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

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

 $\left\{\begin{array}{c}-\Delta u+u=\frac{\alpha }{\alpha +\beta }{a\left(x\right)|v|}^{\beta }{|u|}^{\alpha -2}u\phantom{\rule{1em}{0ex}}\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}{ℝ}^{N}\hfill \\ -\Delta v+v=\frac{\beta }{\alpha +\beta }{a\left(x\right)|u|}^{\alpha }{|v|}^{\beta -2}v\phantom{\rule{1em}{0ex}}\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}{ℝ}^{N}.\hfill \end{array}\right\$
-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 : https://doi.org/10.1051/cocv/2012022
Classification : 35J45,  35J50,  35J60
Mots clés : 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
DA  - 2013///
SP  - 574
EP  - 586
VL  - 19
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2012022/
UR  - https://www.ams.org/mathscinet-getitem?mr=3049724
UR  - https://doi.org/10.1051/cocv/2012022
DO  - 10.1051/cocv/2012022
LA  - en
ID  - COCV_2013__19_2_574_0
ER  - 
Benrhouma, Mohamed. Existence of solutions for a semilinear elliptic system. ESAIM: Control, Optimisation and Calculus of Variations, Tome 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 2250636 | Zbl 1217.35060

[2] V. Benci, Some critical point theorems and applications. Commun. Pure Appl. Math. 33 (1980) 147-172. | MR 562548 | Zbl 0472.58009

[3] J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space. J. Differ. Equ. 163 (2000) 41-56. | MR 1755067 | Zbl 0952.35033

[4] K.-J. Chen, Multiplicity for strongly indefinite semilinear elliptic system. Nonlinear Anal. 72 (2010) 806-821. | MR 2579347 | Zbl 1183.35110

[5] P. Clement, D.G. Figuereido and E. Mitidieri, Positive solutions of semilinear elliptic systems. Commun. Partial Differ. Equ. 17 (1992) 923-940. | MR 1177298 | Zbl 0818.35027

[6] D.G. Costa, On a class of elliptic systems in RN. Electron. J. Differ. Equ. 7 (1994) 1-14. | MR 1292598 | Zbl 0809.35020

[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 2326023

[8] R. Dalmasso, Existence and uniqueness of positive solutions of semilinear elliptic systems. Nonlinear Anal. 39 (2000) 559-568. | MR 1727272 | Zbl 0940.35091

[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 1617988 | Zbl 0938.35054

[10] I. Ekeland, On the variational principle. J. Math. Anal. Appl. 47 (1974) 324-353. | MR 346619 | Zbl 0286.49015

[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 2778702 | Zbl 1240.35176

[12] D.D. Hai, Uniqueness of positive solutions for semilinear elliptic systems. J. Math. Anal. Appl. 313 (2006) 761-767. | MR 2183334 | Zbl 1211.35107

[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 1765572 | Zbl 0962.35052

[14] G.B. Li and J.F. Yang, Asymptotically linear elliptic systems. Commun. Partial Differ. Equ. 29 (2004) 925-954. | MR 2059153 | Zbl 1140.35406

[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 778970 | Zbl 0541.49009

[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 778974 | Zbl 0704.49004

[17] Z. Nehari, On a class of nonlinear second-order differential equations. Trans. Amer. Math. Soc. 95 (1960) 101-123. | MR 111898 | Zbl 0097.29501

[18] W.-M. Ni, Some minimax principles and their applications in nonlinear elliptic equations. J. Anal. Math. 37 (1980) 248-275. | MR 583639 | Zbl 0462.58016

[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 488128 | Zbl 0375.35026

[20] J. Serrin and H. Zou, Nonexistence of positive solutions of Lane-Emden systems. Differ. Integral Equ. 9 (1996) 635-653. | MR 1401429 | Zbl 0868.35032

[21] T.-F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions. Nonlinear Anal. 68 (2008) 1733-1745. | MR 2388846 | Zbl 1151.35342

Cité par Sources :