# Existence of solutions for a semilinear elliptic system

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

top

top
This paper deals with the existence of solutions to the following system:$\left\{\begin{array}{c}-\Delta u+u=\frac{\alpha }{\alpha +\beta }{a\left(x\right)|v|}^{\beta }{|u|}^{\alpha -2}u\phantom{\rule{1.0em}{0ex}}\phantom{\rule{4.0pt}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}{ℝ}^{N}\hfill \\ -\Delta v+v=\frac{\beta }{\alpha +\beta }{a\left(x\right)|u|}^{\alpha }{|v|}^{\beta -2}v\phantom{\rule{1.0em}{0ex}}\phantom{\rule{4.0pt}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}{ℝ}^{N}.\hfill \end{array}\right$/extract_itex]−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. ## How to cite top Benrhouma, 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 -

## References

top
1. [1] Y. An, Uniqueness of positive solutions for a class of elliptic systems. J. Math. Anal. Appl. 322 (2006) 1071–1082. Zbl1217.35060MR2250636
2. [2] V. Benci, Some critical point theorems and applications. Commun. Pure Appl. Math.33 (1980) 147–172. Zbl0472.58009MR562548
3. [3] J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space. J. Differ. Equ.163 (2000) 41–56. Zbl0952.35033MR1755067
4. [4] K.-J. Chen, Multiplicity for strongly indefinite semilinear elliptic system. Nonlinear Anal.72 (2010) 806–821. Zbl1183.35110MR2579347
5. [5] P. Clement, D.G. Figuereido and E. Mitidieri, Positive solutions of semilinear elliptic systems. Commun. Partial Differ. Equ.17 (1992) 923–940. Zbl0818.35027MR1177298
6. [6] D.G. Costa, On a class of elliptic systems in RN. Electron. J. Differ. Equ.7 (1994) 1–14. Zbl0809.35020MR1292598
7. [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. Zbl05169004MR2326023
8. [8] R. Dalmasso, Existence and uniqueness of positive solutions of semilinear elliptic systems. Nonlinear Anal.39 (2000) 559–568. Zbl0940.35091MR1727272
9. [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. Zbl0938.35054MR1617988
10. [10] I. Ekeland, On the variational principle. J. Math. Anal. Appl.47 (1974) 324–353. Zbl0286.49015MR346619
11. [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. B30 (2010) 1917–1936. Zbl1240.35176MR2778702
12. [12] D.D. Hai, Uniqueness of positive solutions for semilinear elliptic systems. J. Math. Anal. Appl.313 (2006) 761–767. Zbl1211.35107MR2183334
13. [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. Zbl0962.35052MR1765572
14. [14] G.B. Li and J.F. Yang, Asymptotically linear elliptic systems. Commun. Partial Differ. Equ.29 (2004) 925–954. Zbl1140.35406MR2059153
15. [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. Zbl0541.49009MR778970
16. [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. Zbl0704.49004MR778974
17. [17] Z. Nehari, On a class of nonlinear second-order differential equations. Trans. Amer. Math. Soc.95 (1960) 101–123. Zbl0097.29501MR111898
18. [18] W.-M. Ni, Some minimax principles and their applications in nonlinear elliptic equations. J. Anal. Math.37 (1980) 248–275. Zbl0462.58016MR583639
19. [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. Zbl0375.35026MR488128
20. [20] J. Serrin and H. Zou, Nonexistence of positive solutions of Lane-Emden systems. Differ. Integral Equ.9 (1996) 635–653. Zbl0868.35032MR1401429
21. [21] T.-F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions. Nonlinear Anal.68 (2008) 1733–1745. Zbl1151.35342MR2388846

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.