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

Delgado, Manuel; Suárez, Antonio

doi:10.1090/S0002-9939-04-07233-8

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