# Existence of solutions for a semilinear elliptic system

ESAIM: Control, Optimisation and Calculus of Variations (2013)

- Volume: 19, Issue: 2, page 574-586
- ISSN: 1292-8119

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topBenrhouma, Mohamed. "Existence of solutions for a semilinear elliptic system." ESAIM: Control, Optimisation and Calculus of Variations 19.2 (2013): 574-586. <http://eudml.org/doc/272767>.

@article{Benrhouma2013,

abstract = {This paper deals with the existence of solutions to the following system:\[\left\lbrace \begin\{array\}\{l\} -\Delta u+u=\frac\{\alpha \}\{\alpha +\beta \}a(x)|v|^\{\beta \} |u|^\{\alpha -2\}u\quad \mbox\{ in \}\mathbb \{R\}^N\\ [0.2cm] -\Delta v+v=\frac\{\beta \}\{\alpha +\beta \}a(x)|u|^\{\alpha \} |v|^\{\beta -2\}v\quad \mbox\{ in \}\mathbb \{R\}^N. \end\{array\}\right. \]−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.},

author = {Benrhouma, Mohamed},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {semilinear elliptic systems; Nehari manifold; concentration-compactness principle; variational methods},

language = {eng},

number = {2},

pages = {574-586},

publisher = {EDP-Sciences},

title = {Existence of solutions for a semilinear elliptic system},

url = {http://eudml.org/doc/272767},

volume = {19},

year = {2013},

}

TY - JOUR

AU - Benrhouma, Mohamed

TI - Existence of solutions for a semilinear elliptic system

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2013

PB - EDP-Sciences

VL - 19

IS - 2

SP - 574

EP - 586

AB - This paper deals with the existence of solutions to the following system:\[\left\lbrace \begin{array}{l} -\Delta u+u=\frac{\alpha }{\alpha +\beta }a(x)|v|^{\beta } |u|^{\alpha -2}u\quad \mbox{ in }\mathbb {R}^N\\ [0.2cm] -\Delta v+v=\frac{\beta }{\alpha +\beta }a(x)|u|^{\alpha } |v|^{\beta -2}v\quad \mbox{ in }\mathbb {R}^N. \end{array}\right. \]−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.

LA - eng

KW - semilinear elliptic systems; Nehari manifold; concentration-compactness principle; variational methods

UR - http://eudml.org/doc/272767

ER -

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