Existence of solutions for a semilinear elliptic system

Benrhouma, Mohamed

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:

$$\left\{\begin{array}{c}-\Delta u+u=\frac{\alpha }{\alpha +\beta }{a\left(x\right)\left|v\right|}^{\beta }{\left|u\right|}^{\alpha -2}u\text{in}{ℝ}^N\hfill \\ -\Delta v+v=\frac{\beta }{\alpha +\beta }{a\left(x\right)\left|u\right|}^{\alpha }{\left|v\right|}^{\beta -2}v\text{in}{ℝ}^N.\hfill \end{array}\right.$$

-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.

Classification : 35J45, 35J50, 35J60
Mots clés : semilinear elliptic systems, Nehari manifold, concentration-compactness principle, variational methods

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