Nonzero positive solutions of systems of elliptic boundary value problems

Author:
K. Q. Lan

Journal:
Proc. Amer. Math. Soc. **139** (2011), 4343-4349

MSC (2010):
Primary 35J57; Secondary 45G15, 47H10

Published electronically:
April 4, 2011

MathSciNet review:
2823079

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**Herbert Amann,*Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces*, SIAM Rev.**18**(1976), no. 4, 620–709. MR**0415432****2.**Herbert Amann,*On the number of solutions of nonlinear equations in ordered Banach spaces*, J. Functional Analysis**11**(1972), 346–384. MR**0358470****3.**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****4.**D. D. Hai,*Existence and uniqueness of solutions for quasilinear elliptic systems*, Proc. Amer. Math. Soc.**133**(2005), no. 1, 223–228. MR**2085173**, 10.1090/S0002-9939-04-07602-6**5.**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**, 10.1017/S0308210500003115**6.**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**, 10.1016/j.na.2003.10.024**7.**D. D. Hai and Haiyan Wang,*Nontrivial solutions for 𝑝-Laplacian systems*, J. Math. Anal. Appl.**330**(2007), no. 1, 186–194. MR**2302915**, 10.1016/j.jmaa.2006.07.072**8.**P.-L. Lions,*On the existence of positive solutions of semilinear elliptic equations*, SIAM Rev.**24**(1982), no. 4, 441–467. MR**678562**, 10.1137/1024101**9.**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****10.**Haiyan Wang,*An existence theorem for quasilinear systems*, Proc. Edinb. Math. Soc. (2)**49**(2006), no. 2, 505–511. MR**2243798**, 10.1017/S0013091504001506**11.**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**

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Additional Information

**K. Q. Lan**

Affiliation:
Department of Mathematics, Ryerson University, Toronto, Ontario, Canada M5B 2K3

Email:
klan@ryerson.ca

DOI:
https://doi.org/10.1090/S0002-9939-2011-10840-2

Keywords:
Systems of elliptic boundary value problems,
sublinear condition,
nonzero positive solutions,
fixed point index

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

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.