Existence of solutions for a semilinear elliptic system

Benrhouma, Mohamed

doi:10.1051/cocv/2012022

Benrhouma, Mohamed

ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 2, pp. 574-586.

Résumé

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

\{\begin{matrix} - Δ u + u = \frac{α}{α + β} {a (x) | v |}^{β} {| u |}^{α - 2} u in ℝ^{N} \\ - Δ v + v = \frac{β}{α + β} {a (x) | u |}^{α} {| v |}^{β - 2} v in ℝ^{N} . \end{matrix}

-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.

MR 3049724

DOI : https://doi.org/10.1051/cocv/2012022
Classification : 35J45, 35J50, 35J60
Mots clés : semilinear elliptic systems, Nehari manifold, concentration-compactness principle, variational methods

BibTeX
RIS

@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/

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