Existence of solutions for a semilinear elliptic system
ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 2, pp. 574-586.

This paper deals with the existence of solutions to the following system:

-Δu+u=α α+βa(x)|v| β |u| α-2 uin N -Δv+v=β α+βa(x)|u| α |v| β-2 vin N .
-Δu+u=αα+βa(x)|v|β|u|α-2u inRN-Δv+v=βα+βa(x)|u|α|v|β-2v inRN. With the help of the Nehari manifold and the linking theorem, we prove the existence of at least two nontrivial solutions. One of them is positive. Our main tools are the concentration-compactness principle and the Ekeland’s variational principle.

DOI: 10.1051/cocv/2012022
Classification: 35J45, 35J50, 35J60
Keywords: semilinear elliptic systems, Nehari manifold, concentration-compactness principle, variational methods
@article{COCV_2013__19_2_574_0,
     author = {Benrhouma, Mohamed},
     title = {Existence of solutions for a semilinear elliptic system},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {574--586},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {2},
     year = {2013},
     doi = {10.1051/cocv/2012022},
     mrnumber = {3049724},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2012022/}
}
TY  - JOUR
AU  - Benrhouma, Mohamed
TI  - Existence of solutions for a semilinear elliptic system
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2013
SP  - 574
EP  - 586
VL  - 19
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2012022/
DO  - 10.1051/cocv/2012022
LA  - en
ID  - COCV_2013__19_2_574_0
ER  - 
%0 Journal Article
%A Benrhouma, Mohamed
%T Existence of solutions for a semilinear elliptic system
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2013
%P 574-586
%V 19
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2012022/
%R 10.1051/cocv/2012022
%G en
%F COCV_2013__19_2_574_0
Benrhouma, Mohamed. Existence of solutions for a semilinear elliptic system. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 2, pp. 574-586. doi : 10.1051/cocv/2012022. http://www.numdam.org/articles/10.1051/cocv/2012022/

[1] Y. An, Uniqueness of positive solutions for a class of elliptic systems. J. Math. Anal. Appl. 322 (2006) 1071-1082. | MR | Zbl

[2] V. Benci, Some critical point theorems and applications. Commun. Pure Appl. Math. 33 (1980) 147-172. | MR | Zbl

[3] J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space. J. Differ. Equ. 163 (2000) 41-56. | MR | Zbl

[4] K.-J. Chen, Multiplicity for strongly indefinite semilinear elliptic system. Nonlinear Anal. 72 (2010) 806-821. | MR | Zbl

[5] P. Clement, D.G. Figuereido and E. Mitidieri, Positive solutions of semilinear elliptic systems. Commun. Partial Differ. Equ. 17 (1992) 923-940. | MR | Zbl

[6] D.G. Costa, On a class of elliptic systems in RN. Electron. J. Differ. Equ. 7 (1994) 1-14. | MR | Zbl

[7] R. Cui, Y. Wang and J. Shi, Uniqueness of the positive solution for a class of semilinear elliptic systems. Nonlinear Anal. 67 (2007) 1710-1714. | MR

[8] R. Dalmasso, Existence and uniqueness of positive solutions of semilinear elliptic systems. Nonlinear Anal. 39 (2000) 559-568. | MR | Zbl

[9] D.G. De Finueirdo and J.F. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems. Nonlinear Anal. 33 (1998) 211-234. | MR | Zbl

[10] I. Ekeland, On the variational principle. J. Math. Anal. Appl. 47 (1974) 324-353. | MR | Zbl

[11] L. Gongbao and W. Chunhua, The existence of nontrivial solutions to a semilinear elliptic system on RN without the Ambrosetti-Rabinowitz condition. Acta Math. Sci. B 30 (2010) 1917-1936. | MR | Zbl

[12] D.D. Hai, Uniqueness of positive solutions for semilinear elliptic systems. J. Math. Anal. Appl. 313 (2006) 761-767. | MR | Zbl

[13] A.V. Lair and A.W. Wood, Existence of entire large positive solutions of semilinear elliptic systems, J. Differ. Equ. 164 (2000) 380-394. | MR | Zbl

[14] G.B. Li and J.F. Yang, Asymptotically linear elliptic systems. Commun. Partial Differ. Equ. 29 (2004) 925-954. | MR | Zbl

[15] P.L. Lions, the concentration-compactness principle in the calculus of variations. The locally compact case, part 1. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1 (1984) 109-145. | Numdam | MR | Zbl

[16] P.L. Lions, the concentration-compactness principle in the calculus of variations. The locally compact case, part 2. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2 (1984) 223-283. | Numdam | MR | Zbl

[17] Z. Nehari, On a class of nonlinear second-order differential equations. Trans. Amer. Math. Soc. 95 (1960) 101-123. | MR | Zbl

[18] W.-M. Ni, Some minimax principles and their applications in nonlinear elliptic equations. J. Anal. Math. 37 (1980) 248-275. | MR | Zbl

[19] P.H. Rabinowitz, Some critical point theorems and applications to semilinear elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5 (1978) 215-223. | Numdam | MR | Zbl

[20] J. Serrin and H. Zou, Nonexistence of positive solutions of Lane-Emden systems. Differ. Integral Equ. 9 (1996) 635-653. | MR | Zbl

[21] T.-F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions. Nonlinear Anal. 68 (2008) 1733-1745. | MR | Zbl

Cited by Sources: