012, Universidad de Sevilla, Spain
Email: suarez@us.es

DOI: 10.1090/S0002-9939-04-07233-8
PII: S 0002-9939(04)07233-8
Keywords: Indefinite weight elliptic problem, nonlinear eigenvalue problem, bifurcation method
Received by editor(s): August 20, 2002 and in revised form, January 24, 2003
Posted: January 20, 2004
Additional Notes: The authors thank the Spanish Ministry of Science and Technology for research support under grant BFM2000-0797.
Communicated by: David S. Tartakoff
Copyright of article: Copyright 2004, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS