Nonzero positive solutions of systems of elliptic boundary value problems
HTML articles powered by AMS MathViewer
- by K. Q. Lan
- Proc. Amer. Math. Soc. 139 (2011), 4343-4349
- DOI: https://doi.org/10.1090/S0002-9939-2011-10840-2
- Published electronically: April 4, 2011
- PDF | Request permission
Abstract:
A new result on existence of nonzero positive solutions of systems of second order elliptic boundary value problems is obtained under some sublinear conditions involving the principle eigenvalues of the corresponding linear systems. Results on eigenvalue problems of such elliptic systems are derived and generalize some previous results on the eigenvalue problems of systems of Laplacian elliptic equations. Applications of our results are given to two such systems with specific nonlinearities.References
- 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
- Herbert Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, J. Functional Analysis 11 (1972), 346–384. MR 0358470, DOI 10.1016/0022-1236(72)90074-2
- H. Berestycki and P.-L. Lions, Some applications of the method of super and subsolutions, Bifurcation and nonlinear eigenvalue problems (Proc., Session, Univ. Paris XIII, Villetaneuse, 1978) Lecture Notes in Math., vol. 782, Springer, Berlin, 1980, pp. 16–41. MR 572249
- D. D. Hai, Existence and uniqueness of solutions for quasilinear elliptic systems, Proc. Amer. Math. Soc. 133 (2005), no. 1, 223–228. MR 2085173, DOI 10.1090/S0002-9939-04-07602-6
- D. D. Hai and R. Shivaji, An existence result on positive solutions for a class of semilinear elliptic systems, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), no. 1, 137–141. MR 2039906, DOI 10.1017/S0308210500003115
- D. D. Hai and R. Shivaji, An existence result on positive solutions for a class of $p$-Laplacian systems, Nonlinear Anal. 56 (2004), no. 7, 1007–1010. MR 2038734, DOI 10.1016/j.na.2003.10.024
- D. D. Hai and Haiyan Wang, Nontrivial solutions for $p$-Laplacian systems, J. Math. Anal. Appl. 330 (2007), no. 1, 186–194. MR 2302915, DOI 10.1016/j.jmaa.2006.07.072
- P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), no. 4, 441–467. MR 678562, DOI 10.1137/1024101
- Roger D. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Kreĭn-Rutman theorem, Fixed point theory (Sherbrooke, Que., 1980) Lecture Notes in Math., vol. 886, Springer, Berlin-New York, 1981, pp. 309–330. MR 643014
- Haiyan Wang, An existence theorem for quasilinear systems, Proc. Edinb. Math. Soc. (2) 49 (2006), no. 2, 505–511. MR 2243798, DOI 10.1017/S0013091504001506
- J. R. L. Webb and K. Q. Lan, Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type, Topol. Methods Nonlinear Anal. 27 (2006), no. 1, 91–115. MR 2236412
Bibliographic Information
- K. Q. Lan
- Affiliation: Department of Mathematics, Ryerson University, Toronto, Ontario, Canada M5B 2K3
- MR Author ID: 256493
- Email: klan@ryerson.ca
- Received by editor(s): June 30, 2010
- Received by editor(s) in revised form: October 9, 2010
- Published electronically: April 4, 2011
- Additional Notes: The author was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada.
- Communicated by: Yingfei Yi
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 4343-4349
- MSC (2010): Primary 35J57; Secondary 45G15, 47H10
- DOI: https://doi.org/10.1090/S0002-9939-2011-10840-2
- MathSciNet review: 2823079