On the existence and multiplicity of positive solutions for some indefinite nonlinear eigenvalue problem
HTML articles powered by AMS MathViewer
- by Manuel Delgado and Antonio Suárez PDF
- Proc. Amer. Math. Soc. 132 (2004), 1721-1728 Request permission
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.References
- Stanley Alama and Gabriella Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations 1 (1993), no. 4, 439–475. MR 1383913, DOI 10.1007/BF01206962
- Herbert Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620–709. MR 415432, DOI 10.1137/1018114
- H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations 146 (1998), no. 2, 336–374. MR 1631287, DOI 10.1006/jdeq.1998.3440
- H. Berestycki, I. Capuzzo-Dolcetta, and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal. 4 (1994), no. 1, 59–78. MR 1321809, DOI 10.12775/TMNA.1994.023
- Henri Berestycki, Italo Capuzzo-Dolcetta, and Louis Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, NoDEA Nonlinear Differential Equations Appl. 2 (1995), no. 4, 553–572. MR 1356874, DOI 10.1007/BF01210623
- 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), no. 2, 201–207. MR 1025173
- Wenxiong Chen and Congming Li, Indefinite elliptic problems in a domain, Discrete Contin. Dynam. Systems 3 (1997), no. 3, 333–340. MR 1444198, DOI 10.3934/dcds.1997.3.333
- S. Cingolani and J. L. Gámez, Positive solutions of a semilinear elliptic equation on $\mathbf R^N$ with indefinite nonlinearity, Adv. Differential Equations 1 (1996), no. 5, 773–791. MR 1392005
- Donald S. Cohen and Theodore W. Laetsch, Nonlinear boundary value problems suggested by chemical reactor theory, J. Differential Equations 7 (1970), 217–226. MR 259356, DOI 10.1016/0022-0396(70)90106-3
- Michael G. Crandall and Paul H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321–340. MR 0288640, DOI 10.1016/0022-1236(71)90015-2
- J. L. Gámez, Sub- and super-solutions in bifurcation problems, Nonlinear Anal. 28 (1997), no. 4, 625–632. MR 1420381, DOI 10.1016/0362-546X(95)00174-T
- 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), no. 1, 36–72. MR 1785114, DOI 10.1006/jdeq.2000.3772
- Peter Hess and Tosio Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations 5 (1980), no. 10, 999–1030. MR 588690, DOI 10.1080/03605308008820162
- Bongsoo Ko and Ken Brown, The existence of positive solutions for a class of indefinite weight semilinear elliptic boundary value problems, Nonlinear Anal. 39 (2000), no. 5, Ser. A: Theory Methods, 587–597. MR 1727270, DOI 10.1016/S0362-546X(98)00223-5
- Julián López-Gómez, The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Differential Equations 127 (1996), no. 1, 263–294. MR 1387266, DOI 10.1006/jdeq.1996.0070
- J. López-Gómez, On the existence of positive solutions for some indefinite superlinear elliptic problems, Comm. Partial Differential Equations 22 (1997), no. 11-12, 1787–1804. MR 1629486, DOI 10.1080/03605309708821320
- Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487–513. MR 0301587, DOI 10.1016/0022-1236(71)90030-9
Additional Information
- 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
- Received by editor(s): August 20, 2002
- Received by editor(s) in revised form: January 24, 2003
- Published electronically: 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 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 1721-1728
- MSC (2000): Primary 35J65; Secondary 35B32, 35P30
- DOI: https://doi.org/10.1090/S0002-9939-04-07233-8
- MathSciNet review: 2051133